Constructal Design Applied to an Oscillating Water Column Wave Energy Converter Device under Realistic Sea State Conditions

Abstract

In this work, we conducted a numerical analysis of an oscillating water column (OWC) wave energy converter (WEC) device. The main objective of this research was to conduct a geometric evaluation of the device by defining an optimal configuration that maximized its available hydrodynamic power while employing realistic sea data. To achieve this objective, the WaveMIMO methodology was used. This is characterized by the conversion of realistic sea data into time series of the free surface elevation. These time series were processed and transformed into water velocity components, enabling transient velocity data to be used as boundary conditions for the generation of numerical irregular waves in the Fluent 2019 R2 software. Regular waves representative of the sea data were also generated in order to evaluate the hydrodynamic performance of the device in comparison to the realistic irregular waves. For the geometric analysis, the constructal design method was utilized. The hydropneumatic chamber volume and the total volume of the device were adopted as geometric constraints and remained constant. Three degrees of freedom (DOF) were used for this study: $H_1/L$ is the ratio between the height and length of the hydropneumatic chamber, whose values were varied, and $H_2/l$ (ratio between height and length of the turbine duct) and $H_3$ (submergence depth of hydropneumatic chamber) were kept constant. The best performance was observed for the device geometry with $H_1/L=$ 0.1985, which presented an available hydropneumatic power $P_{hyd}$ of 29.63 W. This value was 4.34 times higher than the power generated by the worst geometry performance, which was 6.83 W, obtained with an $H_1/L$ value of 2.2789, and 2.49 times higher than the power obtained by the device with the same dimensions as those from the one on Pico island, which was 11.89 W. When the optimal geometry was subjected to regular waves, a $P_{hyd}$ of 30.50 W was encountered.

1. Introduction

Sustainable energy has become one of the most important and urgent challenges of the 21st century, as the world is facing the threats of climate change, environmental degradation, an increasing demand for energy, and the environmental impacts of fossil fuels [1,2]. To achieve sustainable energy levels, it is necessary to develop and implement renewable sources of energy that can provide clean, reliable, and affordable power for various applications and sectors [3]. Renewable sources of energy, such as solar, wind, geothermal, and wave energy, offer promising alternatives that can meet the energy needs of the present and future generations while reducing greenhouse gas emissions and enhancing energy security [4,5].

Renewable energy technologies are considered the most effective way to minimize CO${}_2$ emissions and have significantly evolved over recent decades, contributing substantially to the global energy mix [6]. In this scenario, there has been continuous growth in the share of renewable energy use in electricity production in Europe, which has gone from 20.1% in 2005 to 34.2% in 2015 [1]. In 2022, while the global matrix had 30% of its energy originating from non-renewable sources, the noteworthy commitment to renewable energy sources in Brazil was evident, as they accounted for 87.9% of the country’s domestic electricity supply, corresponding to electricity generation of 677.1 TWh. Remarkably, photovoltaic solar generation reached 30.1 TWh, which represented a growth of 79.8% over the previous year. However, hydropower generation is still the main renewable source, representing 58% of the domestic electricity supply [7,8].

In this context, wave energy is particularly attractive due to its high availability, predictability, and potential to supply a significant amount of electricity to coastal regions. The total theoretical wave energy potential is said to be 32 PWh/y, but it is heterogeneous and geographically distributed, while technology costs are still very high and hinder deployment [1]. This is a promising option that can harness the abundant and consistent energy from ocean waves, which cover more than 70% of the Earth’s surface [9]. The harnessing of wave energy has the potential to not only diversify the renewable energy matrix but also to provide clean electricity to coastal populations around the world [5]. However, realizing this potential presents multifaceted challenges, necessitating innovative approaches and advanced engineering solutions.

One of the most widely used technologies to convert wave energy into electricity is the oscillating water column (OWC) wave energy converter (WEC) device, which consists of a partially submerged chamber with a bottom opening to the sea and an air turbine at the top [10]. As waves enter and exit the chamber, they cause the water level to rise and descend, creating a pressure difference that drives the airflow through the turbine, generating electricity. OWC devices have several advantages over other types of WECs, such as simplicity, robustness, low maintenance, and environmental compatibility [10].

