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The current study aims to perform a geometrical investigation of an onshore Oscillating Water Column (OWC) on a large scale. The Constructal Design method is employed, aiming to maximize its available power. The OWC is subjected to two constraints (areas of the chamber and ramp below the chamber); and three degrees of freedom: height/length ratio of the chamber (H_{1}/L_{1}), height/length ratio of the ramp (H_{2}/L_{2}), and submersion of the frontal wall of the chamber (H_{3}). A laminar, unsteady, incompressible, and twophase flow was adopted, solving conservation equations of mass, momentum, and transport of waterair volume fraction using Finite Volume Method (FVM) and Volume of Fluid (VOF) model. The global optimal geometry led to a twice maximized available power 37.3% higher than the best case without the seabed ramp below the chamber and seven times better than the worst case. Concerning the sensibility of geometry, results indicated that the chamber geometry, given by ratio H_{1}/L_{1}, over the available power (P) was strongly affected by the ramp ratio H_{2}/L_{2}. Moreover, the behavior of the effect of H_{2}/L_{2} over the once maximized available power (P_{m}) and corresponding optimal shape of the chamber, (H_{1}/L_{1})_{o}, changed dramatically for two different magnitudes of H_{3} investigated.
available power, Constructal Design, numerical simulation, OWC, wave energy
Ocean waves have been considered an important study object in the renewable energy field [14]. It is estimated that their potential achieves 2 TW near the continental areas [5]. Because of that, studies related to Ocean Wave Energy Converters (OWEC) have gotten space in literature, and several main operational principles have been investigated [611].
One of the most promising operational principles is the Oscillating Water Column (OWC). The device has simple technology, and it is in an advanced stage to use on a commercial scale. The main operational principle harnesses the oscillating seawater movement due to the passage of waves to push/pull airflow outside/inside a hydropneumatic chamber to convert pneumatic power into electricity. Several important prototypes have been developed as those builtin Tofteshallen, Norway (500 kW); Sakata, Japan (60 kW); Pico, Portugal (400 kW); Islay island, Scotland (500 kW), Mutriku, Spain (296 kW), Lewis island, Scotland (4.0 MW) [7, 1217]. It is worth mentioning that several detailed information about building largescale prototypes, some policies, and important observations about the design can be seen in Refs. [1217]. Even though the OWC seems to be very promising, it still is in the development stage. One important and desired contribution is the reduction of capital costs of the converters regarding the power generated. Therefore, the knowledge about the influence of design over the device performance is also an important subject to improve the power generated in the devices.
In this sense, several efforts have been performed into experimental and numerical frameworks to improve the physical comprehension of the device's flow and improve the device power takeoff (PTO). Into the experimental framework, Dizadji and Sajadian [18] investigated the geometry of the chamber of OWC inserted in a channel of 16.0 × 0.7 × 0.5 m of dimensions, i.e., a laboratory scale. Two different configurations, with an inclined and standard frontal plate of the chamber compared with the main flow direction, were investigated. The authors observed that the device efficiency is highly dependent on the chamber geometry and properties of the incoming water wave.
Mahnamfar and Altunkaynak [19] carried out twenty experimental sets in a wave tank with a length of 15 m. They evaluated fivewave series conditions and four entrance areas of the chambers to maximize the differential pressure in the system. This study found an optimal geometric configuration for all wave series conditions with an opening height of 0.51 m. The best wave series conditions had a height of 23.37 cm, a period of 1.42 s, and a length of 280 cm. Although it was found an optimal case, it is essential to mention that other studied cases seem to be affected by wave series conditions.
Viviano et al. [20] conducted experiments in largescale OWC devices (nearly 1:5 – 1:9) in Hannover, Germany. More precisely, in the Grosse Wellenkanal (GWK) facility. The influence of two parameters was investigated over the reflection and loading of an OWC: the water depth and the orifice that mimics the Power TakeOff (PTO). The authors concluded that the OWC structure could reduce the wave reflection, and it can be integrated into vertical wall breakwaters, replacing other low reflection alternatives. Moreover, it was noticed that the installation of OWC also could be interesting in low energy seas.
Later, Mahnamfar and Altunkaynak [21] compared a numerical OWC model with a physical experimental one concerning the power improvement due to the chamber's geometric aspects (length, width, and the angle of the frontal plate). The comparison between numerical and physical models has considered the NashSutcliffe coefficient of efficiency (NSE) and showed a good agreement between the methodologies. The NSE is a coefficient that uses the variance of the numerical timeseries divided by the experimental timeseries variance to estimate the accuracy of modeled results [22]. The numerical modeling solved continuity, momentum, and VOF model equations carried out by FLOW 3D software.
The geometric study observed that the chamber front wall inclination also led to a significant gain of power. Dimakopoulos et al. [23] investigated air compressibility effects due to the wave interaction with an OWC. It was employed a simplified model that considers a thermodynamic equation for the air phase and potential flow for the water one. The methodology was validated using largescale experimental data, where compressibility was evident. The methodology was employed to show the importance of compressible effects in prototype scale devices. More recently, Çelik and Altunkaynak [24] investigated the influence of the underwater opening height of the chamber, PTO damping, and wave steepness on the efficiency of conversion experimentally. Results indicated that the optimal damping depends not only on wave characteristics but also on the chamber opening height since it affects the water column surface's behavior in the device.
The numerical modeling has been employed to achieve several relevant recommendations about the design of OWC devices. For instance, Conde and Gato [25] performed numerical simulations of the airflow inside an OWC chamber equipped with two verticalaxis turbines. A horizontal baffleplate was proposed to deflect the air from the turbines, finding that its usage reduces the risk of waterspray towards the turbine. Moreover, this proposed configuration does not seem to disturb the airflow outwards in the turbine. The study also argues that waterspray may be originated because of high tangential velocities that are likely to occur at the airwater interface.
