Hydrodynamic Performance of A Porous-Type Land-Fixed Oscillating Water Column Wave Energy Converter

A hybrid, porous breakwater—Oscillating Water Column (OWC) Wave Energy Converter (WEC) system is put forward and its hydrodynamic performance is investigated using the fully nonlinear, open-source computational fluid dynamics (CFD) model, OpenFOAM. The permeable structure is positioned at the weather side of the OWC device and adjoined to its front wall. A numerical modelling approach is employed in which the interstices within the porous structure are explicitly defined. This permits the flow field development within the porous structure and at the OWC front wall to be observed. The WEC device is defined as a land-fixed, semi-submerged OWC chamber. A range of regular incident waves are generated at the inlet within the numerical tank. The OWC efficiency and the forces on the structure are examined. Results are compared for the simulation cases in which the porous component is present or absent in front of the OWC chamber. It is found that the incorporation of the porous component has minimal effect on the hydrodynamic efficiency of the OWC, reducing the efficiency by less than 5%. Nevertheless, the forces on the front wall of the OWC can be reduced by up to 20% at the higher wave steepness investigated, through inclusion of the porous structure at the OWC front wall. These findings have considerable implications for the design of hybrid OWC—breakwater systems, most importantly in terms of enhancing the durability and survivability of OWC WECs without significant loss of operational efficiency.


Introduction
In recent years, there has been an increasing societal awareness of the pollution damage to the environment posed by the continued use of fossil fuels. Furthermore, it is widely accepted that coal, oil and gas are finite resources which will be more expensive to extract in the future. This has led to a concerted effort to develop and improve renewable energy systems (Kåberger, 2018). Harvesting installations for wind energy and solar energy are proven technologies and have been deployed around the globe. However, with both of these technologies the energy-harvesting period is intermittent and highly dependent on variable, localised weather conditions (Zhou et al., 2018). In contrast, marine energy resources are relatively constant and the temporal variability in the energy harvest potential is minimal. According to previous research the world's oceans have the potential to supply between 8000 and 80000 TWh per annum of energy from wave power alone. It is notable that the upper end of this range is larger than the recent annual global power consumption demand (as of 2017), (Melikoglu, 2018).
Marine energy systems have received less research attention than other forms of renewable energy and are still in the developmental phase (Jeffrey et al., 2013;Wilberforce et al., 2019). This is predominantly because previous research has shown that marine energy is relatively inefficient to exploit. This is especially evident when compared with other forms of renewable energy. It has been suggested that most wave power technologies were impractical or economically unfeasible up to a decade ago (Leijon et al., 2006). Therefore, in recent times many studies have focused on augmenting the efficiency of marine renewable energy harvesting systems to increase their commercial viability.
The main technologies that have been studied and developed to prototype stage to harvest marine energy focus on wave energy, tidal energy and to a lesser extent salinity and temperature gradients. This research concentrates on wave energy harvesting by an oscillating water column (OWC) wave energy converter (WEC). An OWC is a conceptually simple device with few moving parts that can be used to extract wave energy. The operating principle relies on a partially submerged chamber that is open to the sea below the free surface. This opening is typically on the prevailing incident wave side. The hollow chamber traps a pocket of air and as the ocean waves interact with the chamber the internal water column oscillates vertically. This vertical motion drives the trapped air out through an orifice to which a turbine is attached. Then, as the ocean wave recedes, the water column inside the chamber drops. This causes the air pocket in the chamber to become decompressed and air is drawn into the chamber through the orifice. A specialised, directionally independent turbine such as a Wells turbine or an impulse turbine is connected to the orifice. This allows energy to be extracted on both the exhalation and inhalation phases of the cycle.
The main drawbacks associated with wave energy exploitation are the low extraction efficiency, the durability and survivability of the harvester technology and the initial outlay expense for the cost of construction (Zhang and Ning, 2019;Ransley et al., 2017;Foteinis and Tsoutsos, 2017;Astariz and Iglesias, 2015). In terms of the low operational efficiency, it has been shown that much of the kinetic energy associated with the incident wave is dissipated at various stages of the extraction process (He et al., 2016). These energy losses can be broken down into wave reflection losses, frictional losses due to the wave loads on the weather side of the OWC front wall, viscous energy losses due to hydrodynamic flow characteristics, and energy losses at the power take-off (PTO) system Das and Samad, 2020). According to a report from Ocean Energy Systems (Nielsen, 2003), the average conversion efficiency for a pneumatically driven PTO system used in an OWC is 54%.
