Numerical energy balance analysis for an onshore oscillating water column–wave energy converter

Energy 116 (2016) 539e557

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Numerical energy balance analysis for an onshore oscillating water columnewave energy converter Ahmed Elhanafi*, Alan Fleming, Gregor Macfarlane, Zhi Leong National Centre for Maritime Engineering and Hydrodynamics, Australian Maritime College, University of Tasmania, Launceston, Tasmania 7250, Australia

a r t i c l e i n f o

a b s t r a c t

Article history: Received 22 March 2016 Received in revised form 27 June 2016 Accepted 25 September 2016

The hydrodynamic performance of Oscillating Water Column (OWC) wave energy converters depends mainly on the behaviour of the waveeOWC interaction. In this paper, a fully nonlinear 2D RANSebased computational fluid dynamics (CFD) model was used to carry out an energy balance analysis of an onshore OWC. Chamber differential air pressure and free surface elevation from published physical measurements were used to validate the CFD model. Additional validation was carried out via PIV data from available modelescale experiments to validate the CFD model's capability in capturing the flow field and the turbulent kinetic energy. The validated CFD model was then used in an extensive campaign of numerical tests to quantify the relevance of different design parameters such as incoming wave height and turbine pneumatic damping to characterise the hydrodynamic performance and wave energy conversion chain of the OWC. To capture the flow field inside the OWC in good agreement, additional refinement was required at the field of view together with utilizing either SST or RSM turbulence models rather than k-3 . It is found that the applied damping has crucial impacts on the energy conversion process. Also, increasing the wave height can lead to a massive drop in the system efficiency. Furthermore, both power takeeoff (PTO) damping and wave height play an important role in vortex formation around the upper and lower chamber's lips during the ineflow and outeflow stages. © 2016 Elsevier Ltd. All rights reserved.

Keywords: OWC Wave energy Energy balance Numerical wave tank

1. Introduction The Oscillating Water Column (OWC) is a wave energy device that generates electricity by utilizing waveetoeair energy conversion via driving an oscillating column of water in a partially submerged chamber open to the ocean as shown in Fig. 1. The air energy is extracted by means of an air turbine connected to the top part of the chamber. Due to water level oscillations in the chamber, mechanical energy is generated via pushing and sucking airflow between the OWC chamber and the atmosphere through an air turbine that continues to rotate in the same direction regardless of the direction of the airflow. Finally, an electric generator converts the turbine mechanical energy into electricity. Compared to other wave energy converters (WEC), the OWC is simple if not the simplest device in its principle of operational as well as having no mechanical parts underwater which is an advantage for lesser maintenance works. Previous theoretical and numerical modelling of waveeOWC

* Corresponding author. E-mail address: Ahmed.Elhanafi@utas.edu.au (A. Elhanafi). http://dx.doi.org/10.1016/j.energy.2016.09.118 0360-5442/© 2016 Elsevier Ltd. All rights reserved.

interaction is heavily based on potential flow theory assumptions. A theoretical model of the hydrodynamics of a fixed OWC device was developed by Evans [1] assuming the chamber free surface moves as a massless rigid piston with no possibility for chamber interior free surface spatial variation. This assumption can be applied for small chamber (interior free surface) width in comparison with the incident wavelength which allowed the application of oscillating body theory. Aiming at developing Evan's model, Falc~ ao and Sarmento [2], Evans [3] and Falnes and McIver [4] improved the rigidebody approach of an OWC by allowing the increase in pressure at the free surface and the possibility of a noneplane surface (surface deformation). However, these analytical models are still based on the potential flow limitations and only applicable for simple OWC geometry under regular waves providing the maximum wave energy to air conversion efficiency. Within the potential flow theory framework, for complex geometries, the waveeOWC interaction can be numerically modelled and usually solved using the boundary element method (BEM). With BEM, the  and solution can be linear in the frequency domain such as Delaure Lewis [5] or nonlinear where the diffraction/radiation problem is solved in the time domain with possibility of describing some

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Fig. 1. OWC device scheme.

nonlinear effects such free surface and power takeeoff systems (PTO) [6]. Extensive review of potential flow models can be found in Baudry et al. [7]. Different from potential flow models, Computational Fluid Dynamic (CFD) solvers that solve the NaviereStokes equations or the Reynolds Averaged NaviereStokes (RANS) equations allow consideration of complex nonlinearities that cannot be handled with potential flow models. These nonlinearities include large amplitude waves/motions, complex viscous, turbulence, vortex shedding, flow separation, wave breaking and wave overtopping/ green water. Although such solvers, apart from being computationally demanding, require experimental validation [8]. An example of a RANS model that is relevant to the present study is the work of Horko [9] who utilized an experimentally validated 2D CFD model using Fluent solver to investigate the effect of OWC chamber's front lip on the device hydrodynamic performance. Horko found that remarkable improvements in the OWC efficiency can be achieved with simple changes to the chamber's front wall aperture shape by either increasing its thickness or providing rounding. Applying a validated Fluent CFD code with physical measurements, Liu et al. [10] studied the nozzle effects of the OWC chambereduct system on the chamber internal free surface oscillation amplitude as well as the airflow rate through the duct. Zhang et al. [11] developed a 2D twoephase numerical wave tank based on a leveleset immersed boundary to replicate the OWC experiments of Morris-Thomas et al. [12]. They reported over prediction of the efficiency around resonance as a result of the complex pressure changes in the chamber. Although, Zhang, et al. [11] presented and discussed numerical flow field results, there was no validation for such detailed analysis. Looking to find the optimum turbineeinpez, et al. [13] developed a 2D duced damping on an OWC device, Lo numerical model based on RANSeVOF using commercial CFD code StareCCMþ and validated their model with experimental measurements for the chamber differential pressure and airflow rate to study under regular and irregular waves. Iturrioz et al. [14] developed a 3D CFD model based on RANSeVOF using open source code IHFOAM to study the hydrodynamics and pneumatics around an OWC after validating their model with flume tank experiments. Simonetti et al. [15] numerically modelled an OWC device with a validated 3D OpenFOAM CFD code against physical wave flume measurements and highlighted the relevant effect of the orifice aperture on the chamber free surface oscillation and the device frequency response. Kamath et al. [16,17] employed a 2D openesource CFD model (REEF3D) to simulate and study the interactions of an onshore OWC model scale under the action of regular waves of different wavelengths and steepness, and also to discover the model response under different linear PTO damping