Teixeira et al. [8] proposed an improvement of an OWC using a developed code (Fluinco), which was verified with Fluent code results. This study simulates a large scale device operational principle inside a 10 m deep wave channel subjected to 1 m high incident waves with periods from 4 s to 15 s. Optimal values of front wall depth equal to 2.5 m, and the chamber's length equals to 10 m were found in terms of pneumatic power. However, chamber height tests indicate little influence over the performance of the device. Recently, Lisboa et al. [26] adopted the general configuration proposed in Teixeira et al. [8] for the numerical analyzes of an OWC installed in a breakwater on the southern Brazilian coast using Fluent software. This study also adds a seabed ramp before the device. The turbine diameter optimization (TDO) regulated the turbine parameters according to incoming wave characteristics. The main focus here was the turbine's investigation, being achieved an optimal performance by using a turbine with a 2.25 m diameter.
In an innovative numerical study, Elhanafi et al. [27] analyzed the performance of floatingmoored offshore OWC considering 2D and 3D cases under the numerical procedure. The effect of lip drought, chamber length, and PTO damping on power efficiency were analyzed under a set of incident wave periods. The floatingmoored led to a slight improvement in device performance. Bouali and Larbi [28] employed a sequential optimization to maximize the PTO subjected to different geometric configurations and wave conditions. The optimization is sequential, with the optimal configurations memorized at each level of investigation. In the initial step, the first parameter is optimized, considering constant the other parameters. In the second step, the first optimal parameter is considered constant, and the second parameter is optimized. In the third step, the first and second optimal parameters are considered constant, and a third parameter is optimized. This process is repeated until the end of the parameters. Results indicated that the OWC has an optimal operating point that depends on the PTO damping, the thickness, and the height of the front wall and wave conditions.
More recently, KharatiKoopaee and FathiKelestani [29] investigated the effect of wave steepness at various chamber lengths and bottom slopes over the hydrodynamic efficiency of an OWC device. Results indicated that the longer chambers are the best configurations when the device is subjected to waves with low frequency, while the opposite was noticed for waves of high frequency. Results also indicated that for the shoreline OWC, the bottom slope angle change has a negligible effect on the device efficiency away from the resonance condition. High magnitudes of bottom slope angle led to the highest efficiencies of the device at the resonance condition. It is also worth mentioning that, recently, essential works concerned with investigating OWC devices with dual chambers have been investigated [30]. This device has two chambers placed in the direction of the incoming wave. Each chamber has its PTO system.
Despite several significant contributions, the employment of Constructal Design for geometrical evaluation of a largescale onshore OWC device with a seabed ramp below the device chamber has not been investigated in the literature. Observing that many studies applied a numerical approach to evaluate OWEC devices, the present study proposes a numerical investigation to maximize the available power of an Oscillating Water Column (OWC) using the Constructal Design method [3135]. An important observation is that Constructal Design is not an optimization method but a method based on the Constructal Law. This thermodynamic law is a first principle and states that for any finitesize flow system to persist in time (to live), its design must evolve freely along the time to facilitate access to its internal currents [3135]. The ease of access to the current work's internal currents is represented by the maximization of the available power in the OWC device.
It is worth mentioning that the investigation is not restricted to found an optimal configuration but to evaluate the effect of the aspect ratios of the chamber and ramp for two different depths of a frontal plate over the performance indicator. The method also indicates the interaction among different degrees of freedom, i.e., how the variation of one degree of freedom (DOF) affects the sensibility of other DOFs. To the best of the authors' knowledge, this kind of analysis was not previously performed in the literature, mainly for OWEC on a large scale.
In this way, it is investigated chamber geometrical arrange combined with a seabed ramp's insertion below it. Therefore, two constraints and three DOFs are applied at the OWC and seabed ramp system, and an exhaustive search technique tracks optimized geometric configuration. It is also considered an OWC without the lower ramp as a reference case to compare the effect of ramp usage. The incident wave characteristics and dimensions of the wave channel follow the values proposed in Martins et al. [9] and Barbosa et al. [10], which represent the conditions found on the south coast of Brazil [4].
Commercial code Fluent is used for present computational modeling. It solves the governing equations with boundary and initial conditions applying Finite Volume Method (FVM) discretization [36, 37] and Volume of Fluid (VOF) model [38] for the treatment of waterair twophase flow. The VOF considers a volume fraction a (0 ≤ a ≤ 1.0) that determines fluids fraction at each elementary control volume in the domain. Continuity, momentum, and volume fraction equations are used to model the laminar, incompressible, unsteady waterair mixture flow, as given by [38, 39]:
$\frac{\partial \rho}{\partial t}+\nabla \cdot(\rho \vec{V})=0$ (1)
$\frac{\partial(\rho \vec{V})}{\partial t}+\nabla \cdot(\rho \vec{V} \vec{V})=\nabla P+\nabla \cdot \overline{\bar{\tau}}+\rho \vec{g}+\vec{F}$ (2)
$\frac{\partial \alpha}{\partial t}+\nabla \cdot(\alpha \vec{V})=0$ (3)
where, ρ is density (kg/m³); $\vec{v}$ is velocity (m/s); p is pressure (N/m²); $\vec{F}$ are external body forces (N/m³); $\overline{\bar{\tau}}$ the rate of deformation tensor (N/m²); and $\vec{g}$ is gravity.
Fluids properties have to be balanced inside each finite volume of the domain as following equations [38]:
$\rho=\alpha \rho_{\text {water}}+(1\alpha) \rho_{\text {air}}$ (4)
$\mu=\alpha \mu_{\text {water}}+(1\alpha) \mu_{\text {air}}$ (5)
where, µ is viscosity (kg/m·s).
2.1 Wave theory used for wave generation
The mathematical model that describes linear waves yields in the following equations for the velocity field and free surface displacement of the waves [40]:
$u=\frac{\pi H}{T} \frac{\cosh (h+z)}{\sinh (k h)} \cos (k x\sigma t)$ (6)
$w=\frac{\pi H}{T} \frac{\cosh k(h+z)}{\sinh (k h)} \cos (k x\sigma t)$ (7)
$\eta=\frac{H}{2} \cos (k x\sigma t)$ (8)
where, H is the wave height (m); h is wave freesurface elevation (m) that is function of x position (wave propagation direction, in m) and time t (s); s = 2π/T is the wave frequency (1/s); k = 2π/λ is the wavenumber (1/m); T is the wave period (s); λ is the wavelength (m), u is the component velocity in xdirection (m/s), and w is the component velocity in zdirection (m/s), and h is the water depth (m).
2.2 Constructal Design
Constructal Design is a method based on a physical principle and used to improve any flow system with finite dimensions. This physical principle is the Constructal Law, responsible for the design generation of inanimate and animate flow systems. The Constructal Law states that for a flow system to persist in time, i.e., to survive, it must evolve in such a way to provide easier access to internal currents flowing through it. For this reason, this is considered a law for the generation and evolution of design over time [3133].