Most OWC installations have been constructed as proof of concept. However, some have reached a fully commercialised operational stage, providing electricity on a small scale to localized regions. Examples of these installations include the Mutriku power plant on the northern coast of Spain and the Pico plant on the Azores Archipelago (Falcão, 2010). However, in many of these cases, at both proof of concept and commercial level, the OWC facilities have suffered damage from incident waves and in some cases, the plants have failed due to fatigue loading from wave impacts (Falcão et al., 2020;Medina-López et al., 2018;Jalón and Brennan, 2020). In order for OWC technology to be widely adopted and for these devices to be considered as a viable renewable energy harvesting technology, it is necessary to increase their efficiency and also tackle the challenges affecting their durability. To address these deficiencies, many researchers have suggested incorporating OWC devices into breakwater structures (Zhao et al., 2019;Ning et al., 2017;Zheng et al., 2019). This has the added benefit of reducing the initial outlay construction costs associated with a standalone OWC.
The concept of integrating an OWC with a breakwater structure has previously been investigated by many researchers. Zheng et al. (2019) investigated a cylindrical OWC chamber integrated into a vertical breakwater wall using a linear potential flow theory model. They found that the efficiency of the integrated model was higher than that of a standalone cylindrical OWC in open water by up to a factor of two within a specific wave number range. Deng et al. (2019) developed a numerical model to study the behaviour of a floating OWC with a perforated horizontal base plate. They investigated the influence of the OWC geometry and the incident wave environment on the wave reflection and transmission coefficients. John Ashlin et al. (2018) employed a scaled experimental approach to examine the influence of spacing on a series of OWCs integrated into a rubble mound breakwater. Zhao et al. (2019) listed a number of previous studies focusing on the integration of a breakwater structure with an OWC in their review paper on breakwater-WEC systems.
In many of these previous studies listed, a common feature is that the OWC-chamber front wall itself acts as the breakwater. In fact, there is no shielding protection provided to the OWC structure and the front wall of the OWC receives the full loading from the incident waves. A study by Mackay et al. (2019) investigated the effects of pulsating wave loads on thin porous plates. In their research the plates acted as breakwaters to reduce the wave forces on the marine structures. The study adopted numerical and experimental approaches to investigate the influence of plate thickness, plate porosity, and hole separation distance on the forces measured at the plate. They found that the porosity of the plate was the dominant parameter to affect the wave loads.
Research focusing on incident wave loading on caisson structures has shown that when a wave impacts a vertical wall, such as the OWC front wall, extreme impulse load effects may occur, (Takahashi et al., 1994;Peregrine, 2003;Lugni et al., 2006;de Almeida and Hofland, 2020). Peregrine (2003) identified a specific dynamic phenomenon and coined the term "wave flip through" to describe the condition in which the free surface of a wave impacting upon a non-porous vertical breakwater wall focuses on an infinitesimal point. This wave focusing process yields considerably higher dynamic impulse pressure loading at the structure. The existence of wave flip through at a vertical seawall has since been observed in a number of studies including an investigation by Lugni et al. (2006) in which they observed the generation of vertical water jets with accelerations exceeding 1500g. The formation of these jets applies significant reaction loads on the vertical walls. Collapsing breaker waves or other extreme wave loading events are not necessary to generate the exceptional loads during these wave flip through events. The effect of wave loading can be reduced, and the possible development of wave flip-through will be disrupted by installing a porous structure at the weather side of the OWC front wall.
There is a lack of existing research on OWC-breakwater systems that investigate the protection of OWC chambers by a porous component at the front wall of the OWC. Instead, many of the prior studies have focused on exploiting a breakwater layout to manipulate the macroscopic flow field and direct the waves to augment the OWC efficiency. Furthermore, previous research that focuses on enhancing the durability and therefore the design life of OWCs is lacking in the published literature. In this study it is shown that a semi-submersed porous section may be used as an effective protective structure when positioned in close proximity to the OWC seaward wall without detracting significantly from the OWC operational efficiency. The wave focussing methods and efficiency improvement techniques achieved through optimised breakwater layouts and OWC positioning presented by previous researchers (John Ashlin et al., 2018;Howe et al., 2020) can still be employed in conjunction with the current proposed system.
In this study the primary objective is to evaluate the design of a porous section positioned at the weather side of the OWC chamber front wall. A reduced-scale model is investigated in order to facilitate future experimental analysis comparisons. The effects of the attached porous structure on the wave loading on the chamber structure and hydrodynamic efficiency are considered. The rest of the paper is organised as follows, Section 2 presents the numerical model based on the open-source CFD software OpenFOAM and validation tests, Section 3 presents the discussion and finally conclusions are drawn in Section 4.