simulated by porous media. In addition, Kamath et al. [18] utilized the same CFD model to investigate the 3D effects on the hydrodynamics of an onshore OWC device. Based on a comparison between 2D flume, 3D flume and 3D basin, they found that 2D wave flume simulations can be effectively used. Although the great attention paid to investigating the OWC overall hydrodynamic performance and geometry optimization [19e22], there is a part of the incoming wave energy assumed to be lost due to turbulence in the vicinity of the OWC device. Limited research has been conducted regarding the energy conversion process either by visualizing and/or quantifying these losses. Müller and Whittaker [23] identified different losses mechanisms happen inside a shoreline wave power station by visualizing the flow behaviour. Tseng et al. [24] experimentally studied the wave energy conversion process for a multieresonant cylindrical caisson at model scale. Aiming to determine the system efficiency, physical measurements such as water surface elevation and pressure inside the caisson, incoming and reflected wave heights were collected. From their study, about 33%e68% energy losses were estimated throughout the energy conversion chain. Mendes and Monteiro [25] conducted a series of wave tank experiments on a shoreline OWC at model scale subjected to regular waves. In addition to quantifying the amplification factor, energy losses, energy absorption efficiency, internal efficiency and the overall efficiency, they used a sequence of videoeframes of the flow to uncover physics of energy dissipation in their model. They found that the energy losses associated with power transmission to takeeoff is above 50% of the absorbed wave energy for wave steepness greater than 1/40, while more than 80% of the absorbed energy is lost with Stokes 3rd order waves. With the advances in measurement techniques, more information about the flow field (velocity field) becomes possible, thanks to PIV (particle imaging velocimetry) that has been widely used in marine applications. Examples that are most relevant to the present study include Morrison [26] who calculated the kinetic energy and viscous dissipation rates in a physical OWC model scale under different wave conditions by using the PIV technique. Graw et al. [27] also investigated the impact of the OWC front lip shape and inclination under four wave periods on the energy losses in the vicinity of the lip. They found that for cornered lip shape at low frequencies, the mean energy losses over one cycle (mean dissipation divided by the mean power) may be as much as 15%. Fleming et al. [28] used PIV with a phaseeaveraging technique to carry out energy balance analyses and investigate the energy sources, stores and sinks for a forward facing bent duct OWC using the obtained  pez et al. [29] investigated the efvelocity field measurements. Lo fects of the turbineeinduced damping and the variation in the tidal level on the performance of an onshore OWC by means of both PIV measurements and Reynolds decomposition technique. Although the PIV technique provides much more essential information on the flow field that cannot be obtained with analytical or potential flow solvers, its application is limited to single plane measurements (2D PIV). This is due to increased complications with optical access when moving from 2D PIV to Stereo 3D PIV, where it can be extremely difficult to obtain appropriate optical access and calibration. Adding to theses technical challenges, PIV is still an expensive tool and involves the use of a high intensity laser (class 4) that can easily cause permanent blindness. In contrast to PIV, once confidence is achieved through suitable validation, CFD can provide a very powerful tool for researchers and designers to use in investigating the flow field inside and around the OWC for further applications in wave energy balance analysis and geometry optimization. The best platform to achieve the required confidence in CFD for such detailed analysis is to adapt PIV data to validate CFD models. To the author's knowledge, this is the first attempt to

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consider such numerical investigation, consequently the objective of this study is split into two parts. Firstly, test the ability of a 2D numerical model based on RANS equations solver with a VOF surface capturing scheme introduced by Hirt and Nichols [30] in predicting the flow field of an OWC in comparison PIV measurements. Secondly, utilise the validated model to uncover the impact of two design parameters: the incident wave amplitude and PTO pneumatic damping on the hydrodynamic performance of an onshore OWC device through the energy conversion chain.

ZT EE ¼

541

DPðtÞ$qðtÞdt


 J m

(4)

0

The instantaneous potential energy stored in OWC heave oscils ) apart from its rest position per unit chamber width is lation (E given by Eq. (5), whereas the average potential/stored energy over one wave cycle (Es) is provided by Eq. (6),[25]: 2
  s ¼ rg h b J E 2 m

2. OWC energy balance coefficients It is becoming increasingly important to acquire a better understanding of the impact that different design parameters such as wave amplitude and pneumatic damping have, not only on the extracted energy, but also on the different components included in the wave energy conversion chain. In the present study, OWC energy balance analysis is performed according to the energy conservation principle (energy can be neither created nor destroyed but can be transformed from one form to another) given in Eq. (1),[24]. This illustrates that the incoming wave energy (EI) entering the OWC (source) equals to the sum of the energy leaving the system (sinks) such as reflected energy (ER) and pneumatic energy extracted by the PTO (EE) and the energy stored (ES) in the water column heave motion as well as energy losses (EL).