Basically, for the Constructal Design application in engineering problems, it is necessary to define constraints, a variation of degrees of freedom, and improvement of a performance indicator. A detailed procedure explaining how to apply the Constructal Design method can be found in Dos Santos et al. [34].
Therefore, in the present study, considering a generic domain depicted in Figure 1, the Constructal Design method was employed. The flow system is a 327 m wave channel with an OWC at the end. Besides, a ramp (representing an inclined seabed) is considered under the OWC.
From Figure 1, the present study aims to determine the optimized geometry dimensions of the hydropneumatic chamber (H_{1} and L_{1}) and for the seabed ramp (L_{2} and H_{2}) in terms of maximizing available power inside the turbineduct connected to the OWC chamber. Two values for H_{3} (front face submergence) were tested. Therefore, for the application of the Constructal Design method, the degrees of freedom H_{1}/L_{1 }and H_{2}/L_{2 }were investigated for tenmeterdepth flume (h = 10 m) subjected to regular incident waves series (wave height H = 1 m, wave period T = 7.5 s, and wavelength l = 65.4 m), considering two H_{3} values. Three simultaneous constraints were respected: A_{1} = 80 m² (area of the chamber), A_{2} = 40 m² (area of the ramp beneath the chamber), and H_{4} < (hH_{3}) (prevents that frontal chamber plate of the OWC from intercepting lower ramp).
Figure 1. Computational domain of the analyzed OWC with current constrains (A_{1}, A_{2}, H_{4}) and degrees of freedom (H_{1}/L_{1}, H_{2}/L_{2}, H_{3})
The geometric evaluation process illustrated in Figure 2, combines each chamber configuration (H_{1}/L_{1} = 0.2; 0.4; 0.8; 1.0; 2.0; 3.0; and 5.0) with ramp configurations (H_{2}/L_{2} = 0.05; 0.1; 0.2; 0.4; 0.8; and 1.0) for the two different values of front face submergence (H_{3} = 2.5 m and 5.0 m). This combination led to 112 cases that are numerically simulated and compared to each other characterizing an optimization using an exhaustive search technique. More precisely, the optimization process performed here is shared in two steps for two different magnitudes of submergence height of the frontal wall (H_{3} = 2.5 m and 5.0 m). In the first optimization step, the ratio H_{1}/L_{1} is varied for a fixed magnitude of H_{2}/L_{2}. The highest OWC available power (P) is the once maximized available power (P_{m}), and the corresponding optimal geometry is the once optimized ratio of H_{1}/L_{1}, (H_{1}/L_{1})_{o}. In the second optimization level, the ratio H_{1}/L_{1} is varied again; however, for another ratio, H_{2}/L_{2}, keeping the submergence height H_{3} constant. This process is repeated for several ratios of H_{2}/L_{2}. The highest magnitude of OWC available power among all cases varying H_{1}/L_{1} and H_{2}/L_{2} is the twicemaximized available power (P_{mm}), and the corresponding shapes are the twice optimized ratio of H_{1}/L_{1}, (H_{1}/L_{1})_{oo}, and the once optimized ratio of H_{2}/L_{2}, (H_{2}/L_{2})_{o}. The same optimization process is repeated two times for the two different submergence heights of the frontal wall (H_{3}).
Figure 2. Flowchart of performed simulations among which OWC optimal design is determined
As mentioned above, the performance indicator adopted was the OWC available power (P), which must be maximized. From the numerical results obtained for each case, P is given by [18]:
$P=\left(\Delta p+\frac{\rho v^{2}}{2}\right) \frac{\dot{m}}{\rho}$ (9)
being Δp the air pressure variation at the OWC turbineduct (kg/m^{3}), between gauge 1 and gauge 2 (see Figure 1); $\rho$ the air density (kg/m³); v the air velocity at the OWC turbineduct (m\s); and $\dot{m}$ the air mass flow rate at the OWC turbineduct. It is essential to highlight that no flow restriction representing the turbine pressure drop was considered.
To solve the different cases proposed by Constructal Design, the Finite Volume Method (FVM) was used for the numerical treatment of governing equations (continuity, momentum, and volume fraction) utilizing Fluent commercial code [41], employing VOF model to treat twophase waterair flow. Moreover, the pressurebased solver has been utilized for the simulations, the First Order Upwind scheme has been used for advective terms, and PRESTO (Pressure Staggering Option) has been adopted for spatial discretization of pressure in the momentum equation. Pressurevelocity coupling adopts PISO (Pressure Implicit Split Operator) algorithm, and the waterair interface is treated with the Georeconstruction method. Additionally, subrelaxation factors of 0.3 and 0.7 were applied for conservations equations of mass and momentum, and convergence occurs for mass and momentum equations residues lower than 10^{6}.
The generation of regular waves in the wave channel is possible due to the left border wavemaker (dashanddot black line in Figure 1) that imposes the transient velocity profile corresponding to the analytical Airy wave theory, i.e., a prescribed velocity boundary condition defined by Eqns. (6) and (7). Continuous black lines are noslip boundary conditions, and dashed black lines are boundary conditions of atmospheric pressure. These last two boundary conditions apply a normal derivative of αn equal to zero for the VOF equation. The initial condition (t = 0 s) takes fluids at rest, being the water depth of h = 10 m.
Results were obtained by taking into account mass flow rate $(\dot{m})$ through the duct of a turbine with a length of 2.0 m next to the OWC chamber, which hydraulic diameter is 0.6 m, and the pressure variation (Δp) between gauge 1 and gauge 2. Water freesurface displacement is monitored at x = 50 m from the wavemaker (see Figure 1).
About the discretization of the computational domains, this study adopts finite rectangular volumes. Figure 3 shows the entire computational domain and also a detailed section around the OWC chamber. It is noticed that refinements are applied near the walls (surfaces with noslip and impermeability boundary conditions), at the waterair interface, and at the outward of the turbine duct (where air mass flow rate is monitored). This study follows the bibliography recommendation concerning finite volume sizes for the current simulation type [8].
Figure 3. Spatial discretization applied to the computational domain
3.1 Timestep convergence test and computational model verification
Due to the unsteady behavior of the problem in the study, a timestep convergence test was performed. More precisely, the water level (η) along the time was monitored for different timesteps. Four timestep values were tested, as indicated in Table 1. Found results are also presented in Table 1, which points out that $\Delta t$ = 2.0 ´ 10^{2} s is an appropriate timestep that manages result accuracy and computational effort control. The instantaneous relative difference between two consecutive simulations (Δt = 1.0 ´ 10^{2} s and Δt = 2.0 ´ 10^{2} s) in timestep convergence test achieves 1% for the instant of time t = 15 s.
Table 1. Freesurface elevation (η) at time t = 15.0 s for the considered timesteps.
Δt (s) 
η (m) 
Relative difference (%) 
5.0 ´ 10^{3} 
10.50779 
 
