Investigations on 3D effects and correlation between wave height and lip submergence of an offshore stationary OWC wave energy converter

Applied Ocean Research 64 (2017) 203–216

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Investigations on 3D effects and correlation between wave height and lip submergence of an offshore stationary OWC wave energy converter Ahmed Elhanafi a,b,∗ , Gregor Macfarlane a , Alan Fleming a , Zhi Leong a a b

National Centre for Maritime Engineering and Hydrodynamics, Australian Maritime College, University of Tasmania, Launceston, Tasmania 7250, Australia Department of Naval Architecture and Marine Engineering, Alexandria University, Alexandria, Egypt

a r t i c l e

i n f o

Article history: Received 7 August 2016 Received in revised form 31 December 2016 Accepted 5 March 2017 Keywords: Offshore oscillating water column OWC Experimental and CFD results Wave energy 3D effects Front and rear lip effects

a b s t r a c t Understanding the hydrodynamic interactions between ocean waves and the oscillating water column (OWC) wave energy converter is crucial for improving the device performance. Most previous relevant studies have focused on testing onshore and offshore OWCs using 2D models and wave flumes. Conversely, this paper provides experimental results for a 3D offshore stationary OWC device subjected to regular waves of different heights and periods under a constant power take–off (PTO) damping simulated by an orifice plate of fixed diameter. In addition, a 3D computational fluid dynamics (CFD) model based on the RANS equations and volume of fluid (VOF) surface capturing scheme was developed and validated against the experimental data. Following the validation stage, an extensive campaign of computational tests was performed to (1) discover the impact of testing such an offshore OWC in a 2D domain or a wave flume on device efficiency and (2) investigate the correlation between the incoming wave height and the OWC front wall draught for a maximum efficiency via testing several front lip draughts for two different rear lip draughts under two wave heights and a constant PTO damping. It is found that the 2D and wave flume modelling of an offshore OWC significantly overestimate the overall power extraction efficiency, especially for wave frequencies higher than the chamber resonant frequency. Furthermore, a front lip submergence equal to the wave amplitude affords maximum efficiency whilst preventing air leakage, hence it is recommended that the front lip draught is minimized. © 2017 Elsevier Ltd. All rights reserved.

1. Introduction Wave energy is one of the most promising renewable energy resources and research on utilizing this energy is being conducted worldwide. The numerous invented techniques for wave energy conversion may be classified by deployment location (shoreline, nearshore and offshore), type (attenuator, point absorber and terminator) and mode of operation (submerged pressure differential, oscillating wave surge converter, oscillating water column and overtopping device) [1]. Of all the proposed and existing wave energy converters, the oscillating water column (OWC), which is the focus of the present paper, is arguably one of the most simple and elegant in design and principle of operation. Fundamentally, an OWC device utilizes ocean waves to drive a water column inside

∗ Corresponding author at: National Centre for Maritime Engineering and Hydrodynamics, Australian Maritime College, University of Tasmania, Launceston, Tasmania 7250, Australia. E-mail address: Ahmed.Elhanafi@utas.edu.au (A. Elhanafi). http://dx.doi.org/10.1016/j.apor.2017.03.002 0141-1187/© 2017 Elsevier Ltd. All rights reserved.

a partially submerged chamber open below the ocean free surface. The free surface oscillations inside the chamber generate mechanical energy via pushing and sucking airflow between the OWC pneumatic chamber and surrounding atmosphere through an air turbine that is designed to rotate in the same direction regardless of airflow direction. An electric generator can be used to convert the mechanical energy into electrical energy. Having no moving parts underwater, an OWC device provides minimal and easier maintenance works. OWCs can be deployed as fixed structures at the shoreline or nearshore, or integrated into breakwaters and floating structures [2]. Several researchers have experimentally, analytically and numerically analyzed the hydrodynamic performance of an OWC device. The majority of the experiments have been performed for validating analytical or numerical models and investigating the impact of different design parameters such as wave conditions, power take–off (PTO) damping and underwater geometry on the power extraction efficiency [3–6]. For simple OWC geometries such as a thin–walled vertical tube and two parallel vertical thin walls,

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Fig. 1. Photos of (left) AMC towing tank (looking towards the beach end) and (right) offshore OWC model.

Fig. 2. Dimensions of an offshore–stationary OWC (scale 1:50).

Evans [7] developed an analytical hydrodynamic model based on linear water wave theory, and treating the free surface motion inside the chamber as a rigid weightless piston. This model was further improved to consider the increase in pressure at the free surface as well as provide the possibility of a non–plane free surface [8–10]. For complex geometries, the wave–OWC interactions can be numerically modelled and usually solved using the boundary element method (BEM) where results can be obtained in a reasonable time on standard computers. An extensive review of using potential flow models in OWCs can be found in Baudry et al. [11] and Folley [12]. With increasing computational resources, researchers started utilizing computational fluid dynamic (CFD) models that solve the Navier–Stokes equations or the Reynolds–Averaged Navier–Stokes (RANS) equations. In contrast to potential flow solvers, CFD models can capture detailed physics such as strong nonlinearity, complex viscous effects, turbulence, vortex shedding and wave breaking. Commonly used CFD codes are either open–source codes such as REEF3D, OpenFOAM and Fluinco or commercial codes as Fluent and Star–CCM+. These models are widely used by several researchers for investigating OWC performance [13–18]. The above–mentioned research has been performed in 2D numerical and/or experimental tanks. Freeman [19] stated that a 2D version of a bottom–mounted OWC is invalid (as the aft wall extends to the seabed and water cannot get around it without the third dimension), but an open ocean version (i.e., an offshore OWC device with partially immersed front and rear walls) is not. From the authors’ point of view, a 2D assumption can only be accepted for OWC devices that are relatively long in the direction parallel to