 J EI ¼ ER þ EE þ ES þ EL m

(1)

The incident wave energy per unit width (EI) is calculated using wave theory as the sum of the wave potential and kinetic energy per unit surface area multiplied by the incident wavelength (L) [31]:

EI ¼


 1 J rgA2 L 2 m

(2)

where A is the incoming wave amplitude. Assuming that both incident and reflected energies are proportional to the square of the wave height, the wave reflection coefficient (CR) is defined as the ratio between the reflected wave height (HR) and the incident wave height (H). In order to resolve the reflected and incident waves, instantaneous wave elevation in front of the OWC structure is measured at three points following the probes spacing requirements for Mansard and Funke [32] resolving technique. Having the reflection coefficient, the reflected wave energy can be defined as EI  CR2 . An inehouse code was developed to calculate the reflection coefficient and validated, obtaining good agreement with experimental results conducted in the wave basin at the Australian Maritime College, using regular waves interacting with a vertical wall of CR ¼ 1.0 (see Fig. 2). The amount of wave energy absorbed by the OWC structure (EA) defined in Eq. (3) represents the maximum available energy to be extracted [25]:

 
J EA ¼ EI  ER ¼ EI 1  CR2 m

(3)

The extracted pneumatic energy at the air turbine (EE) per wave cycle is calculated by integrating the instantaneous extracted power (over one wave period, T) that is defined as the product of the instantaneous differential pressure (DP(t)) between the chamber and the exterior domain, and the airflow rate through the turbine (q(t)) [24,25,33]:

Es ¼

1 T

ZT

rgh2 b 2

(5)

dt ¼

2b rgHw

16


 J m

(6)

0

where h is the vertical oscillation of the water volume displaced from the still water level, Hw is the height of the water column oscillations and b is the chamber length (in wave propagation direction, see Fig. 1). Based on the energy conservation principal in Eq. (1), the overall energy losses in the OWC system (EL) is the rest of the absorbed energy that has not been transmitted to the air for further conversion at the turbine minus the energy stored in the water column's heave motion.

EL ¼ EA  EE  ES

(7)

Having defined every energy component in Eq. (1), nondimensionalization is performed by dividing both sides of Eq. (1) by the incoming wave energy (EI). This provides the different energy balance coefficients used in the present study as illustrated in Fig. 3, and summarized in Table 1. It is important to note that it is common by researchers to use the term overall efficiency (zOverall) to describe the percentage of the timeeaveraged extracted power (PE) defined in Eq. (8) relative to the incoming wave power (PI) given by Eq. (9),[31]. However, in this study, an overall energy extraction coefficient (COverall) is utilized to represent the fraction of the incoming wave energy that is converted into pneumatic energy. Considering that the incoming wave group velocity in Eq. (10),[31] does not equal to the wave celerity (L/T), the power ratio (zOverall) will be different from the energy ratio (COverall), and accordingly, to keep consistency in Eq. (1) and avoid using a combination of power and energy ratios, the overall energy extraction coefficient (COverall) is used.


 W m

PE ¼

EE T

PI ¼


 EI W Cg m L

Cg ¼

u 2k

 1þ

(8)

h i 2kh m sinhð2khÞ s

(9)

(10)

where u is the wave angular frequency, k is the wave number that is 2 calculated based on the dispersion relationship ug ¼ ktanhðkhÞ and h is the water depth. According to Folley and Whittaker [34], the energy in the water column heave can be converted into useful pneumatic power via the PTO, but also some will be radiated in outgoing waves due to its motion and some dissipated as viscous losses. This can be seen in energy balance chart in Fig. 3, which illustrates that there is no direct coupling between the absorbed and the extracted pneumatic

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Fig. 2. Standing waves experimental measurement (left) and reflection code validation (right).

Fig. 3. OWC wave energy balance chart.

3. Numerical model

Table 1 OWC energy balance coefficients. Wave energy balance coefficient

Description

Overall pneumatic energy extraction (COverall)

EE/EI (HR/H)2

Reflected energy (CR2 ) OWC energy absorption (CAbs) OWC internal energy extraction (CInternal) Energy losses (CL) OWC stored energy (CS)

EA/EI EE/EA EL/EA Es/EA

energy, and the only possible way to extract apart of the absorbed energy is through the water column heave motion (i.e., the heave motion is driving the pneumatic energy). Accordingly, it is favourable to the most if not all the incoming energy to go through this heave motion. Here, it is necessary to mention that the fraction of the extracted energy from a given stored heave energy depends mainly on the applied PTO damping, and there is always a part of the stored energy that remains in the system (given by Eq. (6)) without being converted to any other forms as previously seen in Refs. [24,25]. Also, as the water level inside the OWC chamber changes under a constant PTO damping, the energy stored in chamber accumulates and disperses through the wave cycle as given by Eq. (5). This instantaneous change in the stored energy alters the different possible sinks from this energy to the pneumatic, reflected (and radiated) and energy losses.

3.1. Governing equations For the small model scale considered in this study, both water and air compressibility is assumed to be negligible. This assumption is valid for the model scale as air is being compressed by small pressures and almost behaves as an incompressible fluid, however for full scale air compressibility can affect the device performance significantly, especially for a chamber several metres in height [35]. The flow motion of the incompressible fluid is described by the continuity and RANS equations. As a result of decomposing the instantaneous velocity and pressure fields into mean and fluctuating components, and the subsequent timeeaveraging of the set of the NaviereStokes equations, new terms called Reynolds stresses are introduced into NaviereStokes equations. These stresses need to be modelled in order to mathematically close the problem. The most commonly used models among engineers are eddy viscosity models that use Boussinesq assumption to relate the Reynolds stresses to the mean flow quantities by a turbulent (eddy) viscosity. This paper mainly uses a twoeequation Shear Stress Transport (SST) keu turbulence model, however other eddy viscosity models such as standard k-3 and realizable k-3 are considered for comparison together with Reynolds stress turbulence model (EBeRSM). The StareCCMþ (RANSeVOF) solver is used in this paper to solve the flow sets of equations.

