Design Selection and Geometry in OWC Wave Energy Converters for Performance

energies Article Design Selection and Geometry in OWC Wave Energy Converters for Performance Iván López 1,* , Rodrigo Carballo 1, David Mateo Fouz 1and Gregorio Iglesias 2,3   Citation: López, I.; Carballo, R.; Fouz, D.M.; Iglesias, G. Design Selection and Geometry in OWC Wave Energy Converters for Performance. Energies 2021,14, 1707. https://doi.org/10.3390/en14061707 Academic Editor: Duarte Valério Received: 9 February 2021 Accepted: 16 March 2021 Published: 19 March 2021 Publisher’s Note: MDPI stays neutral with regard to jurisdictional claims in published maps and institutional affiliations. Copyright: © 2021 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (https:// creativecommons.org/licenses/by/ 4.0/). 1Departamento de Enxeñaría Agroforestal, Universidade de Santiago de Compostela, EPSE, Rúa Benigno Ledo s/n, 27002 Lugo, Spain; [email protected] (R.C.); davidmateo.fouz.var[email protected] (D.M.F.) 2MaREI, Environmental Research Institute and School of Engineering, University College Cork, P43 C573 Cork, Ireland; gr[email protected] 3School of Engineering, University of Plymouth, Marine Building, Drake Circus, Plymouth PL4 8AA, UK *Correspondence: [email protected] Abstract: Although oscillating water column (OWC) wave energy converters are arguably one of the most studied technologies, it is not clear which chamber geometry, among all of the available alternatives, would provide the best performance at a site of interest. In this work, a numerical model based on the Navier-Stokes equations for two compressible fluids, using a volume-of-fluid interfacecapturing approach, is implemented to determine the best performing OWC geometry in a case study off the Port of Vigo (NW Spain). Four general shapes of OWC are analyzed: classic, stepped-bottom, U-shaped and L-shaped, and geometrical variants are investigated. In total, 18 chamber geometries are studied, considering the same turbine geometry in all of them. It was found that the U-shaped and L-shaped designs are the most easily tuned to resonate at a period of interest. Of these two, the L-shaped performs better. The best performance is achieved for an L-shaped OWC design with a shallow entrance, a high horizontal chamber duct and a wide vertical duct, for which a maximum capture-width ratio of 71.6% was achieved. Keywords: wave energy; wave energy converter; WEC; oscillating water column; OWC 1. Introduction Oscillating water column (OWC) devices are one of the most studied wave energy conversion technologies so far [ 1 ]. An OWC consist of an empty chamber opened to the sea below the surface, and an air turbine. Wave action causes the oscillation of the water column inside the chamber, alternately compressing and decompressing the air in the upper part of the chamber. The pressure difference created between the interior of the chamber and the atmosphere forces the air to move in and out of the chamber, driving the turbine in the process. Due to the bidirectional characteristics of the flow, a special solution is required to harvest the greatest amount of available pneumatic energy per cycle. The most common solution are the self-rectifying air turbines [ 2 ], although there are other alternatives as the vented OWCs, a novel design that, by using passive rectifying air flow valves in the chamber, works with unidirectional turbines [ 3 ]. Among self-rectifying air turbines, Wells (reaction) turbines [ 4 ] and impulse turbines, both axial [ 5 ], radial [ 6 ], or the novel biradial [7], are at the forefront of the developments. However, not only the turbine design is important in order to maximize the performance of an OWC device, the chamber geometry is also an essential aspect. The chamber must be designed to ensure near-resonant conditions with the incident waves and, at the same time, to avoid energy losses. In this context, the geometry of OWC converters has been widely analyzed in recent literature. A special design of OWC, equipped with an extra front wall that causes the OWC chamber to be connected to the sea through a vertical duct, increasing the natural oscillation frequency of the water column was proposed in [ 8 ]. As a result of this configuration this device is known as U-shaped OWC. Rezanejad et al. [ 9 ] Energies 2021,14, 1707. https://doi.org/10.3390/en14061707 https://www.mdpi.com/journal/energies Energies 2021,14, 1707 2 of 18 proposed a new configuration of the OWC by locating a step in front of the chamber. The step adds a new resonant mechanism that takes place when the water volume above the step acts similarly to a rigid piston with its natural frequency of