In this context, Bouali and Larbi [11] discussed the effects of the geometry of an OWC WEC device over its performance by employing progressive waves with constant periods and wavelengths, aiming to maximize the available power in the device. They analyzed the hydrodynamic efficiency of the device when subjected to second-order Stokes waves and reached optimal values with a front wall immersion depth of between 0.38 and 0.44 times the water depth. The width of the hydropneumatic chamber was also evaluated, with optimal values ranging from 0.8 to 1 times the water depth. Jalón et al. [12] used an optimization method to investigate the design of a bottom-fixed OWC device under different time scales—more specifically, a sea state, a season, and a year—to reproduce the intra-annual and interannual climate variabilities at the site. They investigated optimal OWC design parameters combined with an analysis of the performance during the device’s useful lifetime, as well as values of the submergence and rotational speed of the turbine for which the device performance was optimal. Cui et al. [13] analyzed the performance of a hybrid WEC, which combined an OWC and an oscillating buoy (OB) device. They evaluated the wave power extraction capability of the device when subjected to regular and irregular waves generated with the JONSWAP spectrum. The hybrid device was compared to both the OWC and OB cases to investigate the incident wave direction, OWC opening size, OWC opening submergence, distance between OWC and OB, hinge position, and OB radius.

Zhou et al. [14] investigated the hydrodynamic response of a shoreline fixed OWC WEC device under regular waves and various irregular wave conditions with the JONSWAP energy spectrum. The authors found that both pneumatic pressures and hydrodynamic efficiencies were lower under irregular wave conditions, but the maximum instantaneous pneumatic power was approximately 14 times greater than the average values, which indicates that the stability of the power take-off system needs to be taken into consideration in practical applications. Portillo et al. [15] analyzed the spring-like air compressibility effect in both fixed and floating coaxial-duct OWC WEC devices through a non-linear time domain model with regular waves. Their results indicated that air compressibility might significantly affect the OWC performance and that, depending on the wave frequency, air compressibility reduces or increases the energy absorbed by the system.

The performance and efficiency of OWC devices depend largely on their geometries and the characteristics of the incident waves [16]. Therefore, it is essential to optimize the design of OWC devices to maximize their power outputs and minimize their costs. Several studies have conducted investigations regarding the geometry of OWC WEC devices [16,17,18,19], and many have employed the constructal design method for that purpose [20,21]. Ning et al. [18] numerically analyzed the hydrodynamic performance of a fixed U-shaped OWC device, evaluating the effects of geometrical parameters, such as the vertical duct height, vertical duct width, and wall thickness. The authors concluded that the hydrodynamic efficiency and air pressure inside the chamber increase with increases in both the vertical duct height and thickness of the first wall.

Gaspar et al. [19] investigated the performances of two onshore OWC WEC devices with different front and back wall slopes when subjected to regular waves. In terms of the geometry, their first device had vertical front and back walls and their second device possessed wall slopes of 40° in relation to the horizontal plane. Their numerical results indicated that the device with inclined walls was more efficient in comparison with the other geometry. Lima et al. [21] performed a geometrical investigation of an OWC WEC device with one to five chambers by employing the constructal design method and the exhaustive search technique. The authors employed numerical regular waves and did not consider the turbine in their analysis, only the duct without the turbine. Their results indicated that devices with five coupled chambers led to higher magnitudes of hydropneumatic power and that the maximum hydropneumatic power was achieved when the height to length ratio of the hydropneumatic chamber ranged from 0.25 to 0.50.