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3.2. Computational fluid domain In the present study, the computational domain consists of three main zones: domain, free surface and OWC as shown in Fig. 4. The domain length is selected such that the OWC front wall is placed at 6 wavelengths (L) from the wave velocity inlet boundary and one wavelength is left behind the OWC rear wall to avoid unwanted boundary damping on the air flowing through the PTO (simulated by a slot opening). Domain width is set at two cells (2D model). Different cell sizes are applied to each zone according to the required refinement in a computationally costeeffective way. Taking into account the OWC chamber internal free surface amplification and the partial standing waves developing in front of the OWC structure, the free surface zone height was set to three times the input wave height (H). The initial conditions are set as: the water level is defined at the desired level (h), waves are prescribed by the velocity components at the wave velocity inlet boundary and generated throughout the whole domain till the OWC's front lip by specifying that point on the water level, the tank two sides are defined as symmetry plans and atmospheric pressures is defined at the top and end boundaries. Trimmer mesher technique is used in generating the desired mesh. Although the Trimmer mesher permits hanging nodes, it only allows for increasing or decreasing cell size by a factor of 2. This means that to get the specified cell size in a certain zone

543

especially at the free surface zone, the base size of the mesh must be a power of 2 of this required cell size. Otherwise, the cell size within the zone where the mesh refinement is applied will snap to the factor of 2 of the base size nearest to the input value. This restriction, will in turn limit the cell aspect ratios to be a power of 2 as well. A base mesh cell size of 400 mm is selected to be applied throughout the fluid domain unless local (zone) refinement is defined for all simulations in the present work. Considering that ocean waves are the main exciting source acting on offshore structures like OWCs, a proper modelling of these waves is a crucial step for accurate prediction of the hydrodynamic loads, structure's response and performance [36]. Therefore, mesh refinement and time step studies are carried out to find the most efficient meshing parameters. First, a mesh sensitivity study is conducted considering different cell sizes in both x (Dx) and z (Dz) directions resulting in different number of cells per wavelength and wave height, accordingly. From this study illustrated in Fig. 5 (a), it is found that the cell aspect ratio is critical and should not be more than 16 for getting less than 1.5% error in the simulated wave height in comparison with the input wave theory, where the number of cells can at least be 12 and 36 per wave height and wavelength, respectively. In addition to the cell size sensitivity study, a time step study is carried out with secondeorder temporal discretization. Based on the results given in Fig. 5 (b), 1200 time steps per wave period is

Fig. 4. (a): Computational domain, (b): Detailed mesh at the PTO and (c): OWC model.

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Fig. 5. Mesh (a) and time step (b) sensitivity study.

selected in order to generate accurate waves with less than 1.0% deviation from the analytical input wave height. Different from the volumetric refinement assigned for the free surface zone, additional surface refinements are applied with 6.25 mm and 0.78125 mm surface size for the OWC noneslip walls and the PTO slot opening noneslip walls, respectively. For all noneslip walls, 10 prism layers with a first cell height equivalent to yþ y 1 and a growth rate (stretching) of 1.5 are applied. 4. Experiments and validations 4.1. Pneumatic energy parameters As given by Eq. (4), both chamber air differential pressure and airflow rate through the turbine are the two parameters required to evaluate the pneumatic extracted energy. Therefore, in the first validation, CFD results for the chamber differential air pressure and water level inside OWC chamber are compared against physical pez et al. [37]. The differential pressure is experimental data by Lo numerically monitored by measuring the pressure at two points; one inside the chamber and another outside. Also, airflow rate can be simply monitored numerically by measuring the mass flow rate through the PTO slot opening and dividing by the air density (constant for incompressible flow assumption), however

experimentally it is usually calculated as the product of the free surface velocity (rate of change in the water displacement, dh/dt) and the chamber horizontal crossesectional area assuming the air is incompressible. Both ways for measuring the airflow rate are compared in this validation. Fig. 6 shows the numerically measured time series data for the chamber differential air pressure (DP) and chamber free surface elevation at its centre (h) in comparison with experimental measurements for slot opening, e ¼ 2.5 mm and regular waves with wave height, H ¼ 40 mm, period, T ¼ 1.4 s and water depth, h ¼ 420 mm. It is clear that the numerical model agrees well with the experimental data. Furthermore, the measured airflow rate (q) either by directly monitored through the PTO (q1) or calculated from free surface velocity (q2) together with the measured pneumatic power (PE) are illustrated in Fig. 6. It is obvious that there is no significant difference in the two methods used to measure the airflow rate. As a result, the first method is kept for the remaining simulations in this study. It worth mentioning that in the present study, Time ¼ 0 s does not mean zero simulation time, but it refers to the instant of starting a new wave period. 4.2. Incident and reflected waves Measuring the wave elevation in front of the OWC structure

Fig. 6. Comparison between experiment [37] and CFD for (a): Chamber water level (h) and (b): Differential air pressure (DP). CFD results for (c): Airflow rate (q) and (d): Extracted pneumatic power (PE) for H ¼ 40 mm, T ¼ 1.4 s, h ¼ 420 mm and e ¼ 2.5 mm.

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Fig. 7. Free surface elevations along the NWT in comparison with experimental measurements [37]. (a): WP1, (b): WP2 and (c): WP3.