1.0 ´ 10^{2} 
10.50777 
1.9 ´ 10^{4} 
2.0 ´ 10^{2} 
10.50770 
6.6 ´ 10^{4} 
4.0 ´ 10^{2} 
10.49640 
1.1 ´ 10^{1} 
Figure 4a plots freesurface displacement of the analyzed cases with different timestep sizes, being possible to observe the good agreement among solutions except for the one that applies Δt = 4.0 ´ 10^{2} s. In sequence, employing Δt = 2.0 ´ 10^{2} s, the computational model verification was performed comparing in Figure 4b the free surface elevation of the water obtained with the present numerical code and the one reached with the analytical solution given by Eq. (8). One can note a good agreement between numerical and analytical results, with a minor relative difference of 5% after the numerical wave stabilization at t =15 s.
(a) Timestep convergence test (b) Computational model verification
Figure 4. Results for wave flow in a channel
Figure 5. Comparison of velocity profile in xdirection below the wave crest and trough obtained with the present numerical method and analytical solution [40]
In addition to comparing instantaneous behavior of free surface elevation, Figure 5 shows two velocity profiles in xdirection monitored below the wave crest and trough. The monitoring line is placed at x = 50 m from the wavemaker of the channel. The results obtained with the present code are compared with the analytical solution of Dean and Dalrymple [40] for a potential flow. In general, the numerical results are in close agreement with the analytical ones, with a difference lower than 5.0 %, regardless of the profile below the crest or trough. In the lower region of the channel (z ≤ 0.5 m), the differences found are concerned with the imposition of noslip and impermeability boundary conditions in the numerical solution, which is not considered for the analytical solution since it considers a potential flow. In the region of the waterair interface (8.5 m ≤ z ≤ 9.5 m for the trough and 9.5 m ≤ z ≤ 10.5 m for the crest), the differences found can be related to the density gradient which varies in the present numerical solution from water (where α = 1.0) to air (where α = 0). To comply with the mass conservation, the velocity in the interface must vary to avoid discontinuity between the air and water phases. Therefore, the differences found are related to the simplifications employed for the nearwall and waterair interface regions' potential flow. These simplifications are relaxed in the numerical solution. In the other regions, the velocity profiles are in close agreement showing that the present computational method can be adopted to represent the wave flow simulated in this work.
To validate the present numerical method for predicting of velocity and pressure fields into the OWC chamber, it was reproduced a case study investigated numerically and experimentally by Liu et al. [42]. It is worth mentioning that this comparison was previously performed in the work of Gomes et al. [35] with the same numerical code. Consequently, the same results obtained by Gomes et al. [35] are achieved here.
More precisely, it is simulated the wave flow in a channel with length and height of 700.0 m and 30.0 m, respectively. The water column’s depth is 16.0 m, and the generated wave has a height of 2.0 m and nine different periods (in the range 3.5 s ≤ T ≤ 8.0 s). At the end of the channel, it is placed an OWC device with a length of 6.0 m, frontal plate length of 1.0 m, and a height of 5.6 m. In the chamber's upper surface, a chimney is placed with a diameter of 2.5 m and a length of 6.5 m. It is not used for any obstacle or turbine in the chimney region. The averaged magnitudes of positive and negative amplitudes of the air pressure and velocity variations in the chamber were monitored for different wave periods. The results were compared with those obtained numerically and experimentally by Liu et al. [42]. The numerical results of Liu et al. [42] were simulated using the ReynoldsAveraged Navier Stokes (RANS) approach with the k – ε model for the closure of timeaveraged equations. In the present approach, the fluid flow is considered laminar and incompressible.
The comparison between the velocity and pressure fields obtained with the present code and those predicted by Liu et al. [42] are presented in Figures 6(a) and 6(b), respectively. For the velocity field, Figure 6(a), results indicated that the twodimensional cases slightly overestimate the magnitudes of velocities in the chamber in the range 5.5 s ≤ T ≤ 7.0 s. Moreover, the velocities predicted here have intermediate magnitudes between the 2D and 3D results of Liu et al. [42]. For pressure fields, Figure 6(b), results obtained here are in good agreement with experimental and 3D numerical results of Liu et al. [33]. The 2D results of Liu et al. [42] led to an overestimation of pressure fields, similar to that of velocity fields. Despite some differences reached between the results of the present method and those predicted by Liu et al. [42], in general, the results are in close agreement, indicating the validity of the present method for the flow conditions studied in the present work. This is an important observation since considering the threedimensional domain, turbulent and incompressible flows would significantly enlarge the computational effort, making the geometrical evaluation performed in the present study unfeasible.