wavefronts, regardless of being deployed onshore, nearshore, offshore or integrated into long breakwaters. Recently, Simoentti et al. [20–22], Iturrioz et al. [15] and He et al. [23] investigated numerically and experimentally offshore OWC models in a wave flume of width almost equal to the model breadth. In a wave flume, the 3D modelling of airflow through an orifice plate can be represented, but the wave–OWC hydrodynamic interactions may be different in a wider wave tank that allows the incident waves to pass not only underneath the OWC chamber but also around the tested structure considering wave scattering. To the best of the authors’ knowledge, there are no publicly available studies that demonstrate the validity of 2D assumptions or wave flume modelling of offshore OWCs. Accordingly, the first contribution of the present study is to discover the impact of 2D and wave flume modelling on the performance of offshore OWCs through direct comparison with 3D experiments and 3D CFD simulations. The impact of increasing the incoming wave height on the performance of an OWC of a given front wall submergence has been previously studied. For example, using 2D CFD models, Luo et al. [14], Kamath et al. [24] and Anbarsooz et al. [25] found that the hydrodynamic efficiency of onshore OWCs considerably decreases as wave height increases. Conversely, Ning et al. [5,26] observed (experimentally and numerically) that the overall efficiency increases with increasing the wave amplitude to a maximum value and then decreases thereafter. Several researchers have also investigated the influence of changing the lip draught of an OWC device subjected to regular waves. For instance, Sarmento [27] physically tested a bottom–standing OWC with four different front wall draughts and concluded that the maximum efficiency is approximately independent of the front wall draught; increasing the front wall draught shifts the point of maximum efficiency to higher periods but narrows the efficiency bandwidth. Ning et al. [5], Wilbert [28] (experimentally) and Luo et al. [14] (numerically) observed that increasing the front lip draught of an onshore OWC decreases both the peak efficiency and the resonant frequency. In addition, reductions in the hydrodynamic efficiency were found to be significant and negligible for high– and low–frequency waves respectively, which in turn narrows the bandwidth of the efficiency peak. Similar observations, except that the peak efficiency slightly changes were also reported experimentally and numerically in Morris-Thomas et al. [3] and Ning et al. [26], respectively. For offshore OWCs, He et al. [23] (experimentally) and Elhanafi et al. [29] (numerically using a 2D CFD model) concluded that symmet-

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Fig. 3. Experiment layout (not to scale).

Table 1 Coordinates of wave gauges and pressure sensors. Instrument

Coordinates (x, y) in metres

G0 G1 G2 G3 G4 G5 G6 G7 G8 S1 S2

(−8.838, 1.550) (−2.247, 0.100) (−2.023, 0.100) (−1.685, 0.100) (2.572, 0.100) (0.00, −1.350) (0.00, 0.100) (0.109, 0.100) (0.162, 0.040) (0.162, 0.160) (0.227, 0.100)

rically increasing both the front and rear lip draughts reduces the peak efficiency as well as the resonant frequency. Elhanafi et al. [29] also found that increasing the asymmetrical lip submergence ratio (i.e., the ratio between the rear and front lip draughts) can significantly improve the power extraction efficiency. Interactions between an onshore OWC and ocean waves result in a partial standing wave envelope in front of the device, which can significantly escalate the wave height at the chamber front lip. Therefore, the OWC front wall must have a sufficient draught to avoid air leakage between the OWC chamber and the surrounding atmosphere during incoming wave troughs [14,27,30], which leads to negligible extracted power at the PTO [31]. To eliminate such an undesirable event, the seaward wall in previous studies was relatively deep and a non–small fraction of the water depth. However, as mentioned above, these interactions are different for offshore OWCs; and therefore, a large front wall draught may not be important for such devices. In addition, none of the above literature investigated the relation between the incoming wave height and the optimum front wall draught at different wave heights, which is the second contribution of the present paper. The present paper utilizes both experiments and a fully nonlinear 3D CFD model based on the RANS equations with a volume of fluid (VOF) surface capturing scheme to provide a better understanding of the implications of the 2D modelling on the over-

all efficiency of offshore OWCs and to illustrate the correlation between the incoming regular wave height and optimum front wall draught. The rest of this paper is organized as follows. The experimental set–up of a 1:50 scale model of an offshore stationary OWC is described in Section 2. The development and validation of the CFD model are discussed in Section 3. In Section 4, the importance of the 3D modelling is discussed together with the relevance of the regular wave height to the chamber seaward lip draught for maximizing power extraction efficiency. Finally, conclusions of this study are summarized in Section 5. 2. Experimental set–up The physical model tests were performed in the towing tank of the Australian Maritime College (AMC), University of Tasmania, Australia (see Fig. 1). The towing tank is 100 m long, 3.5 m wide and 1.5 m deep. The tank is equipped with a flap–type wavemaker at one end and a wave–absorption beach at the other end. 2.1. Physical offshore OWC model A 1:50 physical model of an offshore OWC device was manufactured using plywood with the dimensions illustrated in Fig. 2. The columns and pontoons attached to the chamber provide the required buoyancy to permit future experiments over a wider range of wave conditions on the same model, but as a floating OWC which is tension moored using a similar concept to that proposed in Lye et al. [32]. For the present experiments, the model was fixed in all six degrees of freedom to simulate a stationary OWC, a logical and necessary step for validating the fully nonlinear 3D CFD model. To collect a sufficient number of wave cycles clear of wave reflection from the tank beach, the OWC model was installed 15 m from the wavemaker. The water depth was fixed during the test at 1.5 m which represents 75 m water depth at full–scale (according to Froude similitude law). Considering that the model width to the tank breadth ratio is almost 0.11, which is less than 0.2, the tank sidewall effects can be ignored [33]. The OWC has an overall draught of 350 mm, and the chamber front (dF ) and rear (dA ) lips

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Fig. 4. Computational fluid domain (not to scale). (a): 3D mesh with boundary conditions, (b): OWC surface mesh and (c): initial conditions.