seaside is required for resolving the incident and reflected waves and then calculating the reflection coefficient as described in Section 2. Wave elevations along the numerical wave tank (NWT) at three probes namely; WP1 (x ¼ 15.3 m), WP2 (x ¼ 15.6 m) and WP3 (x ¼ 16 m) are compared with good agreement in Fig. 7 against pez et al. [37] for the same testing physical measurement by Lo condition used in Section 4.1. 4.3. PIV flow field In addition to the good agreement achieved in Sections 4.1 and 4.2, investigating the capability of the CFD model in estimating wave energy losses in the OWC is performed in this section via comparing the water flow field and turbulent kinetic energy pez et al. [29]. Their experiment data against physical PIV data by Lo used for this validation is for H ¼ 60 mm, T ¼ 1.6 s, h ¼ 470 mm and e ¼ 5.0 mm. Another mesh sensitivity study for the field of view (area of interest) is conducted with cell size, d ¼ 6.25 mm, 3.125 mm and 1.5625 mm as shown in Fig. 8 together with the flow velocity and vorticity fields. Flow field results are presented at four equally spaced instants through one wave cycle starting when the water level inside the chamber is minimum (trough). For clear visualization purpose, an artificial grid (50  50 points) is used to represent the velocity vectors for all simulations in this paper. It can be qualitatively seen that both meshes with d ¼ 3.125 mm and 1.5625 mm give almost the same results, while for mesh with d ¼ 6.25 mm, the vortex developing cannot be captured well especially during the maximum ineflow at 0.25T (b). This can be a result of the complex hydroeaero dynamic process during the compression (waterefilling) stage. In addition to this sensitivity study with SST turbulence model, other eddy viscosity turbulence models such as standard k-3 and realizable k-3 and the Reynolds stress model (EBeRSM) are compared with SST model. For k-3 models, simulations are carried out with two different yþ approaches: yþ y 1 and yþ > ~30, whereas yþ y 1 is used for both SST and EBeRSM. All yþ wall treatment is applied for all turbulence models. It is found that both yþ approaches provide almost the same results and there is no significant difference between the two tested k-3 models, thus only results for realizable k-3 are included in the comparison with other

models shown in Fig. 9. It is obvious that k-3 models fail to predict the flow behaviour throughout the whole wave cycle even with the most refined mesh (d ¼ 1.5625 mm), where SST provides comparable results with EBeRSM. Considering the computation time, a cell size of 3.125 mm with SST turbulence model is used for the rest of simulations in this paper. Figs. 10 and 11 show the numerical velocity (the same vector scale used in the experiment) and vorticity fields for eight phases through one wave cycle (ineflow and outeflow). Although, the numerical results are instantaneous not phaseeaveraged like the experimental data, the numerical model captures the flow field in good agreement. The ineflow stage starts at the lowest water level inside the chamber (trough), as the water begins ascending, small clockwise vortices start developing underneath the front lip, which keep building up and expanding behind the front wall. As the water level moves toward its peak, countereclockwise vortices appear under lip inner edge. When the outeflow stage begins, the clockwise vortices start dissipating, while the antieclockwise vortices strengthen and move downward out of the chamber. Just before the water falls to its minima, the antieclockwise vortices become weak and a small clockwise vortex initiates under the front wall outer corner. Following the satisfactory agreement in terms of velocity and vorticity fields, the proposed energy balance model is applied to check its applicability in estimating the energy losses in the OWC devices. Fig. 12 shows one wave cycle timeeseries for chamber free surface (h), kinetic energy (KE), turbulent kinetic energy (TKE), stored energy (ES), extracted power (PE) and power dissipation rates (ε). It is clear that kinetic energy curve has two maximum and minimum values (peaks/troughs) which agrees with experimental findings by Fleming et al. [28], Graw, et al. [27] and Morrison [26] who all found that there is a typical kinetic energy curve characterised with two peaks and troughs. Further description of Fig. 12 is given below:  Starting where the water level inside the chamber is minimum (Time ¼ 0.0), at this instant, the potential energy stored in the OWC is maximum whereas the velocity field is quite low and the chamber free surface vertical velocity is almost zero due to the negligible chamber water level gradient (slope). As a result, at

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Fig. 8. Velocity and vorticity fields mesh sensitivity study at four instants through one cycle, (a): Time ¼ 0 (trough), (b): Time ¼ 0.25T, (c): Time ¼ 0.50T (crest) and (d): Time ¼ 0.75T.

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547

Fig. 9. Velocity and vorticity fields with different turbulence models through one wave cycle, (a): Time ¼ 0 (trough), (b): Time ¼ 0.25T, (c): Time ¼ 0.50T (crest) and (d): Time ¼ 0.75T.

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Fig. 10. Comparison between PIV experimental velocity and vorticity fields [29] (left) and CFD results (right): Ineflow stage (phase a e phase d).

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Fig. 11. Comparison between PIV experimental velocity and vorticity fields [29] (left) and CFD results (right): Outeflow stage (phase e e phase h).

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Fig. 12. Chamber free surface elevation (h), kinetic energy (KE), turbulent kinetic energy (TKE), dissipation rate (ε), stored energy (ES) and extracted power (PE) (top), velocity and vorticity fields and VOF (middle), wave elevation along NWT, incident and reflected wave power spectral densities (bottom).

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this moment, the kinetic energy, turbulent kinetic energy and power dissipation rate are at their lower values together with no power being extracted. As the water starts rising inside the chamber, both the rate of change in the chamber wateresurface and the velocity field start magnifying and consequently the kinetic energy is growing up and getting its maximum value almost when the water level reaches its initial stillewater level at Time ¼ 0.25T (i.e. where the ineflow is maximum). When the rate of change in the waterelevel is maximum (maximum slope), the extracted power is peaked, while the stored potential energy is minima (zero oscillation). Furthermore, during this stage, power dissipation rate increases gradually as a result of building up clockwise vortices. Further upward water movement results in increasing the stored potential energy associated with the increase in water oscillation. On the other hand, the extracted power diminishes. Moreover, the kinetic energy gradually drops down to about half its peak where the rate of water level changes returns to zero (water level crest), while dissipation rates are almost steady with slight reduction at Time ¼ 0.50T. When the outeflow stage starts at Time ¼ 0.50T, the water level falls and consequently the potential stored energy gradually decreases (disperses). On the other side, both kinetic energy and dissipation rates increase till they get maximum when the water level reaches the mean zero level at Time ¼ 0.75T. At this point the extracted power is maximum. Also during this stage, it seems that there is a slight increase in the extracted and dissipated/losses power peaks in comparison with the ineflow stage. As discussed in Section 2, this may be a result of the stored potential energy in the oscillating water column heave dispersing, and the dispersed energy is contributing to the extracted and losses energy (and also radiated/reflected energy which is not shown in this figure). Following this maximum outeflow, kinetic energy, dissipation rates and extracted power progressively slow down, whereas the potential power escalates until they all restore their initial values when the water level become minimum.