Since the wave characteristics of installation sites for OWC WECs are very singular, one of the main difficulties is providing good rates of hydrodynamic efficiency for highly variable sea states [16]. Therefore, the wave parameters of a possible installation site must be evaluated so that they may present a high efficiency. In this context, Machado et al. [22] introduced the WaveMIMO methodology, a technique for the numerical generation of irregular waves, which reproduces realistic sea state data for a coastal region of interest. This methodology was employed in the present study to assess the wave climate of a coastal region, aiming to reproduce an OWC WEC device subjected to a realistic sea state. In a similar approach, Mocellin et al. [23] used this methodology to analyze the influence of bathymetry on the generation and propagation of realistic irregular waves and to geometrically optimize an OWC WEC device. The authors, however, focused on evaluating the performance of the device under realistic waves in the coastal region of the municipality of Tramandaí, Brazil.

Due to the random and chaotic nature of sea waves, few studies have made use of irregular waves [14,24,25,26], and even fewer have employed realistic sea data [23,27]. In this lies the objective of the present study, which was to perform numerical simulations using irregular waves, originating from realistic sea data, as a means to analyze the geometry and performance of an OWC WEC device located in the coastal region of Rio Grande, Rio Grande do Sul, Brazil by using the constructal design method. The device was also subjected to regular waves representative of these sea data, as a means to evaluate the hydrodynamic performance of the device in comparison to the realistic irregular waves. It should be highlighted that these sea data were obtained through the Tomawac model by employing the WaveMIMO methodology [22] and were used as boundary conditions in the Fluent software.

This study was structured as follows: Section 2 reports the materials and methods employed in this research and describes the mathematical and numerical models, the WaveMIMO methodology, the numerical model verification, and the constructal design method; Section 3 presents the results obtained, along with a discussion about them; and Section 4 describes the conclusions of this research.

2. Materials and Methods

An impressive number of wave energy technologies have been developed over the last 25 years [27]. Among the various types of WEC devices that convert sea wave energy into electrical energy, whether in the design phase, test phase, or pre-commercial phase, the OWC type device is possibly the most extensively studied, due to its simplicity and conversion potential [10]. An OWC WEC device is basically composed of a partially submerged hollow structure, namely a hydropneumatic chamber, which can be fixed or floating. In addition, the device contains a turbine duct, a turbine, and a generator. The turbines used in OWC WEC devices are usually self-rectifying axial flow air turbines [28,29]. There are three types of air turbines, namely Wells turbines, impulse turbines, and Denniss–Auld turbines, of which the most commonly used are Wells turbines and impulse turbines [10,16]. The hydropneumatic chamber of the device possesses an open bottom below the free water surface, retaining the trapped air inside the chamber. The movement of the waves causes the air to be continuously compressed and decompressed, which causes it to flow through a turbine coupled to a generator [10]. Figure 1 depicts this principle.

Several theories have been developed in an attempt to describe ocean waves, where several factors are taken into account. However, not all theories are applicable to every type of situation, since some assumptions and physical simplifications are made. While the use of regular waves is a widely adopted approach to represent ocean waves, irregular waves represent the sea state in a less idealized and more realistic way [30]. Therefore, in this study, the focus was on the incidence of realistic irregular waves over the OWC WEC device. However, the characteristics of representative regular waves were used as parameters for the spatial discretization of the computational model as well as for a final comparison regarding the OWC WEC behavior. The representative regular waves were achieved through a statistical analysis of spectral data, which is explained in detail in Section 2.2.1. Thus, the linear wave characteristics are shown in Figure 2. Another important characteristic is the wave period (T, in s), which represents the time required for two successive crests (or two successive troughs) to reach a specific point [31].

2.1. Mathematical and Numerical Modeling

In order to reproduce an OWC WEC device subjected to realistic sea states, the commercial computational code ANSYS Fluent version 2019 R2 was employed. This is a type of computational fluid dynamics (CFD) software, based on the finite volume method (FVM), which allows broad modeling of faculties for incompressible and compressible fluids as well as laminar and turbulent flow in complex geometries [32].