oscillation being close to the frequency of the incident wave. There exist also L-shaped OWC geometries in which a duct oriented in the direction of wave propagation is used to increase the resonant frequency of the system [ 10 , 11 ]. This concept was also used in the OWC module of the hybrid wave energy converter proposed in [ 12 ] that merges an OWC and an overtopping device. When the OWC is designed to be isolated, the cylindrical shape is one of the most common alternatives given that this solution is not affected by wave direction [ 13 ]. It was proposed both for nearshore devices [ 14 , 15 ] as well as for hybrid wind-wave applications [ 16 ]. Hybrid wave farms combining OWC WECs with other wave energy conversion technologies have also been investigated [ 17 , 18 ]. In addition, many studies investigate the performance improvements that specific geometric modifications introduce in an OWC converter, e.g., variations in the slope of the front and rear walls of the chamber [ 19 ], or the effects of the length and the opening angle of harbor walls (a pair of sidewalls in front of the device) [ 20 ]. Finally, OWC technology lends itself to the installation of multiple chambers, integrated e.g., in floating platforms or deployed along the coast as part of a structure [ 21 , 22 ]. In all the mentioned cases, both on the offshore and the nearshore OWC devices, the optimization of the chamber geometry has to be carried out taking into account the wave climate at the locations of interest, being necessary to characterize the wave conditions that provide the bulk of energy (e.g., [23–25]). Despite the efforts of the last few years to obtain optimized OWC geometries, it is not straightforward to select a design for a new plant at a site of interest. In this work, a bidimensional numerical model is implemented to determine the best performing OWC design for a given site. The numerical model is based on the RANS-VOF numerical wave flume implemented and validated in previous works [ 26 , 27 ] but improved in order to take into account the spring-like effect of the air compressibility, a factor that meaningfully affects the performance of an OWC converter [ 28 ]. The restrictions imposed on the OWC designs in this study are: (i) the design must lend itself to integration into a coastal structure; and (ii) its construction should be easy (avoiding intricate geometries) for containing costs. In this way, the solution achieved would be applicable to a wide range of ports. Four main general shapes of OWC are analyzed: classic, stepped-bottom, U-shaped and L-shaped; and for each of them different geometrical modifications are considered. In total, 18 geometries are studied. The work is illustrated through a case study in the surroundings of the Port of Vigo (NW Spain). 2. Materials and Methods 2.1. Case Study Site The Port of Vigo (Figure 1) is one of the most important ports in the northwest of the Iberian Peninsula. It is located in the Ría de Vigo and constitutes a natural harbor that is sheltered by the Cíes Islands. In this port—as in many others—there is a growing interest in developing sustainable solutions which compensate for the high energy consumption and pollution emissions that take place on its operational areas. In this context, wave energy is one of the most important renewable energy sources at the Ría de Vigo area [ 29 ]. However, due to the natural protection that Cíes Islands provide, most of the port surroundings are wave sheltered areas. One of the most interesting locations is the zone near Cape Estai, just in front of a small island, named Toralla, off the south coast of the Ría, which is identify as Point A in Figure 1. The easting (X) and northing (Y) coordinates of this point are: X= 515,603 m, Y= 4,672,240 m (UTM29N/ETRS89). The reasons for selecting this area are mainly two: (i) it is one of the zones in the port surroundings with more available wave energy as it is exposed both to the northwest and southwest waves, that enter the Ría through the channels between the Cíes Islands and the north and south margins of the Ría; and (ii) an OWC wave energy converter at this location could provide shelter to Toralla Energies 2021,14, 1707 3 of 18 Island, protecting it from the extreme wave events that often produce material damages to the facilities in the island. The annual characterization matrix of the wave climate at Point A, discretized in terms of energy bins (bivariate intervals of spectral significant wave height and energy period), is presented in Figure 2. It can be seen that the sea states which provide the bulk of energy are those with significant wave heights around H m0 = 1 m and energy periods