The average turbulent kinetic energy coefficient defined as the ratio between the turbulent kinetic energy and the flow kinetic energy over one wave cycle is found to be 16.5% which agrees well with the experimental value of 18.7% considering the level of uncertainty in the PIV data was found to be about 14 mm/s in the

551

velocity measurements (error below 7% of the maximum velocities inside the OWC chamber) [29]. Wave elevation at the three wave probes used to find the reflected energy is shown together with the resolved reflected and incident wave spectral densities in Fig. 12 (bottom). Obtaining a high frequency spectrum resolution may be difficult, given that a time series of five cycles is short, however to overcome this, the utilized 5ecycle time series window is selected carefully to ensure that the start and end points of the time series data can be jointed smoothly. After this, the time series window can be repeated to provide long timeeseries data (here, it is repeated 20 times). Fig. 13 illustrates the percentage of different wave energy components included in the energy balance model (Eq. (1)). From this figure, it can be seen that approximately 54% of the incoming wave energy is reflected and only 12.7% is converted into pneumatic energy for further electricity generation. On the other side, about 18.6% of the incident energy is lost during the energy conversion chain. It also can be seen that there is an average of 14.7% of the incident wave energy stored in the OWC heave motion, which has not been converted into radiated, extracted or energy losses (see Section 2, Eq. (6)). 5. Numerical campaign tests After the numerical model is validated with good agreement, in this section the impact of increasing incoming wave amplitude on the energy balance model under various wave heights and damping factors is investigated. For this study the water depth remains constant at 470 mm and wave period T ¼ 1.6 s. Five wave heights, H ¼ 40, 60, 80, 100 and 120 mm are considered with the following eight damping factors, which are simulated by different slot openings, e ¼ 0.65, 0.8, 1.0, 1.2, 1.5, 2.0, 2.5 and 5.0 mm. The chamber differential air pressure (DP), airflow rate (q), chamber free surface response amplitude operator (RAO) or amplification factor (hmax/A) and the time (phase) shift (b) between the water level inside and outside the chamber (defined as a fraction of the wave period, T) for different wave heights (H) under various damping factors, C (b/e), are shown in Fig. 14. The chamber pressure is seen to gradually increase with increasing the damping factor as well as wave steepness. The variation in the chamber pressure becomes almost negligible after a damping value corresponding to slot opening, e ¼ 1.0 mm, which indicates that the pressure variation becomes less sensitive for higher PTO damping. On the other side, airflow rate seems to increase as the wave height

Fig. 13. Wave energy balance for regular wave of H ¼ 60 mm, T ¼ 1.6 s, h ¼ 470 mm and e ¼ 5.0 mm.

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Fig. 14. Impact of incident wave height and PTO damping on (a): Variation in chamber differential air pressure (DP), (b): Airflow rate (q), (c): Chamber free surface amplification (hmax/A) and (d): Phase shift between inside and outside chamber free surface (b).

increases, but it drops down when the damping factor grows up. Completely opposite to the pressure variation, the water amplification factor descends as the damping factor and the wave height increase. It is obvious that the airflow rate trend does not follow the amplification factor in terms of wave height effect. In order to explain the reason for this, the chamber free surface elevation shown in Fig. 15 is investigated. For different PTO damping at a constant wave height, H ¼ 60 mm (Fig. 15a) and for different wave heights with a constant PTO damping corresponding to slot opening, e ¼ 1.5 mm (Fig. 15b), it is clear that the chamber free surface gradient (rate of change or slope) increases as the wave height increases and the damping factor decreases (slot opening

increases) which in turn results in increasing the free surface as well as air velocities considering incompressible airflow. It can be also seen from Fig. 14d that the phase shift between the free surface inside and outside the chamber increases as the PTO damping and incoming wave height increase. The overall energy extraction coefficient (COeverall) of the OWC for different wave heights and eight damping factors is shown in Fig. 16a. Generally, the system overall extraction coefficient starts low (10.6e14.2%) for the lowest damping factor (e ¼ 5.0 mm) and progressively increases as the damping increases till getting its peak value at a certain damping factor which is the optimum value for that condition. For instance, at 40 mm wave height, the peak

Fig. 15. OWC chamber free surface elevation for (a): different PTO damping with a constant wave height, H ¼ 60 mm and (b): different wave heights under a constant damping, e ¼ 1.5 mm.

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Fig. 16. OWC energy balance coefficients for different five wave heights and under eight PTO damping. (a): Overall energy extraction coefficient (COeverall), (b): Reflection coefficient (CR), (c): Energy absorption coefficient (CAbs), (d): Internal energy extraction coefficient (CInternal), (e): Energy stored coefficient (CS) and (f): Energy losses coefficient (CL).