(a) Velocity magnitudes
(b) Pressure magnitudes
Figure 6. Comparison between the present and Liu et al. [42] results in an OWC for various wave periods
Initially, it was numerically analyzed the performance of the reference case in which there is no ramp below the OWC chamber, being A_{2} = H_{2}/L_{2} = 0 (see Figure 1). It was evaluated the H_{1}/L_{1} effect over the available power of the device with H_{3} = 2.5 m. Results are shown for the mass flow rate (Figure 7a) and pressure variation (Figure 7b) of air along time. Figure 7 presents three values of the H_{1}/L_{1} ratio, representing the geometric extremes (H_{1}/L_{1} = 0.2 and 5.0) and the geometric configuration that achieved the maximum available power ((H_{1}/L_{1})_{o} = 0.4).
From Figure 7a, a general symmetric oscillating behavior can be identified in mass flow rate for three H_{1}/L_{1} presented conditions with straight convergence in time, i.e., the results are in phase. It is also observed that the H_{1}/L_{1} = 5.0 case achieved the worse results once the mass flow rate is integrally inferior to two other cases, for example, almost two times less than the best case at instant t = 57.5 s. In this way, it is noticed that H_{1}/L_{1} = 0.2 and 0.4 presented similar comportment. However, H_{1}/L_{1} = 0.4 has a specific performance superiority. Therefore, results indicate that higher chambers hinder that the airflow achieves the turbineduct coupled to the hydropneumatic chamber.
(a) Mass flow rate
(b) Pressure variation
Figure 7. Transient results of the reference case (H_{2}/L_{2} = 0) with H_{1}/L_{1} = 0.2, 0.4, and 5.0 for H_{3} = 2.5 m
Regarding the pressure variation analysis, Figure 7b shows an asymmetric oscillating behavior with more significant magnitudes when the mass flow rate is positive (flowing in the positive xdirection, outward the chamber). That probably occurs because the airflow meets atmospheric pressure when flowing in the positive xdirection (outward), which offers less resistance; since the airflow in the negative xdirection (flow back) is submitted to the more elevated internal chamber pressure. Besides, Figure 7 indicates that the mass flow rate and pressure variation seem to present a similar response to geometry changes. It is observed that geometric configuration with (H_{1}/L_{1})_{o} = 0.4 also offers more considerable pressure variation inside the duct after the first incident wave (after t = 47 s).
Figure 8a presents RMS available power (P_{RMS}) concerning time for the same cases of Figure 7, still considering the no ramp reference case. Hence, the behavior follows the pressure variation aspect above mentioned (see Figure 7b), with higher peaks according to outward mass flow rate, as well as geometry (H_{1}/L_{1})_{o} = 0.4 achieving the best performance while H_{1}/L_{1} = 0.5 worsened the performance of the OWC converter.
In its turn, Figure 8b plots average available power of all studied H_{1}/L_{1} ratios (0.2; 0.4; 0.8; 1.0; 1.5; 2.0; 3.0; and 5.0). One can note that (H_{1}/L_{1})_{o} = 0.4 achieved the best performance, generating a maximized available power of P_{m} = 240.88 W; while geometry H_{1}/L_{1} = 0.2 achieved P = 206.13 W (17% difference) and geometry H_{1}/L_{1} = 5.0 got P = 31.34 W being the worst performance.
(a) Transient RMS available power
(b) Timeaveraged RMS available power
Figure 8. Results of the reference case (H_{2}/L_{2} = 0) for different ratios of H_{1}/L_{1} and H_{3} = 2.5 m
To evaluate the effect of setting a ramp below the OWC chamber, Figures 9a and 9b compare mass flow rate and pressure variation between reference case (without ramp, A_{2} = H_{2}/L_{2} = 0) and Figure 1 described case (with A_{2} = 40 m² and H_{2}/L_{2} = 0.8). The other degrees of freedom adopted for this comparison are H_{1}/L_{1} = 0.2 and H_{3} = 2.5 m.
From Figure 9, it is possible to observe that ramp usage promotes a magnitude increase in both operational parameters (m ̇ and ∆p). Besides, general graphs aspects are very similar to those presented in Figure 7, where higher magnitudes were found at outward airflow instants. After that, the ramp's effect over the converter RMS available power along time can be viewed in Figure 10. Figure 10 indicates an improvement of the RMS available power due to ramp usage. Moreover, more significant peaks are identified at outward airflow instants as well.
Based on these preliminary results, it is relevant to investigate the influence of OWC chamber geometry (H_{1}/L_{1}) over its average available power when associated with different seabed ramp configurations (H_{2}/L_{2}). These results are plotted in Figure 11, considering H_{3} = 2.5 m. The reference case results, earlier presented in Figure 8b, are also plotted in Figure 11, aiming to help the discussion.
One can infer in Figure 11 that the maximum values for the OWC available power were always reached for a specific value of H_{1}/L_{1} ratio, independently of H_{2}/L_{2}. Hence, its once optimized value is (H_{1}/L_{1})_{o} = 0.4, being the same configuration that optimizes the reference case. The H_{2}/L_{2} effect analysis of Figure 11 shows that ramp insertion benefits the device with more meaningful average power for all H_{1}/L_{1} cases compared with the reference case. Additionally, power increment for H_{2}/L_{2} > 0.2 occurs significantly around optimal geometry, i.e., (H_{1}/L_{1})_{o} ≈ 0.4. It is also noticed that higher H_{1}/L_{1} values (H_{1}/L_{1} > 1.5) present very close performances regardless of the H_{2}/L_{2} ratio. Therefore, for H_{3} = 2.5 m, the ramp showed more effectiveness in the optimal region of the ratio H_{1}/L_{1}.