Fig. 5. CFD results vs. physical measurements (EXP.) for H = 0.05 m and T = 1.2 s.

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have a 200 mm submergence. To simulate the pneumatic damping induced by the PTO system during experiments at small–scales, it is common to use either an orifice (or a slot) to simulate an impulse turbine (nonlinear/quadratic pressure–flow rate characteristics) or a porous medium to mimic a Wells (linear characteristics) turbine. During this experiment, an orifice plate of 17.84 mm in radius was used and results were utilized to validate the developed CFD model in Section 3.

repeated runs as recommended by the International Towing Tank Conference ITTC [36] to demonstrate experiment repeatability. Examples of the good experimental repeatability achieved are provided in Elhanafi et al. [34]. Additionally, an experiment uncertainty analysis was performed in line with the method adopted by the ITTC [36–38], and all measurement uncertainties were in the order of ± 5% considering a level of confidence of approximately 95% which is accepted by ITTC [37].

2.2. Measurements and OWC performance

3. CFD modelling and validation

Six resistive–type wave gauges (G0 –G5 ) were used to measure the regular wave instantaneous free surface elevations () at different locations along the tank, as shown in the experiment layout in Fig. 3. The OWC was fitted with three wave gauges. One was centred at the seaward wall to measure the wave run–up (G6 ) and the other two (G7 and G8 ) were situated inside the OWC chamber for spatial averaging the chamber free surface elevation (OWC ). In addition, two pressure sensors (S1 and S2 ) were installed on the OWC top plate and used to measure the differential air pressure inside the chamber. Their average value is regarded as the chamber differential air pressure (P (t)). Coordinates for wave gauges and pressure sensors are summarized in Table 1. All measurements were sampled at 200 Hz, and a low pass digital filter with a 5 Hz cut–off and 10th order Butterworth–filter was applied (using Matlab function filtfilt) during the post–processing stage. The chamber free surface vertical velocity (dOWC /dt) was calculated by numerically differentiating the measured time–series data of the spatial averaged chamber free surface elevation (OWC ). Assuming incompressible air at the small model–scale used in the experiment [34], airflow rate (q (t)) was calculated by Eq. (1), and then the time–averaged extracted pneumatic power (PE ) and the overall hydrodynamic efficiency () were calculated from Eqs. (2) and (3), respectively [5,34].

3.1. Governing equations and numerical settings

q (t) = (dOWC /dt)ba

(1)

where b and a are the chamber length and width, respectively (see Fig. 2). 1 PE = T

T P (t) q (t) dt

(2)

0

where T is the wave period.  = PE / (PI a)

(3)

where PI is the mean incident wave power (energy flux) per unit width that is defined as the product of the wave energy (EI ) (potential  and kinetic) per unit ocean surface area and the group velocity Cg [35]: PI =

1 gA2 Cg 2

 ω

Cg =

2k

1+

A fully nonlinear numerical wave tank (NWT) was developed using Star–CCM+ CFD code [39]. Assuming negligible air compressibility effects at the small model–scale considered in this study [34], the CFD model solves the continuity and RANS equations for the two incompressible phases (water and air) using a finite volume method for discretization and the volume of fluid (VOF) surface capturing scheme [40] to model and track the free surface motion. RANS equations are based on decomposing the instantaneous velocity and pressure fields in the Navier–Stokes equations into mean and fluctuating components, and then time–averaged. This procedure results in additional unknowns called Reynolds stresses that need to be modelled using a turbulence model. In this paper, the two–equation shear stress transport (SST) k−ω turbulence model was implemented to relate Reynolds stresses to the mean flow variables. A summary of the numerical settings used herein can be found in Elhanafi et al. [34]. 3.2. Computational fluid domain Instead of faithfully reproducing the full–length physical tank (see Fig. 3), the numerical set–up illustrated in Fig. 4 has been shown to be effective in collecting eight wave cycles up to a distance of 8 L from the wave inlet boundary before incoming waves interfere with reflected waves from the OWC structure and the assigned outlet boundary at the end of the wave–damping zone of 1 L in length [34,41]. Similarly, the width of the NWT was set to be half–width of the physical tank (i.e., 1.75 m) with a symmetry plane. The assumption of using a symmetry plane and its impact on the numerical results was found to be negligible [34]. Ocean waves are the primary exciting source acting on offshore structures such as OWCs and therefore accurate modelling of these waves is important for providing a good estimation of the hydrodynamic loads and predict the structure response and performance [42]. Utilizing the built–in automatic meshing technique in StarCCM+ of a trimmed cell mesher and surface remesher, a base cell size of 400 mm was applied to the whole fluid domain with progressive refinements at the free surface zone (of 1.5 H in height). Within this zone, the cell aspect ratio was set to be less than 16

(4) 2kh sinh (2kh)

 forintermediatewaterconditions



 L

L 20


1L fordeepwaterconditions h > 2T 2

where A is the incident wave amplitude, ω is the wave angular frequency, k is the wave number given by the dispersion relation2 ship K = ωg = ktanh (kh), L is the incoming wavelength, h is the still water depth,  is the water density and g is the gravitational acceleration. Experimental repeatability is very important to ensure high quality and reliable measurements. Therefore, all measurements have been repeated several times including non–sequentially

L 2

 (5)

with at least 12 and 36 cells per wave height (z–direction, +ve up) and wavelength (x–direction, +ve in wave propagation direction), respectively. This allows generating regular water waves with errors of less than 1.0% (in comparison with the analytical input wave height) considering 1200 time steps per wave period (T) with a second–order temporal discretization scheme [41]. A 100 mm cell size was used as the most cost–effective size for meshing the free surface zone in the transverse (y) direction [34]. Further surface refinements of 6.25 mm and 0.78125 mm cell surface sizes were

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Fig. 6. 2D, wave flumes (A and B) and 3D CFD modelling effects vs. the 3D physical measurements described in Section 2 (EXP.) for H = 0.05 m.