For this study, a non-linear multiphase model was employed to tackle the water–air interactions. For the fluids in the model, the governing equations were the continuity and momentum conservation equations, given by [33,34]:

$$\begin{array}{c}\hfill \frac{\partial \rho }{\partial t}+\nabla ·\rho \overrightarrow{v}=0\end{array}$$

(1)

$$\begin{array}{c}\hfill \frac{\partial \left(\rho \overrightarrow{v}\right)}{\partial t}+(\nabla ·\rho \overrightarrow{v})\overrightarrow{v}=-\nabla p+\nabla (\mu \nabla ·\overrightarrow{v})+S\end{array}$$

(2)

in which $\rho$ is the fluid density (kg/m${}^3$), t is the time (s), $\overrightarrow{v}$ is the velocity vector (m/s), p is the pressure (N/m${}^2$), $\mu$ is the absolute viscosity coefficient (kg/m$·$s), and S is the damping sink term.

The multiphase model used in this study employed the volume of fluid (VOF) technique [35], which enables the reproduction of two or more immiscible fluids and the monitoring of the volume fraction ($\alpha$) associated with these fluids within individual computational cells across the domain. Consequently, within a domain comprising water and air, there can be three distinct scenarios based on the volume fraction values: $\alpha$ = 0 denotes a cell devoid of water and filled entirely with air; $\alpha$ = 1 indicates a cell completely filled with water; and any other $\alpha$ value between 0 and 1 means the presence of the interface between the two phases within the cell. The volume fraction for these dual phases across the two-dimensional domain was governed by a transport equation, defined as [34,36]:

$$\frac{\partial \alpha }{\partial t}+\frac{\partial \alpha u}{\partial x}+\frac{\partial \alpha w}{\partial z}=0$$

(3)

The values of the density and absolute viscosity coefficient, both essential fluid properties within the momentum conservation equation (Equation (2)), were established using the following equations [34]:

$$\begin{array}{c}\hfill \rho =\alpha {\rho }_{water}+(1-\alpha ){\rho }_{air}\end{array}$$

(4)

$$\begin{array}{c}\hfill \mu =\alpha {\mu }_{water}+(1-\alpha ){\mu }_{air}\end{array}$$

(5)

Regarding the flow regime, Gomes et al. [20] performed numerical simulations with OWC WEC devices, comparing them with experimental results. In their study, the mathematical and numerical models employed considered laminar flow, and the numerical results showed good agreement with the experimental results. The authors stated that, in the flow of water and air through the hydropneumatic chamber, the predominant factor is the pressure difference, in which case, for the purpose of analyzing the airflow, turbulence can be disregarded. Therefore, only laminar flow was considered in this study.

Due to the potential interference of wave reflections with the surface elevation within the channel, a numerical beach was introduced in the domain. In this scenario, a damping sink term (S) was incorporated into the momentum equation for the cell zone near to the pressure outlet boundary located at the edge of the wave channel [32]:

$$S=-\left[C_1\rho V+\frac{1}{2}{C}_2\rho \left|V\right|V\right]\left(1-\frac{z-z_{fs}}{z_b-z_{fs}}\right){\left(\frac{x-x_s}{x_e-x_s}\right)}^2$$

(6)

in which $C_1$ and $C_2$ are, respectively, the linear (s${}^{-1}$) and quadratic (m${}^{-1}$) damping coefficients, V represents the velocity in the vertical direction (m/s), and z stands for the vertical distance from the free surface level (m). Additionally, $z_{fs}$ denotes the free surface coordinate along the vertical axis (m) and $z_b$ is the bottom coordinate also along the vertical axis (m). Furthermore, x denotes the horizontal coordinate (m), $x_s$ indicates the beginning of the numerical beach (m), and $x_e$ indicates its ending (m), all in the horizontal direction.