around T e = 10 s. Therefore, an OWC appropriate for being placed at this location should provide its best performance under wave conditions with these characteristics. Energies 2021, 14, x FOR PEER REVIEW 3 of 18 enter the Ría through the channels between the Cíes Islands and the north and south margins of the Ría; and (ii) an OWC wave energy converter at this location could provide shelter to Toralla Island, protecting it from the extreme wave events that often produce material damages to the facilities in the island. The annual characterization matrix of the wave climate at Point A, discretized in terms of energy bins (bivariate intervals of spectral significant wave height and energy period), is presented in Figure 2. It can be seen that the sea states which provide the bulk of energy are those with significant wave heights around H m0 = 1 m and energy periods around T e = 10 s. Therefore, an OWC appropriate for being placed at this location should provide its best performance under wave conditions with these characteristics. Figure 1. Location of the area of study (left) and detail of the Port of Vigo and the point of interest (Point A) (right). Figure 2. Annual characterization matrix of the wave energy resource at the point of interest. Numbers indicate the occurrence in annual hours of the sea states in each energy bin. 2.2. Numerical Modeling In order to select the geometry of the OWC converter best suited for operating under the wave climate in the area of interest, a numerical wave flume—a mirror of the experiFigure 1. Location of the area of study (left) and detail of the Port of Vigo and the point of interest (Point A) (right). Energies 2021, 14, x FOR PEER REVIEW 3 of 18 enter the Ría through the channels between the Cíes Islands and the north and south margins of the Ría; and (ii) an OWC wave energy converter at this location could provide shelter to Toralla Island, protecting it from the extreme wave events that often produce material damages to the facilities in the island. The annual characterization matrix of the wave climate at Point A, discretized in terms of energy bins (bivariate intervals of spectral significant wave height and energy period), is presented in Figure 2. It can be seen that the sea states which provide the bulk of energy are those with significant wave heights around H m0 = 1 m and energy periods around T e = 10 s. Therefore, an OWC appropriate for being placed at this location should provide its best performance under wave conditions with these characteristics. Figure 1. Location of the area of study (left) and detail of the Port of Vigo and the point of interest (Point A) (right). Figure 2. Annual characterization matrix of the wave energy resource at the point of interest. Numbers indicate the occurrence in annual hours of the sea states in each energy bin. 2.2. Numerical Modeling In order to select the geometry of the OWC converter best suited for operating under the wave climate in the area of interest, a numerical wave flume—a mirror of the experiFigure 2. Annual characterization matrix of the wave energy resource at the point of interest. Numbers indicate the occurrence in annual hours of the sea states in each energy bin. 2.2. Numerical Modeling In order to select the geometry of the OWC converter best suited for operating under the wave climate in the area of interest, a numerical wave flume—a mirror of the experimental wave flume at the University of Santiago de Compostela (USC)—was implemented. This approach has been followed in previous works both for the study of OWC wave Energies 2021,14, 1707 4 of 18 energy converters [ 26 , 27 ] and other coastal engineering problems [ 30 ], obtaining excellent results. The numerical model used was OpenFOAM [ 31 ], an open source computational fluid dynamics (CFD) package organized as a series of toolboxes that encompass different numerical solvers, meshing tools and preand post-processing utilities. 2.2.1. Governing Equations In order to take into account the effects of air compressibility—which is known to play a significant role in the performance of prototype OWC devices (e.g., [ 32 ])—the compressibleInterFoam solver was used. It is a pressure-based solver for two compressible, non-isothermal immiscible fluids using a volume-of-fluid interface capturing approach. The governing equations are the mass (1), momentum (2) and energy (3) conservation equations, together with an equation of state (6) or (7): ∇·(ρu)+∂ρ ∂t=0 , (1) ∂ρu ∂t+∇·(ρuu)=−∇p+∇·τ+ρg+F, (2) ∂ρCpTf ∂t+∇·ρCpTfu=∇Ψe∇Tf+ST, (3) where ρ is the fluid density; uis the fluid velocity vector; τ is the viscous stress tensor; gis the gravitational acceleration; pis the