Mathematical formulation
The numerical simulations were performed by using the open-source CFD software OpenFOAM, which is a collection of compiled C++ libraries that use the finite volume method to model various fluid flow simulation scenarios. In this study, the dual phase, incompressible solver inter-FOAM is used to model the flow field. The incompressible solver is a reasonable selection as the air compressibility effects will not manifest strongly at the analysed model scale. The interface between the air phase and the water phase is computed by using the Volume of Fluid (VOF) method (Hirt and Nichols, 1981). Recent versions of OpenFOAM releases are compiled with wave modelling functionality incorporated. OpenFOAM version V1812 is used in this investigation.

Governing equations
OpenFOAM uses the Navier−Stokes equations to control the advection of fluid during the simulation. The mass conservation equation in vector form is ∇ · U = 0 (1) and the momentum conservation equation is ∂ρU ∂t where is the fluid flow velocity vector, is the fluid density, is a stress tensor representing the surface forces acting on a fluid particle, and is a body force which acts on the volumetric mass of the fluid particle.
The free surface development is tracked using the VOF method. In the VOF method, a function is introduced at each grid cell in the domain. The value of the function is set to unity at each cell that is entirely occupied by water, and zero at other cells entirely occupied by a second fluid. Those cells which form the interface between the two fluids have a value of . Then, the temporal advection of the free surface is governed by the transport equation u v x y where and are the velocities in the component and directions respectively.

Pneumatic model
The hydrodynamic efficiency of an OWC can be computed from the relationship between the pneumatic power absorbed from the incident waves by the OWC device and the time-average energy flux of the incident waves. The hydrodynamic efficiency is given by where the numerator, represents the pneumatic power absorbed by the OWC, is the average energy flux per unit width of the incident wave and is the width of the OWC chamber.
The pneumatic power absorbed by the OWC can be computed from the time averaged product of the air flow rate through the orifice multiplied by the air pressure in the chamber as follows: is the wave period, is the air volume flux through the orifice and is the air pressure in the chamber.
According to linear wave theory the average power per unit width of incident wave is given by where is the gravitational acceleration, is the incident wave amplitude and is the group velocity of the incident wave packet given by k h c where is the wave number, is the static water depth and is the incident wave velocity given by ω where is the angular frequency computed from The hydrodynamic efficiency of the OWC is bounded in the range . The lower bound indicates that the device does not harvest any of the incident wave energy and the upper bound represents a closed system in which there is no loss in the energy transfer process from the wave loading to the energy extracted by the power take-off unit.

Forces on the OWC front wall
In order to determine the influence of the porous section on the force reduction at the front wall, the total horizontal forces at the wall are calculated according to Eq. (10).
where is the vertical wetted surface area of the wall, is the pressure applied to the wall surface and is the normal vector to the wall in the x direction. On the external wall surface, the normal force is applied from the incident and reflected waves; however, the oscillating free surface level dictates that the trapped air pocket within the chamber will undergo pressurization. Then, the air phase in the chamber also applies a force component normal to the internal wall surface.

Geometry and boundary conditions
The numerical model set-up follows an experimental investigation carried out at The State Key Laboratory of Coastal and Offshore Engineering at Dalian University of Technology by Ning et al. (2016a). A three-dimensional model is created by using the open-source CFD software OpenFOAM. The numerical wave tank measures 22.0 m long, 0.4 m wide and 1.4 m deep. A Cartesian coordinate system is set at the base of the left end of the tank with the x-axis pointing in the longitudinal direction of the tank and the z-axis pointing vertically upwards. At the left end of the wave tank there is a wave inlet boundary condition. An active absorption condition is also specified at this boundary to damp the waves reflected from the OWC structure. This removes wave re-reflection that would otherwise occur at the wave boundary generation zone. The OWC is installed with the seaward side of the chamber front wall located 20.0 m from the inlet. The chamber is 0.55 m wide and the chamber walls are 0.04 m thick. The water depth in the tank is set to 0.8 m. The immersion depth of the front wall of the OWC is 0.14 m and the internal chamber height is 0.2 m above the initial water depth. A circular orifice is located centrally along the y-axis with its centre point 0.26 m from the front face of the chamber front wall. The diameter of the orifice opening is 0.042 m. A sketch of the tank is presented in Fig. 1 and the geometric parameters of the simulation are summarised in Table 1. The tank base, the sidewalls and the OWC walls have a no-slip boundary condition applied. The top of the tank is considered to be open to the atmosphere and has a prescribed boundary condition which permits the outflow of the air phase.  The OWC dimensions are selected to reflect previous experimental work that has been carried out at The State Key Laboratory of Coastal and Offshore Engineering at Dalian University of Technology by Ning et al. (2016a). This facilitates the validation process and also allows for the comparison of the non-porous simulation results with previously generated experimental data. The numerical modelling study is conducted at a 1:10 scale and the OWC chamber dimensions are selected accordingly. Froude scaling techniques can be employed to extrapolate these reduced scale observations to full scale projections. The current study does not include the effects of air compressibility within the chamber. These effects are negligible at the small scale considered in this investigation. However, at full scale, the air compressibility effects may be significant. Further information on the scaling effects when modelling OWC WECs can be found in Dai et al. (2019).