(a) Mass flow rate
(b) Pressure variation
Figure 9. Comparison between transient results of reference case and the case with H_{2}/L_{2} = 0.8, considering H_{1}/L_{1} = 0.2 and H_{3} = 2.5 m
Figure 10. Comparison between transient results of the reference case and the case with H_{2}/L_{2} = 0.8, considering H_{1}/L_{1} = 0.2 and H_{3} = 2.5 m for RMS available power
Figure 11. Effect of H_{1}/L_{1} over available power for different H_{2}/L_{2} ratios and H_{3} = 2.5 m
Still looking at Figure 11, in a general way, the increase of the H_{2}/L_{2} ratio promotes an augmentation in OWC available power. Therefore, it is possible to define the twice optimized chamber geometry and the once optimized ramp geometry given, respectively, by (H_{1}/L_{1})_{oo} = 0.4 and (H_{2}/L_{2})_{o} = 0.8; reaching a twice maximized average power of P_{mm}= 331.57 W. If this value is compared with the maximum average power for the reference case (P_{m} = 240.88 W), an improvement of 37.7% was achieved.
Beyond evaluated operational work parameters, it was also presented phase fraction and velocity fields. It was adopted the time instant of 68 s, the optimized OWC chamber geometry of (H_{1}/L_{1})_{o} = 0.4, and the submergence of frontal plate chamber of H_{3} = 2.5 m. In addition, three values for the seabed ramp geometry were considered: H_{2}/L_{2} = 0.05 (Figure 12), H_{2}/L_{2} = 0.2 (Figure 13), and (H_{2}/L_{2})_{oo} = 0.8 (Figure 14).
(a) Phase fraction field (b) Velocity field
Figure 12. Geometric configuration with (H_{1}/L_{1})_{o} = 0.4, H_{2}/L_{2} = 0.05, and H_{3} = 2.5 m for t = 68.0 s
Figures 12a, 13a, and 14a (where red and blue colors represent, respectively, water and air phases) show a wave crest interacting with the optimized OWC configuration, causing the water level elevation inside the hydropneumatic chamber, which forces airflow passage outward the turbineduct. One can observe this airflow in the velocity fields of Figures 12b, 13b, and 14b, respectively. It is also possible to notice that the H_{2}/L_{2} ratio increases promote an increase of the velocity magnitude, showing a consistent behavior with previous analyzes about power performance.
The presented results so far were obtained for H_{3} = 2.5 m. Now, these same analyses were carried out using an OWC frontal wall with H_{3} = 5.0 m, aiming to identify how this geometric parameter influences the fluiddynamic behavior of the converter associated with the seabed ramp. Therefore, Figure 15 plots the H_{1}/L_{1} ratio's effect over average available power for all tested H_{2}/L_{2} ratios, considering H_{3} = 5.0 m.
As depicted in Figure 15, for the reference case (H_{2}/L_{2} = 0), the H_{1}/L_{1} effect over P is similar to the effect over the reference case with H_{3} = 2.5 m (see Figure 11), but here a lower value for the maximized available power is obtained. Therefore, the optimal chamber configuration suffers the influence of the front wall magnitude, being observed that the reference case achieves a maximum power of 145.10 W, i.e., 39.7% lower than the best case with H_{3} = 2.5 m. Moreover, the optimized H_{1}/L_{1} ratio is changed from 0.4 (for H_{3} = 2.5 m) to 1.0 when H_{3} = 5.0 m is applied.
(a) Phase fraction field (b) Velocity field
Figure 13. Geometric configuration with (H_{1}/L_{1})_{o} = 0.4, H_{2}/L_{2} = 0.2, and H_{3} = 2.5 m for t = 68.0 s
(a) Phase fraction field (b) Velocity field
Figure 14. Geometric configuration with (H_{1}/L_{1})_{o} = 0.4, (H_{2}/L_{2})_{oo} = 0.8, and H_{3} = 2.5 m for t = 68.0 s
Figure 15. Effect of H_{1}/L_{1} over available power for different H_{2}/L_{2} ratios and H_{3} = 5.0 m
(a) Phase fraction field (b) Velocity field
Figure 16. Geometric configuration with H_{1}/L_{1} = 0.8, H_{2}/L_{2} = 0.05, and H_{3} = 5.0 m for t = 68.0 s
Figure 15 also indicates that most of the analyzed cases with a ramp obtained worse available power response than the reference case without a ramp. Considering the maximum available power of reference case, one can note that the only configurations that improve the OWC performance due to seabed ramp usage are composed of H_{2}/L_{2} = 0.8 in the range 0.2 £ H_{1}/L_{1} £ 0.8. It is important to mention that the H_{4} < (hH_{3}) constraint (as described in section 2.3) restricted interval values of H_{1}/L_{1} for H_{2}/L_{2} > 0.1. Moreover, it is observed that there is no global H_{1}/L_{1} optimal ratio. However, twice maximized power (P_{m m} = 163.42 W) is achieved for (H_{1}/L_{1})_{oo} = 0.4 and (H_{2}/L_{2})_{o} = 0.8 and, being nearly three times superior to obtained power with the worst geometry. Lastly, the twice maximized power obtained with H_{3} = 5.0 m is 50.7 % lower than the previously obtained twice maximized available power considering H_{3} = 2.5 m (see Figure 11), what means that increasing the vertical dimension of the OWC frontal wall conducted to poor performance for the presented conditions.