pressure; tis the time; Fis the source of momentum due to surface tension; T f is the temperature; C p is the fluid specific heat; Ψe is the fluid thermal conductivity; and STis the source term in the energy equation. The viscous stress tensor is: τ=µe f f h∇u+(∇u)Ti+2 3µe f f (∇·u)I, (4) where Iis the identity tensor and µeff is the effective dynamic viscosity: µe f f =µ+ρνt, (5) where µis the dynamic viscosity of the fluid and νtis the turbulent kinematic viscosity. The equation of state for the air, which is assumed as a perfect gas, is: ρair =p RTf , (6) where Ris the specific gas constant. In the case of the water, a barotropic equation of state is used [33,34], defined as: ρwater =ρwater,0 +ψp, (7) where ρwater,0 is the initial reference water density; and ψ = 1/c 2 is an isothermal compressibility factor, with cbeing the speed of sound in water. For the closure of the RANS equations, the k-omega-SST turbulence model [ 35 ] for compressible flow was adopted. For the capturing of the air–water interface, the volume-offluid (VOF) technique [ 36 ] was used. The VOF technique is a free surface tracking method that makes use of a phase-fraction function ( α ) which represents the fractional volume of a cell that is filled with water (e.g., α = 1 water, α = 0 air and 0 < α < 1 for an interfacial cell). Finally, the waves2Foam toolbox was used for wave generation [ 37 ]. It enables both the generation and (passive) absorption of waves based on the relaxation method. 2.2.2. Scaling Given that the numerical model replicates the wave flume of the USC (with dimensions of 0.95 m high, 0.65 m wide and 20 m long) it is necessary to downscale the OWC designs Energies 2021,14, 1707 5 of 18 in order to fit them in the numerical flume. Following previous works carried out in the flume of the USC [28,38,39], a 1:25 scale ratio was adopted. When dealing with free surface flows—as in the case of a wave energy converter—the Froude number (ratio of inertia to gravity forces) in model and prototype must be maintained [ 40 ]. In addition, in the case of OWC devices, air compressibility effects inside the chamber are significant. Therefore, the ratio of the inertia to air compression forces must also be equal in model and prototype [ 41 ]. The best option for the simultaneous satisfaction of both ratios of forces is to make use of a conveniently shaped air chamber that satisfies the required scaling of the air chamber volume according to [42]: Vm Vp =1 kp δ−1ε2, (8) where the subscripts mand pdenote model and prototype, respectively; Vis the air chamber volume in the absence of waves; kis the polytropic exponent of the full-size turbine, for which a value of k p = 1.2 is assumed [ 42 ]; δ is the water density ratio ( δ = ρm / ρp ); and ε is the scale ratio (in this case ε = 1/25). For satisfying Equation (8) the air chamber was connected to an air reservoir of appropriate volume [ 28 , 42 ]. Details of the calculation of the air reservoir dimensions are presented in Appendix A. 2.2.3. Computational Domain As aforementioned, the computational domain reproduced the wave flume of the USC with the OWC model to be analyzed located at its end (Figure 3). In this case, a bidimensional set-up was considered. Thus, wave crests are assumed to be parallel to the OWC front wall. In order to implement the wave generation, an upstream relaxation zone of length equal to one wavelength was implemented. The air reservoir necessary to correctly consider the air compressibility effects was also reproduced in the numerical model. Energies 2021, 14, x FOR PEER REVIEW 5 of 18 2.2.2. Scaling Given that the numerical model replicates the wave flume of the USC (with dimensions of 0.95 m high, 0.65 m wide and 20 m long) it is necessary to downscale the OWC designs in order to fit them in the numerical flume. Following previous works carried out in the flume of the USC [28,38,39], a 1:25 scale ratio was adopted. When dealing with free surface flows—as in the case of a wave energy converter— the Froude number (ratio of inertia to gravity forces) in model and prototype must be maintained [40]. In addition, in the case of OWC devices, air compressibility effects inside the chamber are significant. Therefore, the ratio of the inertia to air compression forces must also be equal in model and prototype [41]. The best option for the simultaneous satisfaction of both ratios of forces is to make use of a conveniently shaped air chamber that satisfies the required scaling of the air chamber volume according to [42]: 𝑉 𝑉=1 𝑘𝛿𝜀 , (8) where the subscripts m and p denote model and prototype, respectively; V is the air chamber volume in the absence of waves; k is the polytropic