3.3. CFD vs. experimental results Although CFD modelling represents an efficient tool that provides more detailed information than experiments, it is computationally demanding and requires experimental validation [2]. Accordingly, the physical tests described in Section 2 were utilized to validate the developed 3D CFD model. Fig. 5 illustrates the good qualitative agreement between the CFD and experimental results in the time domain for wave conditions of H = 0.05 m, T = 1.2 s. Good agreement was also achieved at the other wave periods investigated. In addition, a further validation of the overall hydrodynamic efficiency () for the full range of tested wave periods (T = 0.9, 1.0, 1.1, 1.2, 1.3, 1.4, 1.5, 1.6, 1.8 and 2.0 s) under a constant wave height of H = 0.05 m is given in the following Section. Quantitative comparisons were also performed by using the normalized root mean square deviation (NRMSD), defined by Eq. (6) [18,34]. The NRMSD for all CFD results were found to be within the experiment uncertainty mentioned in Section 2.2.

NRMSD =

1 xmax − xmin

  N 1

(xi − yi )2 N

(6)

i=1

where xi , yi , are the measured and estimated value form the experiment and CFD, respectively, N is the number of data point (here data only from five wave cycles are considered), xmax , x min are the maximum and minimum measured values from the experiment, accordingly. 4. Results and discussion The well–validated CFD model described in Section 3 is employed in this section to highlight the implications of testing offshore OWCs in narrow wave flumes as well as investigate the relation between the incoming wave height and the chamber lip submergence for improving device efficiency. Fig. 7. 2D, wave flume (A and B) and 3D CFD results for H = 0.05 m and T = 1.1 s (Kb = 1.0). (a): chamber free surface elevation (OWC ), (b): airflow rate (q), (c): chamber differential air pressure (P) and (d): extracted pneumatic power (PE ).

performed for all non–slip walls of the OWC surfaces and the PTO orifice opening, respectively as shown in Fig. 4b. In the initial conditions, the still water level was set at the desired level of z = 1.5 m by defining the volume fraction of the two phases (water and air). Waves were prescribed by the velocity components at the wave velocity inlet boundary; however, to reduce the computation time, the waves were fully generated until the OWC seaward lip by initially specifying that point (x = 5 L) on the water level instead of x = 0.0 (see Fig. 4c).

4.1. 3D effects During physical experiments in wave tanks, interference from tank sidewalls should be avoided. This necessitates either adopting small model–scales in narrow wave flumes or testing in expensive and not always available 3D wider wave basins or towing tanks. On the other side, CFD modelling can be used to perform 2D or 3D numerical tests depending on the validity of 2D assumptions and the available computational resources and time, which significantly increases for 3D simulations. Thus, it is important to ensure the validity of the 2D and wave flume modelling of an offshore OWC device. Accordingly, the width of the developed NWT was reduced to the model full–width (referred as Flume B) of 200 mm (with a symmetry plane). A further reduction in the tank width to only

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Fig. 8. Free surface spatial variations for the OWC model in wave Flume A (left) vs. wave Flume B (right) for H = 0.05 m and T = 1.1 s (Kb = 1.0).

include the OWC chamber width of 100 mm (referred as Flume A) was also considered to highlight the effects of the attached columns and bottom pontoons (see Fig. 4b). Results from all models are presented in Fig. 6 for a 0.05 m wave height and the entire wave period range used in Section 3.3 (which is represented by a dimensionless parameter Kb, where K = ω2 /g and b is the chamber length shown in Fig. 2). These wave conditions represent a wave steepness (H/L) range of 0.009–0.040. Starting with the results of wave Flume A in comparison with the 2D modelling, both models provide nearly the same results. The relatively negligible differences between these models can be assigned to the methods used in simulating the PTO damping. Although the 2D simulations were performed under a PTO damping represented by a slot opening of 5 mm that provides the same opening ratio of the orifice plate (17.84 mm in radius) used in wave Flume A, He and Huang [31] and Elhanafi [43] found that modelling the PTO damping via an orifice plate rather than a slot of the same opening ratio induces a higher chamber differential air pressure, which agrees with the results illustrated in Fig. 7c. This higher damping decreases the chamber free surface oscillation amplitude (Fig. 7a) and its rate (or slope, which is defined as (OWC,max − OWC,min )/T) for a given wave period, which consequently decreases the airflow rate (Fig. 7b) according to Eq. (1). In Eq. (2), it is not the chamber differential air pressure or the airflow rate that controls the pneumatic power, but instead, the resultant product of the two parameters. Thus, there are insignificant impacts of modelling the PTO damping in wave flume tests using a slot (2D) opening or an orifice plate of the same opening ratio on the extracted power (Fig. 7d) and the overall efficiency. This is in agreement with the comparison between the experimental tests performed in a wave flume and the 2D numerical potential flow simulations where a PTO damping was simulated by a slot opening of the same opening ratio with the circular orifice used in the experiments [5]. In comparison with wave Flume A, the columns and pontoons attached to the OWC in wave Flume B lead to a significant increase in the device efficiency over the entire tested frequency. This can be explained by considering the additional vertical sidewalls of the columns’ frontal area as well as the deeper pontoons that work as side beaches, which further reduce the incoming wave energy passing underneath the device. This effect is illustrated in the tank free surface spatial variations for eight instants through one wave cycle of a period T = 1.1 s (Fig. 8) that starts when the chamber free surface level is minimum (trough). Results show the minor transmitted