efficiency of about 39.4% is achieved at a damping factor C ¼ 85.33 (e ¼ 1.5 mm). Following this apex, the efficiency drops down with extra increase in the applied PTO damping (COeverall ¼ 18.7% at e ¼ 0.65/C ¼ 196.92). It is also clear that increasing the wave height has a little increasing effect under smaller damping factors (up to e ¼ 2.5 mm), while discrepancies become more obvious for intermediate damping (2.5 mm < e < 1.0 mm). Although the extracted energy’ variables (pressure and flow rate) increase with increasing the wave height as shown in Fig. 14a and b, which means that the pneumatic energy also increases as the wave height escalates, the overall extraction coefficient declines with increasing the wave height for e < 2.5 mm (C > 51.2) as seen in Fig. 16a. As the wave height increases from 40 mm to 120 mm, the overall maximum extraction coefficient dramatically falls down by about 7.5% at the optimum damping (e ¼ 1.5 mm). Increasing the incoming wave height three times (from H ¼ 40e120 mm) magnifies the incident energy 8.6 times (see Eq. (2) considering the resolved/measured incident wave heights instead of the input theoretical values). By recalling the overall energy extraction coefficient definition which is the ratio between the extracted pneumatic and incident wave energy (see Fig. 3 and Table 1), maintaining the same extraction coefficient under bigger waves of H ¼ 120 mm requires escalating the output pneumatic energy also 8.6 times. On the other hand, the output pneumatic energy under H ¼ 120 mm and e ¼ 1.5 mm only increases 6.88 times that under 40 mm. This can be a result of increasing the reflected energy, which in turn limits the available absorbed energy. In addition, viscous and turbulence losses and other nonlinear effects on the chamber internal free surface may have impacts on the extracted energy. This can be explained through the following detailed wave energy balance analysis. While Fig. 16b shows that the reflected energy (represented by the reflection coefficient) decreases with increasing both the damping factor from e ¼ 5.0 mm up to 1.5 mm and the wave height till a minimum reflection coefficient of about 0.66 is achieved at e ¼ 1.5 mm for 40 mm wave height, additional increase in the damping factor leads to increasing the reflection coefficient to more than 0.85 for the largest damping (e ¼ 0.65 mm). Similarly, after the

minimum reflection coefficient is achieved, the influence of increasing the wave height is reversed such that the higher wave height, the higher reflected energy (CR ¼ 0.89 at H ¼ 120 mm and e ¼ 0.65 mm). On the other hand, the system capability in absorbing the incident wave energy is complementary to the energy reflected by the OWC structure (see Eq. (3)), thus, the energy absorption coefficient trend is totally reversed in comparison with the reflection coefficient as demonstrated in Fig. 16c. The OWCewave interaction consists of two subeinteractions; namely radiation and scattering/excitation. While the radiation problem concerns the radiation of waves caused by an oscillating dynamic air pressure above the interface, the excitation problem concerns the oscillation caused by an incident wave when the dynamic air pressure is zero [38]. Therefore, the reflected waves under pneumatic damping consist of pure reflected waves from the front wall as well as radiated waves from the inside chamber water heave oscillation. Looking in more details on the damping and wave height effects not just on the chamber water level amplification factor but also on the time (phase) shift (b) between the water level inside and outside the chamber is important for better understanding the reflection and in turn the absorption efficiency trends. It can be seen from Fig. 14d that a phase shift of about 0.098T is found for the lowest damping (e ¼ 5 mm, H ¼ 40 mm), whereas a phase shift of bout 0.25T is measured with the highest damping (e ¼ 0.65 mm, H ¼ 120 mm). This phase shift together with the amplification factor can uncover the reflection coefficient trend which indicates that the system tends to radiate large in phase waves under lower PTO damping and in turn, for higher imposed damping, relatively small out of phase waves are generated. Despite being inversely proportional to the absorbed energy (see Table 1), the energy stored coefficient is proportional to the energy stored in the chamber free surface heave motion which is a function of its oscillation as given by Eq. (5). As shown in Fig. 16d, the system is likely to relatively store less energy as the PTO damping and wave height increase, which is almost the same trend for the amplification factor. This indicates that the system capability for storing incident wave energy heavily depends on the

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amplification factor than the amount of energy being absorbed. Also, the system tends to relatively store less energy for higher waves especially under lower PTO damping. The system internal energy extraction coefficient (CInternal) is presented in Fig. 16e for different wave heights and PTO damping factors. It is clear that this coefficient is following to some degree the same general trend of the overall energy extraction coefficient, except that the peak value is shift to a higher damping factor and shows lesser sensitivity to further increasing in the pneumatic damping in comparison with the overall extraction coefficient. Additionally, increasing the wave height slightly improves the internal energy extraction coefficient that is more pronounced under intermediate damping. The overall declining and growing trends of the energy stored coefficient (Fig. 16d) and the internal energy extraction coefficient (Fig. 16e), respectively as the PTO damping and wave height increase support the strong coupling between these two variables explained in Section 2. While increasing the PTO damping and/or wave height reduces the energy stored in the column heave motion, a part of this energy is converted into pneumatic energy, and accordingly the internal energy extraction coefficient increases. For instance, under a wave height H ¼ 60 mm, enlarging the damping factor from C ¼ 25.6 (e ¼ 5.0 mm) to C ¼ 106.7 (e ¼ 1.2 mm), the stored energy (ES) drops by almost 4.6 times (from 1.865 to 0.404 J/m), whereas the extracted energy (EE) increases by about 2.6 times (from 1.609 to 4.263 J/m). Another example under 40 mm wave height and the same damping factors

(from e ¼ 5.0 mme1.2 mm) revealed that there is a reduction of 4 times (from 0.943 to 0.236 J/m) in the stored energy, while the extracted energy escalates from 0.596 to 2.049 J/m (3.4 times). In addition to the impact on the extracted energy, the change in the stored energy may also contribute to the change in the reflected energy as presented by the reflection coefficient in Fig. 16b and the energy losses in Fig. 16f. Regarding the losses coefficient shown in Fig. 16f, it is found that the losses are dramatically high at low damping and this is a result of the corresponding high oscillation motion (amplification factor, Fig. 14c) together with the high airflow rate passing through the PTO (Fig. 14b). As both parameters (amplification and airflow rate) sequentially descend as the PTO damping increases, the losses also cascade to their minimum at a specific damping factor (e ¼ 1.5 mm) and remain almost steady up to e ¼ 1.0 mm. However, the losses coefficient seems to slightly increase after getting a minimum value especially at higher damping factors and this is due to the large reduction in the absorbed energy while the losses change slightly at these damping factors. Additionally, for damping factors (C) less than 85.33, as the wave height increases, the losses coefficient also extends. However, after this damping limit (C ¼ 85.33), the more increase in the wave height, the lower losses coefficient, which can be a result of increasing the internal energy extraction coefficient (Fig. 16e) at a given absorbed energy for a certain PTO damping factor. It is important to note that the energy losses coefficient is an indicator to the energy losses relative to the absorbed energy

Fig. 17. Velocity and vorticity fileds for different wave heights and PTO damping during ineflow (waterefilling) stage.