exponent of the full-size turbine, for which a value of k p = 1.2 is assumed [42]; δ is the water density ratio (δ = ρ m /ρ p ); and ε is the scale ratio (in this case ε = 1/25). For satisfying Equation (8) the air chamber was connected to an air reservoir of appropriate volume [28,42]. Details of the calculation of the air reservoir dimensions are presented in Appendix A. 2.2.3. Computational Domain As aforementioned, the computational domain reproduced the wave flume of the USC with the OWC model to be analyzed located at its end (Figure 3). In this case, a bidimensional set-up was considered. Thus, wave crests are assumed to be parallel to the OWC front wall. In order to implement the wave generation, an upstream relaxation zone of length equal to one wavelength was implemented. The air reservoir necessary to correctly consider the air compressibility effects was also reproduced in the numerical model. Figure 3. Sketch of the computational domain. The computational domain was discretized by means of the meshing tools blockMesh and snappyHexMesh available in the OpenFOAM package. First, a hexahedral base mesh was created using blockMesh. Second, snappyHexMesh was used to adapt the mesh to the OWC model surface by iteratively refining and morphing the base mesh. Four different mesh regions can be differentiated (Figure 3): (i) propagation region; (ii) relaxation zone region; (iii) transition region; and (iv) model-testing region. The propagation region constitutes a meshing zone parametrized based on the wave conditions, i.e., its discretization changes depending on the wave condition to be simulated. It goes from x = 0 (end of the relaxation zone region) to the start of the transition region. In this section, the mesh is uniform along the x-axis, ensuring 50 cells per wavelength [43], and non-uniform along the z-axis, establishing a refined region in the vicinity of the air–water interface [26,28] that ensures H/Δz ≥ 8 [44]. The relaxation zone and the Figure 3. Sketch of the computational domain. The computational domain was discretized by means of the meshing tools blockMesh and snappyHexMesh available in the OpenFOAM package. First, a hexahedral base mesh was created using blockMesh. Second, snappyHexMesh was used to adapt the mesh to the OWC model surface by iteratively refining and morphing the base mesh. Four different mesh regions can be differentiated (Figure 3): (i) propagation region; (ii) relaxation zone region; (iii) transition region; and (iv) model-testing region. The propagation region constitutes a meshing zone parametrized based on the wave conditions, i.e., its discretization changes depending on the wave condition to be simulated. It goes from x= 0 (end of the relaxation zone region) to the start of the transition region. In this section, the mesh is uniform along the x-axis, ensuring 50 cells per wavelength [ 43 ], and non-uniform along the z-axis, establishing a refined region in the vicinity of the air–water interface [ 26 , 28 ] that ensures H/ ∆ z ≥ 8 [ 44 ]. The relaxation zone and the transition regions are, again, meshing zones parametrized based on the wave conditions. In the case of the relaxation zone region, its length is equal to the wavelength ( λ ) of each test [ 44 , 45 ]. Therefore, it goes from x= −λ to x= 0. The same cell size settings used for the propagation Energies 2021,14, 1707 6 of 18 region were maintained with the exception of a growth rate that increase the cell size along the x-axis towards the left wall of the model, established to help in the dissipation of reflected waves and to reduce the total number of cells. Regarding the transition region, its length is equal to 30% of the wavelength. It goes from the end of the propagation region to x= 10.30 m. The cell size settings in the transition region are the same of those in the propagation region except for a decreasing cell size along the x-axis towards the cell size in the next region. Finally, the model-testing region constitutes a non-parametrized meshing zone. It goes from the end of the transition region to the rear end of the air reservoir. Starting from a regular mesh of cell size 0.04 m, up to six refinement levels (minimum cell size of 6.25 × 10 −4 m) were used to accommodate the shapes of the models. For illustration, the computational mesh for one of the wave conditions tested is presented in Figure 4. Energies 2021, 14, x FOR PEER REVIEW 6 of 18 transition regions are, again, meshing zones parametrized based on the wave conditions. In the case of the relaxation zone region, its length is equal to the wavelength (λ) of each test [44,45]. Therefore, it goes from x = −λ to x = 0. The same cell size settings