In this study, the OpenFOAM incompressible, dualphase solver interFOAM is used to generate the solution to the governing flow equations. This solver uses the PIMPLE algorithm to solve the pressure velocity coupling to gener-ate the flow field solution. The equation discretization schemes for the temporal derivative, the gradient divergence and Laplacian are presented in Table 2. The preconditioned conjugate gradient (PCG) solver was used together with the diagonal incomplete-Cholesky (DIC) preconditioner for the solution of the pressure field. The preconditioned bi-conjugate gradient (PBiCG) solver was used with the diagonal incomplete-Lower-Upper (DILU) preconditioner for the solution of the velocity field. Simulation time stepping was limited by the Courant-Friedrichs-Lewy (CFL) condition to ensure stability of the solution. In the simulations with the porous structure adhered at the weather side of the OWC front wall, a permeable section is generated which consists of an assembly of packed spheres of diameter 0.05 m. This structure is an idealized representation of a porous breakwater. The spheres are arranged into a cubic packing system. This yields a geometry with an overall porosity of approximately 47%. The bounding extents of the packing arrangement measures 0.25 m× 0.4 m×0.4 m in the x, y and z directions respectively. This results in a porous structure that measures five sphere diameters in the x direction, and eight sphere diameters in both the y and z directions, producing a porous assemblage consisting of a total of 320 spheres. The porous structure extends 0.15 m below the SWL. This cubic packed sphere porous configuration has previously been demonstrated to efficiently dissipate wave impact energy (Mayon, 2017). Fig. 2 shows an image of the porous assembly in conjunction with the OWC chamber.
The simulations were performed in parallel using a 32 core Intel®Xeon Silver 4110, 1.1 GHz processor, with max-imum wall clock time of approximately 80 hours in the case of the porous simulations with higher wave steepness.

Model verification and validation
An initial model verification study was performed by undertaking a grid discretization study. A number of model tests were completed with varying levels of mesh resolution applied to the domain geometry. The domain mesh was progressively refined using a grid-halving approach. The wave height, H i , at the longitudinal mid-point along the numerical flume was analyzed using three different mesh resolutions. By using the free surface elevation values at this location along the numerical tank, it was determined that a mesh with grid elements measuring 0.017 m in the x-direction, 0.02 m in the y-direction and 0.04 m in the z-direction produced results in the asymptotic range of convergence according to the grid convergence method of Roache (1997).
An image of the volumetric grid elements generated for the simulation without the porous structure is presented in Fig. 3a. As shown in the figure, the region around the OWC structure has been further refined using a grid halving technique. Additionally, the grid has been refined at the free surface interface between the water and air phases with the elements measuring 0.00425 m in the x-direction, 0.005 m in the y-direction and 0.01 m in the z-direction. This yields approximately 1.52 million cells for the entire non-porous simulation domain. Fig. 3b shows a longitudinal section  Robert MAYON et al. China Ocean Eng., 2022, Vol. 36, No. 1, P. 1-14 through the OWC structure with the porous assembly present. The porous interstices were meshed using the native OpenFOAM snappyHexMesh utility and this mesh consists of approximately 1.74 million cells. Fig. 4a shows a comparison of the simulated free surface elevation at the chamber centre with the experimental data by Ning et al. (2016a) for the case of wave amplitude A i =0.03 m and wave period T=1.366 s. It should be noted that the experimental study by Ning et al. (2016a) was limited to an OWC without a porous section located at the front wall. The present numerical validation model is identical to this non-porous experimental setup of Ning et al. (2016a). This non-porous comparison is performed to gain confidence in the veracity of the current non-porous model and the results obtained from the follow-up porous simulations.