energy on the OWC lee side in both wave flumes, especially with Flume B. In addition, wave Flume B seems to reflect more energy than wave Flume A, which is demonstrated by the larger crests and deeper troughs in the wave envelope (partial standing waves consisting of nodes and antinodes) constructed in front of the OWC structure in wave Flume B. While a fraction of the saved transmitted energy in wave Flume B is reflected and dissipated by the pontoons, the rest of this energy enters the OWC chamber, which can be seen in the higher chamber free surface oscillations in Fig. 7a. Considering that both wave flumes use the same orifice plate (damping), increasing the oscillation amplitude in wave Flume B results in a higher oscillation rate (or airflow rate, Fig. 7b), chamber differential air pressure (Fig. 7c) and extracted pneumatic power (Fig. 7d). As mentioned in the Introduction (Section 1), most of the previous investigations on onshore and offshore OWCs were performed using a 2D modelling or narrow wave flumes (more like wave Flume A used in the present study). The above–discussed results show that in a wave flume, most of the incoming energy is reflected, especially under the high– frequency zone, which agrees with others published experimental results [31]. However, none of the previous research highlighted the impact of the 3D modelling/testing on the hydrodynamic efficiency of an OWC. Results in Fig. 6 reveal the importance of testing offshore OWC devices in 3D tanks to avoid overestimating the device efficiency, especially at wave frequencies higher than the chamber resonant frequency (at which the maximum efficiency occurs). As wavelength shortens (Kb increases), wave diffraction becomes important. According to Chakrabarti [44], and the model characteristics in Fig. 2, wave diffraction is important for the tested conditions of a wavelength less than 2.51 m that corresponds to Kb = 0.749, which is almost the same value in Fig. 6 where 3D effects start to be more noticeable. For instance, the free surface spatial variations illustrated in Fig. 9 for a wavelength of Kb = 1.0 verify that the hydrodynamic interactions between an offshore OWC and the incident waves are completely different in a 3D domain where a large portion of the incident waves is transmitted behind the device in comparison to the wave flume results (Fig. 8). In addition, wave scattering is obvious in the deformed wave profile induced by the presence of the OWC model. These scattered waves represent a fraction of the incoming wave energy, and therefore, it is expected to cause a reduction in the energy absorbed by the OWC structure (which is defined as the rest of the incoming wave energy after deducting the energy reflected, transmitted and scattered). Assuming the internal energy extraction efficiency

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Fig. 10. Effects of the OWC front lip draught (dF ) on (a): chamber differential air pressure (P), (b): airflow rate (q) and (c): overall efficiency () for H = 0.05 m and dA = 0.2 m.

4.2. OWC lip submergence effects

Fig. 9. Free surface spatial variations for 3D OWC model for H = 0.05 m and T = 1.1 s (Kb = 1.0).

(the device’s ability to convert the absorbed energy into pneumatic energy) is constant, the overall efficiency declines due to the drop in the absorbed energy. The relevance of this drop can be seen in the shrinkage of the chamber free surface oscillations (Fig. 7a) and its consequences on reducing the oscillation rate or airflow (Fig. 7b), the chamber differential air pressure (Fig. 7c) and the extracted pneumatic power (Fig. 7d). It is worth mentioning that the linear theory of Evans [8] predicts a maximum hydrodynamic efficiency of 0.5 for a 2D chamber with symmetrical and partially immersed front and rear lips (i.e., an offshore OWC device), and a maximum efficiency of 1.0 for a chamber with a rear lip that extends to the seabed (i.e., an onshore or a bottom–standing OWC) [27]. However, Sarmento [27] theoretically and experimentally predicted a maximum power extraction efficiency of less than 0.5 and 0.4 respectively for an offshore OWC model. This difference between experimental and theoretical results was assigned to viscous and eddy losses. Similarly, among the three experimentally tested symmetrical lip draughts of an offshore OWC, He et al. [23] found that the smaller draught provides the maximum power extraction efficiency of 0.36. Accordingly, the maximum efficiency (about 0.26) in Fig. 6 of the tested (experimentally and using 3D CFD) offshore OWC model in this study is comparable with the results presented in the literature. A detailed energy balance analysis of an offshore OWC device of the same chamber dimensions tested in this study can be found in Elhanafi et al. [41].

It is well known that the dynamic part of the wave pressure undergoes an exponential decay in amplitude with increasing the distance below the still water level. Accordingly, increasing the immersed depth of the chamber front lip reduces the available pressure head at the lip tip, which in turn affects the differential air pressure amplitude inside the pneumatic chamber. In real situations, the importance of the front lip draught further increases due to variations in wave conditions (i.e., irregular waves) and water depth due to local tide conditions. However, in this section only regular wave conditions and constant water depth (1.5 m) are utilized to provide a better understanding of the relevance of changing the chamber front lip draught (dF ) with respect to the incoming wave height (H) to the device performance for two different wave heights and ten wave periods (T = 0.9–2.0 s). Additionally, under a given front lip draught, the influence of changing the aft lip draught (dA ) on the device performance is also discussed. Elhanafi et al. [29] utilized a 2D CFD model of an OWC with the same chamber dimensions (i.e., chamber length and lips draught and thickness) illustrated in Fig. 2 as a parent device to investigate the impact that lip draught and thickness have on the device performance including detailed numerical flow field analysis. In that study, the maximum efficiency was increased from about 0.30–0.79 for a certain combination of lip thickness and asymmetrical lip ratio (i.e., the ratio between the rear and front lip draughts). Although noticeable improvement in efficiency was achieved by shortening and extending the front and rear lip draughts respectively, the minimum lip draught tested was constrained to two times the incoming wave height (which was fixed at 0.05 m) without explaining the limitations on this draught and the influence of increasing the wave height, which are discussed in this section. Furthermore, having defined the implications of the 2D and wave flume modelling on the performance of the presented offshore OWC (see Section 4.1), all simulations in the present study were performed using a 3D model of the parent OWC described in Section 2.1.