used for the propagation region were maintained with the exception of a growth rate that increase the cell size along the x-axis towards the left wall of the model, established to help in the dissipation of reflected waves and to reduce the total number of cells. Regarding the transition region, its length is equal to 30% of the wavelength. It goes from the end of the propagation region to x = 10.30 m. The cell size settings in the transition region are the same of those in the propagation region except for a decreasing cell size along the x-axis towards the cell size in the next region. Finally, the model-testing region constitutes a nonparametrized meshing zone. It goes from the end of the transition region to the rear end of the air reservoir. Starting from a regular mesh of cell size 0.04 m, up to six refinement levels (minimum cell size of 6.25 × 10 −4 m) were used to accommodate the shapes of the models. For illustration, the computational mesh for one of the wave conditions tested is presented in Figure 4. Figure 4. General view of the computational mesh (top) and detail of the testing-zone region (bottom) for a wave condition with T = 1.2 s and H = 0.04 m (in model dimensions). 2.2.4. Testing Program In total, 18 different OWC geometries were tested (Figure 5). The selected geometries were based on OWC configurations analyzed in previous studies, reproduced at a 1:25 scale. As mentioned, the design and dimensioning of these geometries follow two main criteria: (i) ease of integration into a coastal structure; and (ii) ease of construction. Model 1 represents a classic OWC design, intended for integration into the breakwater of the nearby Port of A Guarda (NW Spain) [27,46]. Models 2 to 6 are based on the U-shaped OWC [8]. The geometrical variants analyzed for the U-shaped designs encompass the width of the vertical duct, the submergence of the U-duct opening and the width of the chamber. Model 7 follows the stepped-bottom OWC design [9], and Model 8 combines the stepped and L-shaped designs. Finally, Models 9 to 18 are L-shaped geometries [10,11]. In Model 10 the effects on the capture width ratio of a vertical wall added to the design to prevent wave transmission past the entrance of the OWC are analyzed. In Model 11 the influence of the depth of the entrance to the chamber on the performance of the Figure 4. General view of the computational mesh ( top ) and detail of the testing-zone region ( bottom ) for a wave condition with T= 1.2 s and H= 0.04 m (in model dimensions). 2.2.4. Testing Program In total, 18 different OWC geometries were tested (Figure 5). The selected geometries were based on OWC configurations analyzed in previous studies, reproduced at a 1:25 scale. As mentioned, the design and dimensioning of these geometries follow two main criteria: (i) ease of integration into a coastal structure; and (ii) ease of construction. Model 1 represents a classic OWC design, intended for integration into the breakwater of the nearby Port of A Guarda (NW Spain) [ 27 , 46 ]. Models 2 to 6 are based on the U-shaped OWC [ 8 ]. The geometrical variants analyzed for the U-shaped designs encompass the width of the vertical duct, the submergence of the U-duct opening and the width of the chamber. Model 7 follows the stepped-bottom OWC design [ 9 ], and Model 8 combines the stepped and L-shaped designs. Finally, Models 9 to 18 are L-shaped geometries [ 10 , 11 ]. In Model 10 the effects on the capture width ratio of a vertical wall added to the design to prevent wave transmission past the entrance of the OWC are analyzed. In Model 11 the influence of the depth of the entrance to the chamber on the performance of the device is analyzed. Through Models 12, 13 and 14 the influence on the capture width ratio of a sloping bed in front of the model is analyzed. Model 15 evaluates the effects on the efficiency of an L-shaped OWC of the height of the entrance to the chamber. Finally, Models 16, 17 and 18 enable the evaluation of the effects of the width (dimension orthogonal to the wave fronts) Energies 2021,14, 1707 7 of 18 of the vertical duct and the depth of the entrance of the chamber on the performance of the device. For each of the designs studied the length of the water column was adjusted, to the extent possible, to achieve near-resonant conditions at the period of interest (T= 10 s). Energies 2021, 14, x FOR PEER REVIEW 7 of 18 device is analyzed. Through Models 12, 13 and 14 the influence on the capture width ratio of a sloping bed in front of the model is analyzed. Model 15 evaluates the effects on the efficiency of an L-shaped OWC of the height of the entrance to the chamber. Finally, Models 16, 17 and 18 