There have been few three-dimensional studies performed on wave impacts on porous structures and therefore, we establish credence in the results of the porous simulation by ensuring the non-porous simulation is behaving accurately. The comparison results show excellent agreement in terms of the periodicity of oscillation. There are some discrepancies in the minimum elevations reached by the oscillating free surface which may be explained by the air jet interacting with the free surface during the inhalation phase of the cycle. Fig. 4b shows the comparison of the chamber air pressures recorded at a position close to the orifice within the OWC chamber. It can be seen that there is very good agreement with the results from the experiment.

0.5ρgH i
The pressures on the front wall have been recorded at positions S1, S2, S3 and S4, as shown in Fig. 1. Points S1 and S3 are located 15 mm from the bottom of the front wall on the front and back face respectively. Points S2 and S4 are located at the SWL on the front and back face of the OWC chamber front wall respectively. Fig. 5 shows the maximum pressures observed at these positions versus the wave number, kh, for the non-porous simulations with wave height, H i = 0.06 m. Also shown on these figures are the maximum pressures recorded by experimental approach in the study by Ning et al. (2016b). The pressures have been normalized by dividing by . The maximum pressures recorded at the bottom edge, external face and internal faces of the front wall are in good agreement at low wave number values. However, as the wave number increases, these pressures deviate. The maximum pressures at the wall internal face, bottom edge decrease, this indicates that the shorter wavelength waves cannot penetrate into the chamber and are reflected at the front wall. This deviation in maximum recorded pressures is also evident at Points S2 and S4. The observed maximum pressures show good agreement with the previous study especially at the still wa-  ter level (Points S2 and S4). Notwithstanding this, the pressure comparisons at the other positions on the wall are also reasonable. The bottom edge of the front wall is a region of high vorticity as the fluid flows into the chamber as shown in Fig. 6. This effect can account for the discrepancies between the numerical and experimental results at Points S1 and S3 especially at low and high wave number conditions. The maximum pressures on the outer face of the front wall are larger than the pressures on the inner face in all cases in the current study.

Numerical results and discussion
In this section, the influence of the porous structure located at the side of the OWC chamber on the model efficiencies is analyzed. The free surface development at the OWC chamber is examined. The results from an investigation into the horizontal wave loading on the OWC chamber front wall are also presented. The simulations are carried out for three different incident wave amplitudes, 0.03 m, 0.06 m and 0.09 m. In each case, the simulations are repeated with 15 varying wave periods in the range of 0.683 s to 9.000 s. This allows a range of wave steepness to be investigated. These wave conditions were selected to simulate scaled physical wave conditions and the previous experimental work (Ning et al., 2016b). All other model geometric parameters are maintained constant throughout the simulations.
3.1 Influence of porous section on free surface elevation Fig. 7 shows the development of the free surface along the flume length adjacent to the OWC structure at varying simulation times. The free surface for wave height, H i = 90 mm, wave period 1.366 s is presented in Fig. 7a whilst the result for wave height, H i = 90 mm, wave period 3.000 s is shown in Fig. 7b. The front face of the front wall of the OWC is located at 20 m from the inlet. The influence of the porous component on the free surface can be observed between 19.75 m and 20 m on the porous simulation plots. In this region the free surface is significantly distorted. It is noticeable that the profile of the waves become slightly irregular as the simulations progress. This is due to interference between the reflected and incident wave. In the later stages of the simulations there is a slight shift in wave phase in the comparison between the porous and non-porous model results. This can be explained by the interaction between the waves and the porous structure and also by the effective shortening of flume length due to the inclusion of the porous section. The free surface profile within the chamber is also shown to be significantly distorted, especially in the re- gion beneath the orifice. This is due to the formation of an air jet as the free surface drops in the chamber and the chamber pressure decreases. This inflow air jet interacts with the water surface resulting in a localized dip in elevation below the orifice location. The effect is notably stronger in the longer period waves as the chamber airvolume oscillation range is larger in this case. This effect is discussed further in the Supplementary Data section. Fig. 8 shows the water free surface profile at the external face of the front wall of the OWC at simulation time 40 s. This plot demonstrates the effect of the porous structure on the wave reflection at the front wall of the OWC. At shorter wave lengths, the porous structure is demonstrated to alter the wave profile. The porous structure damps the incident wave preventing wave reflection. In the porous structure simulation, the incident wave is shown to oscillate with regular periodicity and constant amplitude. In contrast, the free surface profile is oscillating irregularly, and the wave amplitude is variable in the non-porous simulation. The free surface irregularity in