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4.2.1. Front lip effects Starting with the front lip effects on the device performance under a constant wave height H = 0.05 m, the seaward lip draught was varied as dF = 0.2 m (i.e., four times the incoming wave height or dF /H = 4.0), 0.1 m (dF /H = 2.0), 0.05 m (dF /H = 1.0), 0.025 m (dF /H = 0.5 or dF /A = 1.0, where A is the wave amplitude) and 0.0125 m (half the wave amplitude, dF /A = 0.5 or dF /H = 0.25) while the aft lip draught was fixed at its default value of dA = 0.2 m. Results in Fig. 10 show that shortening the front lip draught relative to the incident wave height significantly improves the chamber differential air pressure (Fig. 10a), which accordingly under a constant PTO damping, increases the airflow through the PTO (Fig. 10b). Subsequently, the extracted power and the overall efficiency (Fig. 10c) significantly improve, especially for the intermediate– and high–frequency zones. The relevance of the front wall submergence over the low–frequency zone is negligible, which can be assigned to the minimal variations of the wavelength with respect to the front wall immersed depth (L/dF ) under this long wave region [5]. Furthermore, an OWC with a front lip draught equal to the wave amplitude (dF /H = 0.5) or half the amplitude (dF /H = 0.25) provides almost the same results. Adding to the higher–efficiencies achieved with reducing the front lip draught, the peak efficiency is shifted to a higher–frequency corresponding to the chamber new resonant (natural) frequency that increases with decreasing the mass of the water column inside the chamber due to the smaller draught [3,5]. This effect may not be noticeable for OWC models with a front lip draught less than dF /H = 1.0 due to the low–frequency resolution between Kb = 1.207 and 1.49. The peak efficiency ( max ) in Fig. 10c increases substantially from 0.26 with dF /H = 4.0 at Kb = 0.714 to 0.82 with dF /H = 0.25 at a shorter wavelength (higher–frequency) of Kb = 1.207. According to Eq. (2), the available wave power reduces with increasing wave frequency as a result of reducing the group velocity (Cg ) in Eq. (3). For instance, at a 0.05 m wave height, the wave power in the longest wave tested (Kb = 0.30) is about 2.6 times that in the highest wave frequency investigated of Kb = 1.49. However, the enormous improvements achieved in the device efficiency over the intermediate– and high–frequency zones with a front lip draught of dF /H = 0.5 or 0.25 provide a possibility of extracting an even pneumatic power over a broader frequency bandwidth. The same general trend of improving the device efficiency by minimizing the front lip draught from dF /H = 4.0 to dF /H = 2.0 (see Fig. 10c) was also reported in Elhanafi et al. [29] using a 2D CFD model. However, a quantitative comparison between the results illustrated in Fig. 10c and those presented in Elhanafi et al. [29] reveals that a 2D model of the same OWC geometry significantly overestimates the predicted efficiency, which supports the importance of considering the 3D effects as discussed in Section 4.1. This also indicates that for an offshore OWC device like the one investigated in this study, a 2D version can be employed to provide a general insight on the relevance of different wave conditions and various underwater geometries to device efficiency, but accurately predicting the efficiency requires 3D modelling. Considering that water particle orbital motion becomes smaller as wavelength decreases, which leads to smaller particle excursions, the shorter front lip provides an ideal path for these small particle excursions to smoothly enter the OWC chamber. Accordingly, more incoming wave energy is allowed to enter the chamber rather than being reflected or dissipated at the lip tip in forms of vortex formation, which leads to breaking the rigid piston–like motion of the chamber free surface (water sloshing) that negatively influences the airflow rate [16]. For example, the time–series results of the chamber free surface elevation (OWC ), the chamber differential air pressure (P), the airflow rate (q) and the pneumatic power (PE ) are illustrated in Fig. 11 for a wavelength of

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Fig. 11. Effects of OWC front lip draught (dF ) on (a): chamber free surface oscillations (OWC ), (b): airflow rate (q), (c): chamber differential air pressure (P) and (d): extracted pneumatic power (PE ) for H = 0.05 m, Kb = 1.0 (T = 1.1 s) and dA = 0.2 m.

Kb = 1.0. The rise in the chamber free surface oscillation amplitude in Fig. 11a with reducing the ratio dF /H confirms the reasonable passage provided by the shorter front lip draught for the small particle excursions under short waves. As discussed in Section 4.1, increasing the chamber free surface oscillation amplitude under a given wave period results in an increase in airflow rate (Fig. 11b) and the chamber effectively utilizes the higher dynamic wave pressure at the front lip tip, which is reflected in the higher–pressure amplitudes seen in Fig. 11c. It also worth mentioning that the OWC with a front lip draught equal to the wave amplitude (dF /H = 0.5) provides almost the same results as the shortest front lip tested (dF /H = 0.25) during the inhalation stage (de–pressurizing or air entering the chamber through the PTO). Conversely, the effectiveness of the shortest draught (dF /H = 0.25) is more noticeable during the exhalation process (pressurizing or air leaving the chamber through the PTO), which in turn increases the pneumatic power extraction (Fig. 11d). 4.2.2. Rear lip effects The findings in Section 4.1 shows that for a 3D offshore OWC, a significant part of the incoming wave energy is transmitted and scattered, whereas the above–discussed results (Section 4.2.1) reveal that with properly selecting the front lip draught, the device efficiency can be significantly improved. In this section, another design parameter for an offshore OWC, which is the rear lip draught, is considered. For a constant wave height H = 0.05 m and ten wave