As previously mentioned, regular waves representative of the sea state were also generated in order to evaluate the hydrodynamic performance of the device in comparison to realistic irregular waves. Therefore, the horizontal (u) and vertical (w) wave velocity components and the water surface elevation were calculated by [31]:

$$\begin{array}{cc}u=\frac{H}{2}\frac{gk}{\omega }\frac{coshk(h+z)}{cosh\left(kh\right)}\hfill & cos(kx-\omega t)+\frac{3H^2\omega k}{16}\frac{cosh\left[2k\right(h+z\left)\right]}{si{n}^4\left(kh\right)}cos2(kx-\omega t)\hfill \end{array}$$

(7)

$$\begin{array}{cc}w=\frac{H}{2}\frac{gk}{\omega }\frac{sinhk(h+z)}{cosh\left(kh\right)}\hfill & sin(kx-\omega t)+\frac{3H^2\omega k}{16}\frac{sinh\left[2k\right(h+z\left)\right]}{si{n}^4\left(kh\right)}sin2(kx-\omega t)\hfill \end{array}$$

(8)

$$\begin{array}{c}\eta =\frac{H}{2}cos(kx-\omega t)+\frac{H^{2}k}{16}\frac{cosh\left(kh\right)}{sin{h}^3\left(kh\right)}[2+cosh\left(2kh\right)]cos2(kx-\omega t)\hfill \end{array}$$

(9)

in which g is the gravitational acceleration (m/s${}^2$), k represents the wave number (m${}^{-1}$), and $\omega$ is the wave angular frequency (rad/s).

In addition, some numerical parameters were considered for the simulations conducted in this study. The numerical simulations employed a pressure-based solver. To address the challenge of the velocity–pressure linear interdependence, the pressure-implicit with splitting of operators (PISO) coupling scheme was used. Spatial derivatives within the momentum equations were discretized using a first-order upwind scheme, and time discretization was achieved through a first-order implicit formulation. The geo-reconstruct method was employed for the volume fraction calculation, while for the pressure interpolation at the volume faces, we employed the pressure staggering option (PRESTO!) scheme. This scheme is similar to the staggered-grid schemes used with structured meshes [37,38].

Finally, to analyze all the different geometries of the OWC WEC device, the hydropneumatic power ($P_{hyd}$), which depends on the static pressure, air velocity, and cross-sectional area of the duct, was calculated using the equation [17,39]:

$$P_{hyd}=\left(p_{air}+\frac{{\rho }_{air}{v}_{air}^2}{2}\right)\frac{\dot{m}}{{\rho }_{air}}$$

(10)

in which $p_{air}$ is the air static pressure at the turbine duct (N/m${}^2$), $\dot{m}$ is the air mass flow rate inside the turbine duct (kg/s), and $v_{air}$ is the air velocity through the turbine duct (m/s), which, in turn, is obtained as

$$v_{air}=\frac{\dot{m}}{A_{duct}{\rho }_{air}}$$

(11)

in which $A_{duct}$ is the two-dimensional area of the turbine duct (m${}^2$).

2.2. WaveMIMO Methodology

Machado et al. [22] introduced the WaveMIMO methodology, which transforms sea state spectra into free surface elevation series. These data can be subsequently utilized as boundary conditions within a hydrodynamic model (such as Fluent software), allowing us to numerically generate irregular waves.

It is important to highlight that the WaveMIMO methodology allows us to always numerically generate the same sequence of irregular waves, making it useful for studies focused on the geometric configuration analysis of WECs (or other naval, coastal, and oceanic structures). Therefore, the application of the WaveMIMO methodology makes it possible to investigate different geometric configurations of a WEC under the incidence of the same sequence of irregular waves. This aspect promotes a geometric optimization study with a fairer performance comparison. Other approaches, for instance, the JONSWAP spectrum or the Pierson–Moskowitz spectrum, cannot generate the same sequence of irregular waves in different simulations. In addition, the WaveMIMO only needs the water free surface elevation of the sea state, i.e., if the elevation of the free surface is known (obtained in situ by an acoustic Doppler current profiler (ADCP) or a pressure sensor or by means of a numerical simulation (for example, in Tomawac), this sea state can be accurately reproduced with a CFD code. On the other hand, a limitation of the WaveMIMO methodology is that the high-frequency waves are not fully propagated along the wave channel. However, concerning the fluid dynamics behavior of WECs (the focus of the present work) this limitation in reproducing high-frequency waves does not significantly affect the analysis, since low- and medium-frequency waves are the most relevant for wave energy conversion. More details about the advantages, limitations, and applications of the WaveMIMO methodology can be found in Machado et al. [22].