enable the evaluation of the effects of the width (dimension orthogonal to the wave fronts) of the vertical duct and the depth of the entrance of the chamber on the performance of the device. For each of the designs studied the length of the water column was adjusted, to the extent possible, to achieve near-resonant conditions at the period of interest (T = 10 s). Figure 5. Geometries of the OWC designs studied in the numerical model (dimensions in model scale in [mm]). Figure 5. Geometries of the OWC designs studied in the numerical model (dimensions in model scale in [mm]). The eighteen models were tested under six different regular wave conditions resulting from the combination of four wave periods T= 1.2, 1.6, 2.0 and 2.4 s (T= 6, 8, 10 and 12 s, in prototype dimensions) with a wave height of H= 0.04 m (H= 1 m, in prototype Energies 2021,14, 1707 8 of 18 dimensions); plus two wave conditions with wave heights of H= 0.08 and 0.12 m and a wave period of T= 2.0 s (H= 2 and 3 m and T= 10 s, in prototype dimensions). The tests were carried out for a water depth of h= 0.42 m (h= 10.50 m, in prototype dimensions). Consequently, the wavelengths that correspond with the four tested periods, in order from smallest to largest, are λ = 1.96, 2.89, 3.77 and 4.63 m ( λ = 49.06, 72.22, 94.31 and 115.81 m, in prototype dimensions). These wave conditions were selected based on the characterization matrix presented in Section 2.1. The influence of the turbine on the system, i.e., the turbine-induced damping, was modelled by means of an orifice. As the numerical model is bidimensional, a rectangular orifice (slot) was used [ 26 , 47 ]. The flowrate through the slot (provided that turbulent flow is achieved) is proportional to the square-root of the pressure drop. For illustration, the data points of pressure drop versus flow rate for a tested case are presented in Figure 6 together with their corresponding fitted parabola of the form ∆ p/q 2 = const. It can be seen that there is a good agreement. The slot emulates, therefore, the behavior of non-linear air turbines such as the self-rectifying axial impulse or the biradial turbine, which present a quadratic pressure-drop-versus-flowrate relation [ 48 ]. Furthermore, given that the damping provided by these turbines is not significantly influenced by rotational speed changes [ 1 ], the slot appropriately simulates the damping of the turbine at different operating conditions. In this work, a slot of width w s = 1.93 mm was used, a value which provides a pressure-dropversus-flowrate relation that has shown good performance in previous works (e.g., [ 39 ]). By maintaining the slot width constant, the turbine-induced damping does not vary, i.e., it simulates the same the turbine for all the analyzed geometries. Note that better performance figures could be achieved if the turbine-chamber coupling is optimized. 0 0.005 0.01 0.015 0.02 q (m3s m ) 0 50 100 150 200 p (Pa) data parabolic fit Figure 6. Pressure drop ( ∆ p) vs. flow rate (q) data points and corresponding fitted parabola ( ∆ p/q 2 = const) for a wave condition with T= 1.6 s and H= 0.04 m, tested in Model 1 (data in model dimensions). 2.3. Numerical Model Validation In order to validate the numerical model, the experimental data from López et al. [ 46 ] were used. Four wave conditions were considered: (a) H= 0.04 m, T= 1.6 s; (b) H= 0.04 m, T= 2.0 s; (c) H= 0.04 m, T= 2.4 s; and (d) H= 0.08 m, T= 2.0 s. The water depth was set to h= 0.42 m. The damping of the turbine was emulated by means of a slot of width w s = 2.50 mm. A comprehensive description of the OWC model and the experimental set-up can be found in [46]. The time series of pressure drop between the interior of the chamber and the atmosphere and the oscillations of the water column inside the chamber are compared for both models, experimental and numerical, in Figure 7. For clarity, only the first 20 s of simulation are presented. In general, the agreement between both time series is excellent. The numerical model correctly reproduces the period and amplitude of the pressure drop and free surface elevation inside the chamber. Energies 2021,14, 1707 9 of 18 Energies 2021, 14, x FOR PEER REVIEW 9 of 18 numerical model correctly reproduces the period and amplitude of the pressure drop and free surface elevation inside the chamber. Figure 7. Pressure drop (left-hand panels) and free surface elevation (right-hand panels) time series for four different wave conditions: (a) H = 0.04 m, T = 1.6 s; (b) H = 0.04 m, T = 2.0 s; (c) H = 0.04 m, T = 2.4 s; (d) H = 0.08 m, T = 2.0 s (○ laboratory experiment, —numerical model). 