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Fig. 12. Effects of OWC rear lip draught (dA ) on (a): chamber differential air pressure (P), (b): airflow rate (q) and (c): overall efficiency () for H = 0.05 m, dF = 0.05 m (left) and dF = 0.025 m (right).

periods, the impacts of changing the leeward wall draught relative to the seaward wall draught on the device performance are shown in Fig. 12 left and 12 right for two different front lip draughts of dF = 0.05 m and 0.025 m, respectively. Even with a short front lip of dF /H = 1.0 or 0.5, the rear wall draught is an important factor that needs to be considered when optimizing such devices. For instance, when both lips have the same immersed depth (dA /dF = 1.0), most of the incoming energy is transmitted on the OWC lee side because of the reduction in the capability of the device to absorb the incoming energy. Accordingly, the power extraction parameters (pressure (Fig. 12a) and airflow (Fig. 12b)) as well as the device efficiency (Fig. 12c) are minimal. Increasing the aft lip draught reduces energy transmission, which increases the energy absorption efficiency and escalates the hydrodynamic efficiency from about 0.24 with dA /dF = 1.0 (Fig. 12c) to about 0.94 (Fig. 12 left c) and 0.97 (Fig. 12 right c) for dA /dF = 7.0 and 14.0, respectively. In contrast to reducing the front lip draught, the chamber resonant frequency declines with increasing the rear lip draught that can be seen in pushing the maximum efficiency in Fig. 12 left c from 0.68 with dA /dF = 4.0 at Kb = 1.207 to 0.94 with dA /dF = 7.0 at Kb = 1.0. This effect may not be clearly visible for the results of the smaller front lip draught dF = 0.025 m in Fig. 12 right c that is again due to the coarse frequency resolution. As mentioned in Section 4.1, a maximum efficiency of 1.0 can theoretically be achieved for an onshore or a nearshore (bottom–standing) OWC [8]; however, due to viscous and eddy losses, this maximum value was experimentally found to be about 0.90–0.95 [27]. In comparison with onshore and nearshore devices, results in this section show that a maximum efficiency of 0.97 for an offshore OWC can be achieved over a broad wave frequency range with a rear lip draught of dA = 0.35 m that represents only 23% of the water depth. 4.2.3. Limitations on the front lip draught Although adjusting the front and rear wall draughts improves the extracted time–averaged pneumatic power and the overall efficiency (Figs. 10c and 12c), the instantaneous results at different frequencies may highlight the probability of having a zero (negli-

gible) extracted pneumatic power for a certain duration due to air escaping under the chamber lip (referred hereafter as air leakage). The results in Fig. 13 compares the instantaneous impacts that the submergence of the front and rear walls have on (a): the chamber free surface elevation, (b): the airflow rate, (c): the chamber differential air pressure and (d): the extracted pneumatic power for a constant wave height H = 0.05 m and two wave periods corresponding to Kb = 0.62 (Fig. 13left) and Kb = 0.30 (Fig. 13right). For the conditions with a rear wall draught of 0.2 m, no significant air leakage events were observed for a front lip draught larger than 0.25H, whereas air leakage was more pronounced for the OWC with dF = 0.25H as demonstrated by the negligible (almost zero) chamber differential air pressure, airflow (slightly changes close to zero) and extracted pneumatic power. The air leakage duration is higher during the inhalation stage for both wave periods tested, which can be physically explained by tracking the free surface oscillations inside and outside the chamber and the changes in the chamber air pressure as illustrated in Appendix A. The air leakage event during the inhalation stage lasts for about 0.114T for Kb = 0.62 (Fig. 13left) that is close to 0.103T for Kb = 0.30 (Fig. 13right), but it is higher than the duration of only 0.02T for Kb = 1.0 (Fig. 11), which indicates a shorter air leakage duration for the high–frequency zone. Considering the small difference of about 5% (average) in the hydrodynamic efficiency (Fig. 10c) between the OWC devices with a front lip draught of 0.5H and 0.25H, a front lip draught equal to the incoming wave amplitude (dF /H = 0.5) can be used without causing significant air leakage events. With this draught, even with the largest tested aft wall draught of 0.35 m (dA /dF = 14.0 in Fig. 12 right) that significantly increases the device efficiency, no air leakage events were observed over the entire frequency range as illustrated in Fig. 13left (Kb = 0.62) and 13 right (Kb = 0.30). Furthermore, the time–series results in Figs. 11 and 13 illustrate the device capability of extracting more pneumatic power during the inhalation stage as the wavelength increases (Kb decreases), which is highly obvious for the longest wavelength tested of Kb = 0.30 in Fig. 13 right.

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Fig. 13. Effects of OWC lip submergence on (a): chamber free surface oscillations (OWC ), (b): airflow rate (q), (c): chamber differential air pressure (P) and (d): extracted pneumatic power (PE ) for H = 0.05 m, Kb = 0.62 (left) and Kb = 0.30 (right).

Fig. 14. Influence of the OWC lip draught on the overall efficiency () for two different wave heights of H = 0.05 m (H50 ) and 0.10 m (H100 ).