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Fig. 18. Velocity and vorticity fileds for different wave heights and PTO damping during outeflow (watereemptying) stage.

(Table 1), and the reduction in this coefficient does not necessarily mean reducing the energy losses. As it may be expected, increasing the wave height develops nonlinear effects that increase the losses in the system. For instance, increasing the wave height from 40 mm to 120 mm at e ¼ 5.0 mm (C ¼ 25.6) expands the energy losses (EL) from 0.63 J/m (CL ¼ 0.29) to 13.26 J/m (CL ¼ 0.53). Similarly, under a higher damping of e ¼ 1 mm (C ¼ 128) the energy losses magnifies from 0.55 J/m (CL ¼ 0.22) to 2.66 J/m (CL ¼ 0.18) as a result of increasing the wave height from 40 to 120 mm. Finally, in order to visualize the losses happening inside and around the OWC chamber and show the impact of both incident wave height and PTO damping, the flow field inside the chamber at 8 phases through oneecycle is shown in Figs. 17 and 18 for ineflow and outeflow stages, respectively. It is shown that, as the applied damping increases, the vortex formation and dissipation become weak, which supports the results presented in Fig. 16f. Furthermore, the higher wave height, the stronger the vortices generated. It is also obvious that either increasing the damping or the wave height, results in formation of additional vortices at the lower lip. The higher losses illustrated by the vortices magnitude and size under the higher waves demonstrate the heavy obstruction of the flow to pass underneath the front lip and access the pneumatic chamber, which in turn reduces the amplification factor (Fig. 14c) and the coefficient of the energy stored in the water column heave (Fig. 16d) that according to [34] is the only destination directly coupled to the PTO mechanism, which further explain the

reduction in the overall energy extraction coefficient as the wave height increases. 6. Conclusions A previously proposed wave energy balance model was amended in this paper to account for the instantaneous stored potential energy in oscillating water column chamber's heave motion when considering long term averaged results over certain wave cycles. A 2D CFD model based on RANSeVOF was developed and utilized to conduct a wave energy balance for an oscillating water columnewave energy converter. The model is validated in good agreement with experimental measurements for chamber differential air pressure, chamber water level and wave elevations. In addition to these main parameters, the capability of the CFD model in capturing the details of the flow field compared well with PIV data confirming its ability to accurately predict the vortex formation and dissipation processes inside and around the OWC underwater chamber. Various turbulence modes were also examined including the Reynolds stress model (EBeRSM) and eddy viscosity models such as standard k-3 , realizable k-3 and SST. Among these models, k3 models could not capture the flow field even with a very refined mesh, while SST provides a reasonable agreement with EBeRSM. Aiming at uncovering the impact of increasing the incoming wave amplitude and the turbine pneumatic damping on the energy conversion process and the OWC hydrodynamic performance, the

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validated model is used to carry out 40 wave energy balance analyses for five wave heights under eight PTO damping simulated by different slot openings. From the investigations carried out, the following conclusions are drawn: 6.1. Impact of the PTO damping  Increasing the PTO damping leads to higher chamber differential air pressure, lower airflow rate, lower chamber free surface oscillation and lesser energy storage coefficient. Under high damping factors, influence on the pressure trend becomes small with tendency to plateau.  The overall and internal energy extraction coefficients increase as the PTO damping increases till getting maxima and then reduce with a further increase in damping. The internal energy extraction coefficient looks to be less sensitive to the PTO damping than the overall energy extraction coefficient.  The reflection coefficient declines while the PTO damping increases up to a certain damping value at which point the reflection is minimum and the overall energy extraction coefficient is maximised. The further increase in PTO damping, the higher the reflected energy.  The phase shift between the water elevation inside and outside the OWC chamber increases with increasing the applied damping.  The absorbed energy is complementary to the energy being reflected, thus the energy absorption coefficient trend is completely contrary to the reflection coefficient.  The losses coefficient descends with increasing the PTO damping.  The lower PTO damping applied on the chamber free surface, the stronger vortices generation inside the chamber.

6.2. Impact of the incoming wave height  Increasing the incident wave height results in building up more differential air pressure, and airflow rate, whereas decaying the relative water heave motion inside the OWC chamber as well as the energy storage coefficient.  Approximately 7.5% of the overall energy extraction coefficient is lost as a result of escalating the incident wave height three times from 40 mm to 120 mm.  Generally, as the wave height increases, the reflection coefficient increases and the energy absorption coefficient decreases except for the lowest PTO damping.  The phase lag between the water inside and outside the OWC chamber increases with increasing the wave height.  As the wave height increases, energy losses build up, but the losses coefficient increases up to a given damping factor before start decreasing with a further increase in PTO damping.  The larger the wave height is, the larger the vortices being developed. Towards the author's global research aim of investigating the hydrodynamic performance of an offshore moored OWC device, a next step includes utilizing the methodology adopted in the present study by extending the energy balance analysis to consider the transmitted wave energy underneath the OWC structure. Different design parameters will be considered including environmental conditions such as wave height and period, pneumatic damping and underwater structural geometry. Also, the 2D CFD model utilized herein will be extended to a 3D domain to provide more insight into the device performance considering the impact of disturbing the flow by the chamber’ side walls as well as wave

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