2.4. Data Analysis The performance of the different OWC designs was characterized based on the capture-width ratio which is defined as: 𝐶 =𝑃 𝑤𝑃 , (9) where P p is the pneumatic power; w c is the width of the OWC chamber (direction parallel to the wave fronts); and P w is the wave power per meter of wave front. The pneumatic power is calculated as follows: 𝑃=1 𝑡Δ𝑝𝑄𝑑𝑡    , (10) where t max is the time duration of each test; ∆p is the pressure drop between the interior of the OWC chamber and the atmosphere; and Q is the air flow rate through the orifice that emulates the turbine. Both variables, flowrate and pressure drop, were directly measured in the numerical model. Finally, the wave power per meter of wave front is calculated as: 𝑃=1 8𝜌𝑔𝐻𝐶 , (11) Figure 7. Pressure drop (left-hand panels) and free surface elevation (right-hand panels) time series for four different wave conditions: ( a )H= 0.04 m, T= 1.6 s; ( b )H= 0.04 m, T= 2.0 s; ( c )H= 0.04 m, T= 2.4 s; ( d )H= 0.08 m, T= 2.0 s ( # laboratory experiment, —numerical model). 2.4. Data Analysis The performance of the different OWC designs was characterized based on the capturewidth ratio which is defined as: CWR =Pp wcPw, (9) where P p is the pneumatic power; w c is the width of the OWC chamber (direction parallel to the wave fronts); and Pwis the wave power per meter of wave front. The pneumatic power is calculated as follows: Pp=1 tmax Ztmax 0 ∆pQdt , (10) where t max is the time duration of each test; ∆ pis the pressure drop between the interior of the OWC chamber and the atmosphere; and Qis the air flow rate through the orifice that emulates the turbine. Both variables, flowrate and pressure drop, were directly measured in the numerical model. Finally, the wave power per meter of wave front is calculated as: Pw=1 8ρwgH2 iCg, (11) Energies 2021,14, 1707 16 of 18 project administration. D.M.F.: investigation, writing—review and editing. G.I.: writing—review and editing, supervision. All authors have read and agreed to the published version of the manuscript. Funding: This research was funded by the PORTOS project—Ports Towards Energy Self-Sufficiency— reference number EAPA_784/2018, co-financed by the Interreg Atlantic Area Program through the European Regional Development Fund and ‘Axudas para a consolidación e estruturación de unidades de investigación competitivas nas universidades do Sistema Universitario Galego (2020-22)’ with reference number ED341B 2020/25. During this work I. López was supported by a postdoctoral grant of the ‘Programa de Axudas áetapa posdoutoral da Xunta de Galicia’ with reference number ED481D 2019/019. Institutional Review Board Statement: Not applicable. Informed Consent Statement: Not applicable. Data Availability Statement: Not applicable. Conflicts of Interest: The authors declare no conflict of interest. The funders had no role in the design of the study; in the collection, analyses, or interpretation of data; in the writing of the manuscript, or in the decision to publish the results. Appendix A The calculation of the dimensions of the air reservoir for each chamber was made as follows. Given a prototype air chamber of volume V p = ∆ x p×∆ y p×∆ z p , the volume of the chamber at model scale considering perfect geometrical similarity is obtained as: Vm,G=Vp×ε3, (A1) where ε is the scale ratio. However, for correctly representing the spring-like effect of air compressibility, the volume of the chamber at model scale must be: Vm=Vp1 kp δ−1ε2, (A2) where k p is the polytropic exponent of the full-size turbine; and δ is the ratio between the model and prototype water densities ( δ = ρm / ρp ). Thus, the extra volume needed can be calculated as: Vr=Vm−Vm,G. (A3) Given that the numerical model is 2D, the volume of the air reservoir per meter width of converter is obtained as: Vr2D=1 ∆ypεVr−Vc, (A4) where V c is the volume of the connection between chamber and reservoir in the numerical model; and ∆ y p is the width of the prototype air chamber (in the direction perpendicular to wave propagation). Finally, the length ( ∆ x r ) and height ( ∆ z r ) of the air reservoir can be easily estimated to fulfill Vr2D=∆xr×∆zr. Following this procedure, the width of the air reservoir varies linearly with the width of the chamber, enabling 2D modelling. References 1. Falcão, A.F.O.; Henriques, J.C.C. Oscillating-water-column wave energy converters and air turbines: A review. Renew. Energy 2016,85, 1391–1424. [CrossRef] 2. Falcão, A.F.O.; Henriques, J.C.C.; Gato, L.M.C. Self-rectifying air turbines for wave energy conversion: A comparative analysis. Renew. Sustain. Energy Rev. 2018,91, 1231–1241. [CrossRef] 3. Rajapakse, G.; Jayasinghe, S.; Fleming, A. 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