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4.2.4. OWC performance in more energetic seas In this section, the impacts of changing the front and rear lip draughts on the device performance are investigated under a more energetic sea characterized by a 0.10 m wave height (referred as H100 ) that contains a wave energy four times that with the 0.05 m wave height (referred as H50 ) investigated in the previous sections. Doubling the incoming wave height increases wave nonlinearity as represented by increasing the wave steepness range to 0.018–0.080. The effect of changing the front lip draught through various dF /H100 ratios (2.0, 1.0, 0.5 and 0.25) while the rear lip draught remains constant at dA = 0.2 m are presented in Fig. 14. Also shown in this figure are results that show the impact of increasing the rear lip draught to dA = 0.35 m on the overall efficiency for two different front lip draughts of dF /H100 = 0.5 and 0.25. First, increasing the wave height twofold slightly reduces the maximum efficiency from 0.26 (in Fig. 10c and re–presented in Fig. 14) to 0.24 for the OWC with the default lip draught (i.e., dF = dA = 0.2 m) at the same frequency of Kb = 0.714. Second, like the OWC performance under the smaller wave height, shortening the front lip draught increases the overall efficiency up to 0.64 and 0.90 at Kb = 1.207 for the smallest front lip draught tested (dF /H100 = 0.25) with an aft lip draught of dA = 0.2 m and 0.35 m, respectively. Similar to the small wave height, it is only the front lip draught of 0.25H that results in air leakage events, but the higher wave height extends the air leakage duration (time–series results are not presented, but they have the same general trends as the results for the smaller wave height shown in Fig. 13). For example, under a 0.10 m wave height, utilizing a rear lip draught of dA = 0.2 m or 0.35 m with a seaward lip draught of dF = 0.25H100 (0.025 m) resulted in air leakage durations of about 0.16T and 0.17T for the wavelength of Kb = 0.62 and 0.30, respectively. Fig. 14 also shows that an increase in the wave height for any given front lip draught reduces the overall efficiency in the intermediate–high–frequency zone. However, there are noticeable improvements in the device efficiency with the higher wave height (H100 ) for the low–intermediate–frequency region. For a constant rear lip draught, reducing the front lip draught broadens the frequency range over which the more energetic sea provides a higher efficiency. For instance, with front and rear lip draughts of 0.2 m, the larger wave height (H = 0.10 m) results in a higher efficiency than that of the smaller height (H = 0.05) up to a wave frequency of Kb = 0.58 that extends to Kb = 0.80 as the front lip draught shortens to dF = 0.025 m. 5. Conclusions Experiments evaluating the hydrodynamic performance of a 1:50 offshore stationary OWC model–scale in regular waves were presented. The experimental set–up, time–series measurements of the wave elevation along the tank, the OWC chamber free surface elevation and the chamber differential air pressure were provided for selected conditions together with the overall efficiency curve. In addition, a 3D fully nonlinear CFD model was developed, validated in good agreement against the experimental measurements, and then employed to investigate the importance of the 3D modelling, the relevance of both front and rear lip draughts to the device performance and the relation between the chamber front lip draught and the incoming wave height for maximizing efficiency. From a total of 190 CFD simulations, it was found that modelling offshore OWCs in a 2D domain or a narrow wave flume significantly overestimates the device hydrodynamic efficiency, especially for

wave frequencies higher than the chamber resonant frequency, whereas a 3D CFD model can capture the OWC performance in very good agreement within experimental uncertainties. The immersed depth of both front and rear lips represents an important design parameter that needs to be considered when optimizing offshore OWCs for a higher efficiency. This was demonstrated by the significant improvement in the overall hydrodynamic efficiency up to 0.97 via shortening and extending the front and rear lips respectively. Additionally, for a given lip draught, increasing the incoming wave height improves the device efficiency up to a certain frequency that increases with further shortening the seaward lip draught. Unless having a mechanism that allows adjusting the front lip draught with respect to the incoming wave height or utilizing a device with multiple chambers of different draughts, a constant front lip draught should carefully be selected. To avoid air leakage underneath the chamber lip while maintaining a higher efficiency, a front lip draught equal to the amplitude of the maximum expected wave height during the operational conditions is recommended. Acknowledgements The authors gratefully acknowledge the technical support during the experiments from Mr Liam Honeychurch, Mr Tim Lilienthal and Mr Kirk Meyer, Australian Maritime College (AMC), University of Tasmania, Australia. Appendix A. Illustration of an air leakage event The contours of the volume fraction of water and the air pressure shown in Fig. A1 through one wave cycle of H = 0.05 m and T = 1.4 s are used to discuss the air leakage event. Initially, at time = 0.0 (Fig. A1), where outside free surface is slightly higher than the chamber internal water level that is minimum (trough), the front lip is completely out of the water (Fig. A1a). Accordingly, the air pressure inside and outside the chamber is atmospheric as shown in Fig. A1b. The time required to immerse the front lip tip is less than 0.071T as seen in Fig. A1a at which the chamber air pressure begins to build–up (see Fig. A1b at 0.071T), and exhalation process starts. From this instant, the PTO (orifice plate) induces damping/resistance on the chamber free surface and accumulates a higher chamber air pressure (see Fig. A1b at 0.086T), which delays the chamber water column oscillations relative to the outer free surface that is subjected to atmospheric pressure. Consequently, the phase–shift between the inside and outside free surface levels becomes clear and gradually increases almost till the maximum pressure is achieved (about at 0.25T in Fig. A1b), and then diminishes closer to the maximum chamber free surface oscillation (amplitude) at time = 0.5T (Fig. A1a) while the chamber air pressure tends to zero (Fig. A1b). Afterwards, the inhalation stage initiates, and the phase–shift re–increases almost up to the maximum suction (negative) pressure at (0.75T in Fig. A1b). Again, the time lag reduces as the chamber water column continues to drop down, but the water level outside the chamber reaches the front lip tip earlier at 0.786T (Fig. A1a). Accordingly, once the chamber free surface falls below the front lip (after time = 0.857T), the chamber air pressure returns to zero (see Fig. A1b at 0.857T and 0.886T). As the internal free surface keeps declining, the outer free surface starts moving up until immersing the lip tip again.

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Fig. A1. Contours of (a): the volume fraction of water and (b): the air pressure at the symmetry plane for H = 0.05 m, Kb = 0.62, dF = 0.0125 m and dA = 0.2 m.

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