Therefore, in the present work, the water free surface elevation of the sea state of interest was obtained from the Tomawac spectral model version 7.1 (part of the Open Telemac-Mascaret modeling suite) [40]. Tomawac is a scientific software package that models the changes in the power spectrum of waves and the wave agitation in the oceanic domain in both intracontinental seas and coastal zones [41]. To obtain the sea state data used in this study, the WaveMIMO methodology was employed, which corresponds to a sequence of procedures performed to obtain velocity profile data equivalent to realistic irregular waves, and then subsequently apply these in numerical simulations. An illustrative flowchart depicting all the steps necessary to employ the WaveMIMO methodology can be observed in Figure 3.

The first step in applying the methodology is a sea state simulation, which was performed with Tomawac by Oleinik et al. [43]. The computational domain of the Tomawac simulation was the continental shelf extending from the southern region of the state of Rio Grande do Sul at the southern border of Brazil to the northern part of the state of Santa Catarina.

In this simulation, three types of boundary conditions were used: surface, bottom, and oceanic. For the surface wind boundary condition, data from the Reanalysis 1 project from the National Oceanic and Atmospheric Administration (NOAA) were used [44,45]. For the bottom boundary condition, two types of bathymetric data were used: those for the seabed on the continental shelf and those for the deep ocean. The bathymetry of the continental shelf was obtained from nautical charts published by the Directorate of Hydrography and Navigation of the Brazilian Navy [46] and digitized by Cardoso [47], and the bathymetric data for the deep ocean were acquired from the General Bathymetric Chart of the Oceans (GEBCO) [48,49]. The ocean boundary condition was achieved using historical data sourced from the Wave Watch III wave model [45].

For the simulation with Tomawac, a mesh that encompassed the Brazilian coast was employed, totaling approximately 735 km of coastline. The mesh also extended approximately 300 km into the ocean. The shape of the coastline used was obtained from the Global Self-consistent, Hierarchical, High-resolution Geography Database (GSHHG), developed by Wessel and Smith [50].

In the oceanic region, the edges of the elements were approximately 10 km in length, gradually decreasing to 1.3 km on the continental shelf, and down to 250 m on the coastline. The mesh had a total of 165,548 nodes and 323,929 triangular elements.

Lastly, the Tomawac simulation was performed from 1 January to 31 December 2014. Calibration and validation of the Tomawac wave model for this region were performed by Oleinik et al. [43,45].

For the present work, the point corresponding to node 1545 of the mesh of triangular elements was chosen for the analysis (shown in Figure 4), because it is a point located on Cassino beach in the city of Rio Grande, in the state of Rio Grande do Sul, Brazil. This point is located at coordinates 52°17${}^{\prime }$47.25${}^{″}$ W, 32°22${}^{\prime }$30.95${}^{″}$ S, at a distance of 2 km from the coast and with a depth of 9.52 m.

2.2.1. Spectral Data Conversion and Temporal Location

The software used for the numerical simulation of the OWC WEC device does not use spectral data as an input; therefore, conversion of these data was necessary before they were imposed as boundary conditions. For this, wave spectra were converted into a time series of the free surface elevation, which was processed by employing the methodology outlined in previous studies [22,42] and subsequently converted into orbital velocity components. The resulting time series encompassed significant wave heights and mean wave periods for the year 2014, as shown in Figure 5 and Figure 6.

Utilizing the spectral data extracted from Tomawac, it became possible to obtain wave parameters for Cassino Beach for the year 2014. Nonetheless, conducting numerical simulations of an OWC WEC device demands substantial computational resources, requiring the simulation of a relatively brief time span that was incompatible with the volume of available data spanning an entire year. Therefore, it was necessary to carry out a statistical analysis to determine the wave parameters and the time interval that best represented the studied year.