the case of the non-porous simulation is caused by wave interference between the incident and reflected waves. At longer wave lengths, the porous and nonporous simulation free surface profiles show good agreement. A larger proportion of the incident wave energy is transferred into the OWC chamber in the case of the longer wave length simulations and less wave energy is reflected at the OWC front wall. Fig. 8 demonstrates that the porous structure is effective in dissipating the incident wave energy at short wave lengths and the bulk of the incident wave energy is transferred into the chamber at longer wave length conditions. Fig. 9 shows the free surface elevation inside the OWC chamber for both the porous and non-porous simulations with wave height, H i = 90 mm and wave periods 1.366 s and 3.000 s respectively. The free surface elevation is recorded at the chamber centre. In the 1.366 s period simulations the median free surface elevation increases by approximately 5 mm inside the chamber. Initially, the water is forced into the chamber by the high frequency pulsating wave loads and gradually the stable internal free surface reaches a higher elevation than the external free surface level. This occurs because the water does not have sufficient time to exit the chamber before the onset of the following waves. This effect is clearly shown in Fig. 9a between 8 s and 16 s simulation time. As the free surface stabilizes to a higher elevation, the internal and external hydrostatic pressure differential regulates the free surface fluctuation and the internal chamber surface elevation oscillates with the same frequency as the incident wave. The maximum surface elevation is slightly higher for the case of the non-porous simulation, and the free surface also drops to a lower elevation in the non-porous simulation case. Then, the free surface oscillation range is larger in the case of the non-porous simulations when a short wavelength is specified. This demonstrates that the porous structure has a notable effect on the chamber free surface flow development at this specific wavelength, which can effectively reduce the strong wave loads.  In the case of the longer wavelength simulations a different flow field behaviour is observed in the OWC chamber as shown in Fig. 9b. In the 3.000 s wave period simulations, the wavelength is such that the water is allowed to flow into and out of the chamber with the same frequency. The influence of the porous structure is not as distinct in this case and the free surface maxima and minima is almost identical for the case of the porous and non-porous simulations. The internal chamber free surface oscillation range is also much larger than the short wave length simulations even though the incident wave amplitude is 0.045 m in both cases. It also means that the porous section outside the chamber has less influence on the long wave transmission into the OWC chamber, which benefits the wave energy capture. Fig. 10 shows the frequency spectra for the free surface oscillation within the OWC chamber. The spectra have been non-dimensionalised by dividing frequencies by the incident wave frequency. The larger, low-frequency energy region in the spectrum in Fig. 10a indicates that the initial free surface elevation increases before the free surface begins to oscillate with the same regularity as the incident wave, marked by the peak frequency. The existence of higher order harmonics is also evident both in the porous and nonporous simulations in each of the incident wave periods studied. These secondary free surface oscillations which produce the higher order harmonics are discussed further in the Supplementary Section. Whilst the existence of the porous structure slightly reduces the observed amplitudes, it is evident that the free surface oscillation frequency is unaltered. In the 3.000 s period simulations, it is also noticeable that the total area under the peak frequency is much larger than that in the 1.366 s period simulations. This confirms that the longer waves have better transmission ability and can penetrate into the OWC chamber more effectively in contrast to the shorter waves. Much of the shorter wave energy is reflected at the OWC chamber front wall. Therefore, the longer waves have better energy harvesting potential.
3.2 Influence of porous component on air pressure in the chamber Fig. 11 shows the time series record of air pressure in the chamber for both porous and non-porous simulations for wave height, H i = 90 mm and two different incident wave periods. The chamber air pressure is measured at positions P1 and P2 located 20 mm from the orifice edge and then averaged. The pressures recorded in the case of the shorter wave length simulations are significantly smaller than those observed in the longer wave length simulations. The nonporous pressures exhibit a larger variation in the shorter wave length simulations. As the wave surges and recedes  Robert MAYON et al. China Ocean Eng., 2022, Vol. 36, No. 1, P. 1-14 from the OWC front wall, the pressures fluctuate accordingly. In the case of the short wave period and porous simulation there is not sufficient time for all the water to percolate out of the porous structure as the wave recedes before the subsequent wave interacts with the front wall. Furthermore, a steadier pressure with a smaller oscillation between the maximum and minimum recorded pressures is observed in the porous structure with short wave length simulations. In contrast, there are stronger dynamic pressure effects observed in the non-porous simulations which contribute to slightly more irregular maxima and minima pressure records. The inefficiency of the shorter waves to penetrate into the OWC chamber may also contribute to this effect.