To understand the fluiddynamic behavior of the OWC converter having a frontal plate with H_{3} = 5.0 m, the phase fraction and velocity fields were illustrated. Again, the time instant of 68.0 s was adopted. Figure 16 presents the geometric configuration with H_{1}/L_{1} = 0.8 and H_{2}/L_{2} = 0.05; while Figure 17 shows the optimized geometry with (H_{1}/L_{1})_{o} = 0.4 and (H_{2}/L_{2})_{oo} = 0.8.
(a) Phase fraction field (b) Velocity field
Figure 17. Geometric configuration with (H_{1}/L_{1})_{o} = 0.4, (H_{2}/L_{2})_{oo} = 0.8, and H_{3} = 5.0 m for t = 68.0 s
(a) H_{3} = 2.5 m
(b) H_{3} = 5.0 m
Figure 18. Effect of H_{2}/L_{2} ratio over once maximized available power (P_{m}) and once optimized height/length chamber ratio (H_{1}/L_{1})_{o}
The same trend is perceived in Figures 12, 13, and 14, and it can also be observed in Figures 16 and 17. Therefore, in a general way, the results show that longer ramps do not benefit the OWC purpose, causing hydropneumatic effect loss (see Figure 16). On the other hand, when the seabed ramp is smaller than the device chamber, the water level elevations are intensified, and airflow velocity magnitude increases inside the turbineduct (see Figure 17).
Optimal results plotted in Figures 11 and 15 for both front wall submersions (H3 = 2.5 m and H3 = 5.0 m) are summarized in Figures 18a and 18b, respectively. More precisely, Figure 18a depicts the effect of H2/L2 ratio over once maximized power (Pm) and once optimized (H1/L1)o ratio for H3 = 2.5 m, while Fig 16b depicts the same effect for H3 = 5.0 m.
From Figure 18a, it is observed that optimal chamber configuration is not affected by the ramp's configuration, i.e., the optimal H_{1}/L_{1} ratio is constant ((H_{1}/L_{1})_{o }= 0.4) for all tested values of H_{2}/L_{2}. Moreover, as earlier mentioned, the twice maximized average available power of P_{mm}= 331.57 W can be identified with twice optimized OWC chamber geometry of (H_{1}/L_{1})_{oo} = 0.4 and once optimized ramp geometry of (H_{2}/L_{2})_{o} = 0.8 for H_{3} = 2.5 m.
Otherwise, Figure 18b shows that optimal chamber geometry varies significantly due to ramp configuration, achieving maximum local performance for the reference case (H_{2}/L_{2} = 0.0). However, for H_{3} = 5.0 m the maximum global performance of P_{mm}= 163.42 W is also obtained when (H_{1}/L_{1})_{oo} = 0.4 and (H_{2}/L_{2})_{o} = 0.8.
As in the present study, only two values of H_{3} were adopted, it is not possible to indicate an optimized value for this degree of freedom. However, results obtained here indicated that the wall length has a strong influence over the device performance and the effect of other degrees of freedom (H_{1}/L_{1} and H_{2}/L_{2}) over the available power.
Based on the Constructal Design method, the geometric configuration of an onshore OWC with large scale dimensions was evaluated using H_{1}/L_{1} ratio (height/length ratio of the chamber) and H_{3} (Submersion of the frontal wall of the device) associated with the variation of H_{2}/L_{2} ratio (height/length ratio of the seabed ramp). Two constraints were also adopted: the chamber's internal area (A_{1}) and the area of the seabed ramp (A_{2}). As a reference, a case with no seabed ramp (H_{2}/L_{2} = 0) below the OWC was adopted for each H_{3} value. A finitevolume computational model was used to solve the fluiddynamic problem concerning the regular wave incidence over the OWC converter. The results of all proposed geometries were compared, aiming to maximize the available power (P) of the device, characterizing an optimization employing the Exhaustive Search technique.
Among studied geometric configurations, the global optimized geometry is defined by (H_{1}/L_{1})_{oo} = 0.4, (H_{2}/L_{2})_{o} = 0.8, and H_{3} = 2.5 m which reaches P_{mm}= 331.57 W. This bestcase improves 37.3% and 50.7% the OWC performance if compared, respectively, with its reference case (with no ramp) and with the best case for H_{3} = 5.0 m defined by (H_{1}/L_{1})_{oo} = 0.4 and (H_{2}/L_{2})_{o} = 0.8.
It is worth to highlight that H_{1}/L_{1} = 0.4 conducts to superior performances for all values of H_{2}/L_{2} ratio when H_{3} = 2.5 m (see Figures 11 and 18a), as well as for the best case among geometries with H_{3} = 5.0 m (see Figures 15 and 18b).
Concerning to H_{2}/L_{2} effect, in a general way, it is possible to affirm that the employment of a seabed ramp below the OWC converter can improve its performance. This trend is evident for cases having H_{3} = 2.5 m (see Figure 11). However, when H_{3} = 5.0 m, only some geometries presented improvements if compared with its reference case.
Therefore, the influence of OWC geometry over its fluiddynamic behavior is a relevant research subject. Regarding the H_{3} parameter, its variation has a significant effect over the OWC available power and the effect of ratios H_{1}/L_{1} and H_{2}/L_{2} over the available power, deserving further investigation in future work.
R. G. Hübner thanks CNPq for the master sciences scholarship. L. A. O. Rocha (307791/20190), L. A. Isoldi (306012/20170), and E. D. dos Santos (Process 306024/20179) thank CNPq for the research grants. L. A. Isoldi thanks FAPERGS for the financial support (Process 17/255100011112). E. D. dos Santos thanks CNPq for the financial support (Process: 440010/20195). The authors also thank CAPES for financial support. E.D. dos Santos thank CNPq for financial support.
A_{1} 
area of the chamber, m^{2} 