By analyzing the longer wave length chamber pressure results, a contrasting effect is observed. In this case, the minimum pressures recorded within the chamber show good agreement between the porous and non-porous simulations; however, the porous simulation results show consistantly higher maximum pressures. Again, because some of the water phase is retained within the porous structure, there is a higher pressure head differential between the maximum pressures internally and externally at the OWC chamber, this together with the more effective penetration of the waves into the chamber contributes to the higher pressure observed in the porous simulations. Fig. 12 shows the frequency spectra for the oscillatiory pressure results presented in Fig. 11. The abscissa axis has been non-dimensionalized by dividing the frequencies by the incident wave frequency. The pressures within the chamber oscillate at the same frequency as the incident wave, and there is no discrepancy in the frequency of oscillation between the porous and non-porous simulation results. This indicates that the porous structure does not have any influence on the air pressure oscillation frequency. The existance of higher order pressures oscillations are also obvious in the results as presented in Fig. 12, which are similar to those in Fig. 10. 3.3 Influence of porous component on hydrodynamic efficiency Fig. 13 shows the variation of the hydrodynamic efficiency with and without the porous structure located at the front wall for wave amplitudes 0.045 m, 0.03 m and 0.015 m, respectively. It can be seen that the efficiency at the resonant frequency is slightly lower for those simulations where the porous structure is present. This decrease is very slight and is of the order of 1%−5%. The maximum efficiency discrepancy between the porous and non-porous simulations is the largest in the 0.045 m wave amplitude simulations. A larger proportion of the energy component in these higher amplitude waves may be contained in higher order frequency components. These higher order frequencies have a lesser ability to penetrate the OWC chamber and the wave energy may be reflected or dissipated by the por-  ous structure. The optimal operating frequency bandwidth is generally slightly wider for the case of the non-porous simulations. The resonance frequencies for the non-porous simulations occur at non-dimensional wave numbers 1.61, 1.38 and 1.46 for the 0.045 m, 0.03 m and 0.015 m amplitude waves respectively. These results compare favourably with the resonance frequency observed by Ning et al. (2016b) in their experimental analysis of OWC efficiency with similar non-porous geometry. The resonance frequencies for the porous simulations occur at non-dimensional wave numbers 1.35, 1.3 and 1.51 for the 0.045 m, 0.03 m and 0.015 m amplitude waves, respectively. These small shifts in resonance frequency can be explained due the introduction of the porous structure effectively thickening the OWC chamber front wall. Most significantly, Fig. 13 shows that the inclusion of a porous section at the front face of the OWC chamber structure has a minimal negative influence on the operational efficiency of the WEC.

Influence of porous section on wave loading on the chamber front wall
The transient total normal force on the front wall of the OWC is calculated and plotted in Fig. 14 for the case of the 1.366 s and the 3.000 s period simulations with wave height H i = 90 mm. By integrating the normal pressure on the wall over the wetted wall area, the total force on the front wall is computed. The force data presented in Fig. 14 have been normalized by the initial hydrodynamic force. The porous structure located at the front face of the OWC front wall results in a reduction in the dynamic force applied to the front wall. The mean static force on the OWC front wall increases in the case of the shorter wavelength simulations. The shorter wave length has the effect of forcing the water down the numerical wave flume towards the OWC structure. This causes the static water level at the front face of the OWC to increase slightly as the shorter waves cannot efficiently penetrate into the OWC chamber. In the case of the longer wavelengths, the wave period is such that the water level can ebb and flow against the front face of the OWC chamber without an increase in the SWL. By comparing Figs. 14a and 14b it is shown that the porous structure has a greater effect to reduce the forces on the front wall at shorter wave lengths than it does at longer wave lengths. The porous structure dissipates the energy of the high frequency waves more effectively than that of the low frequency waves. This demonstrates that longer waves can infiltrate through the porous structure into the OWC chamber more effectively. Notably, it is obvious from Figs. 14a and 14b that the porous structure has a significant influence on reducing the transient forces on the front wall of the OWC. In the 1.366 s period simulation, the normal force applied to the OWC front wall from the pulsating wave load is reduced by approximately 20% and in the 3.0 s period simulation the force is reduced by about 15.5% by the introduction of the porous structure. Crucially, these reductions in force against the front wall are achieved without a significant reduction in the operational efficiency of the OWC as shown in Fig. 13.
Further analysis was conducted in order to investigate the influence of the introduction of the porous structure at the front wall versus wave steepness. For each amplitude simulation with varying wave period, the averaged minimum normal force applied to the front wall was subtracted from the averaged maximum normal force applied to the front wall. This yielded the averaged force range applied to the OWC front wall for each wavelength. This averaged force range value was then plotted against the wave number. The results are presented in Fig. 15 for three investigated wave heights. It is shown that as the wave number increases, the averaged force range decreases in an exponential manner. This observation holds for both the porous and non-porous simulations. Furthermore, by plotting both the porous and non-porous results in the same figure, it can be seen that the effect of introducing the porous structure yields a reasonably consistent reduction in force range applied to the front wall. This is deduced from the parallelism of the traces for the porous and non-porous simulations. By reducing the range between the maximum and minimum forces applied to the front wall, the peak positive and negative bending moment loadings applied to the front wall can be reduced, enhancing the durability of the structure. Furthermore, smoother operational performance can be achieved by reducing the front wall loading forces thus increasing the OWC efficiency. The reduction in force range can contribute to increasing the serviceability design life of the OWC structure. Finally, to determine the influence of the porous structure on the OWC front wall force range and wave steepness, the porous simulation averaged force results were subtracted from the non-porous simulation results for each of the different wave height data sets. The results are plotted in Fig. 15. It can be seen that the porous structure has a remarkably similar influence on the trend of front wall force reduction for each of the incident wave heights. Initially, as the wave number increases, the influence of the porous structure on the front wall force range decreases. This is signified by a reduction in the value of the difference between the porous and non-porous front wall force. Then, at the resonant wave frequencies, the influence of the porous structure on the force range applied to the front wall reaches the minimum which is indicated by a local minimum at kh = 1.4 for the 90 mm wave height simulation, at 1.54 for the 60 mm wave height simulation and at 1.48 for the 30 mm wave height simulation respectively. These minima values are in good agreement with the resonant wave numbers observed in Fig. 13. The influence of the porous structure on the front wall forces for each set of simulation results then increases until local maxima are reached in Fig. 16, and these local maxima occur at the values of kh corresponding to the second harmonic resonant frequency. Thereafter the influence of the porous structure on the front wall forces decreases again. These results indicate that there is a direct relationship between the influence of the porous structure on the front wall forces and the peak hydrodynamic efficiency resonant frequencies.

Conclusions
In the present work, the performance of a fully integrated OWC-porous breakwater system has been analyzed through a numerical modelling approach. The model has been validated through comparison with previously published studies. A number of test cases with varying wave heights and wave periods have been analyzed both with or without the porous section adhered to the front wall of the OWC. The hydrodynamic efficiencies in the two cases when the porous section is present and when the porous section is absent have been examined. The OWC hydrodynamic performance observed in this study shows good agreement with the results from previously published studies. Significantly, it is shown that the incorporation of the porous region has relatively little effect on the operational efficiency. The efficiency was only reduced by a maximum of 5% by the inclusion of the porous section in the model.
By analyzing the longitudinal free surface development along the numerical flume adjacent to the OWC chamber at various time points, wave reflection effects are observed. The free surface oscillatory elevations in the chamber are relatively unaffected by the inclusion of the porous section. Whilst the normal forces on the OWC front wall are reduced by the porous section, the majority of the wave energy that passes under the porous structure is transmitted into the OWC chamber. The inclusion of the porous section is shown to significantly reduce the forces on the front wall of the OWC. The forces applied to the OWC chamber front wall are reduced by up to 20% by the incorporation of the porous section in the case of the longer wavelengths analyzed. Notwithstanding this, the porous section is also shown to be effective at reducing the forces at the front wall with shorter period waves.

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Supplementary Data
Free surface distortion within the chamber Fig. A1 shows a sectional contour plot for the air phase velocity within the chamber at different time steps. Also indicated in this figure is the free surface interface represented by a white line. The obstructing effect of the turbine machinery at the OWC duct has not been included in these simulations and the duct allows the free flow of air through the opening. It is shown that during the exhalation and inhalation phases of the OWC operational cycle, a strong air jet is formed. This jet has the effect of highly distorting the free surface within the chamber through the formation of point source radiating waves during the inhalation phase. Subsequently, these concentric radiating waves formed be-low the duct propagate outwards from the source position under the turbine opening. Furthermore, localized sloshing at the free surface in the chamber is also observed due to these radiating waves. This has the effect of reducing the operational efficiency of the OWC. The formation of the point source radiating waves at different time steps are shown in Fig. A2. This disturbance of the free surface also contributes to the formation of spray and the development of water vapour within the chamber, which is also detrimental to the OWC durability and may influence the OWC operational efficiency. These radiating waves may also contribute to the higher order harmonics observed in Fig. 9 in the main body of the manuscript.