A_{2} 
area of the ramp, m^{2} 

$\vec{F}$ 
external body forces, N/m^{3} 

g 
gravity, m/s^{2} 

h 
water depth, m 

H 
wave height, m 

H_{1} 
height of the chamber, m 

H_{2} 
height of the ramp, m 

H_{3} 
submersion of the frontal wall of the chamber, m 

H_{4} 
distance from the ramp surface to the frontal wall of the chamber, m 

k 
wave number, 1/m 

L_{1} 
length of the chamber, m 

L_{2} 
length of the ramp base, m 

$\dot{m}$ 
mass flow rate, kg/s 

p 
pressure, N/m^{2} 

P 
available power, W 

t 
time, s 

T 
wave period, s 

u 
velocity component in the xdirection, m/s 

$\vec{v}$ 
velocity vector of the flow, m/s 

x 
horizontal direction, m 

w 
velocity component in the zdirection, m/s 

z 
vertical direction, m 

Greek symbols 

a 
volume fraction, dimensionless 

Δt 
timestep, s 

Δp 
pressure variation, N/m^{2} 

h 
freesurface elevation, m 

λ 
wavelength, m 

µ 
fluid dynamic viscosity, kg/(m·s) 

$\rho$ 
fluid density, kg/m^{3} 

$\begin{array}{l}= \\\tau 
deformation rate tensor, N/m^{2} 

$\sigma$ 
wave frequency, 1/s 

Subscripts 

m 
once maximized 

mm 
twice maximized 

o 
once optimized 

oo 
twice optimized 

Abbreviations 


DOF 
Degree of Freedom 

FVM 
Finite Volume Method 

NSE 
NashSutcliffe coefficient of efficiency 

OWC 
Oscillating Water Column 

OWEC 
Ocean Wave Energy Converters 

PTO 
Power Take Off 

RMS 
Root Mean Square 

VOF 
Volume of Fluid 
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