In this context, a bivariate histogram representing the time series of the significant wave height ($H_s$) and mean wave period ($T_m$) for the year 2014 was outlined. This allowed the identification of the most frequent values for these variables, ensuring an accurate portrayal of the sea state.

The histogram presented in Figure 7 shows, for every pair of variables ($H_s$, $T_m$), the number of occurrences of that sea state in the year 2014, where it was possible to identify the consistency of the sea state attributes over the studied time. It can be observed that the predominant significant wave height was 0.66 m and the wave period was 6.30 s, with an occurrence of over 2000 times, which represents 6% of the total for the year 2014.

Although the values for $H_s$ and $T_m$, which represent the wave climate for the coastal region of Rio Grande in the state of Rio Grande do Sul, Brazil, were calculated, it was not yet known when they occurred in the year 2014.

It is worth noting that each spectrum generated by Tomawac encapsulates a 15 min data span. Consequently, numerous variance spectra for the year 2014 were examined in a previous study [49]. The aim was to determine the time interval that most accurately portrayed the wave climate for the year 2014. Subsequently, the spectrum that closely matched $H_s$≈ 0.66 m and $T_m$≈ 6.30 s was selected. Remarkably, this spectrum corresponded to 23 August at 00:00 (Figure 8).

Therefore, this is the time interval that is representative of the sea state for the city of Rio Grande which was adopted in the numerical simulations of the OWC WEC device. As previously mentioned, the time series of the free surface elevation was appropriately processed using a methodology outlined in previous studies [22,42]. This processing resulted in the transformation of the data into the orbital velocity components of water particles. These components were subsequently employed as boundary conditions within the hydrodynamic model.

2.3. Numerical Model Verification

Before the hydrodynamic model could be used to simulate an OWC WEC device with a realistic sea state, it needed to be verified and validated. It should be noted that Maciel et al. [51] accomplished the verification and validation of the WaveMIMO methodology, while Maciel et al. [52] performed a further investigation regarding the aspects involved in applying this methodology to generate numerical waves. However, an additional verification of the generation and propagation of waves for the present study was essential. For this, only the wave channel was numerically simulated and evaluated for 5 min of the simulation. In this case, wave velocity data were imposed as boundary conditions in Fluent, and then the results were compared with the realistic irregular waves obtained from the Tomawac wave model.

The length of the wave channel was determined by calculating the characteristic wavelength of the realistic sea data for the 15 min of data used in the numerical simulations. Machado et al. [22] computed the waves by counting and measuring them using the zero-up crossing method [31] and then obtained their periods and wavelengths by solving the dispersion relation for k, which is given by [30]:

$${\omega }^2=gktanh\left(kh\right)$$

(12)

As recommended in a previous study [20], the length of the wave channel was five times the characteristic wavelength of the realistic sea data for the 15 min period, which is $\lambda$ = 51.6 m. Hence, the computational domain was 258 m long, 12 m high, and 9.52 m deep, which is the same as the water depth of the point at Cassino beach, corresponding to node 1545 (see Figure 4) of the mesh used in the spectral model simulation [22,43,45].

As for the numerical beach, Machado et al. [22] applied the WaveMIMO methodology and compared two lengths for this region of the domain: 2$\lambda$ and 2.5$\lambda$. They obtained better results with the latter length; therefore, 2.5$\lambda$ on the end of the wave channel was assigned to the numerical beach, as shown in Figure 9 as a means to dampen waves and prevent reflection. As for the calibration of the numerical beach, the values for the linear ($C_1$) and quadratic ($C_2$) damping coefficients were adopted as recommended in previous studies [51,53], in which $C_1=25$ s${}^{-1}$ and $C_2=0$ m${}^{-1}$.

It should be noted that, in the WaveMIMO methodology, the prescribed velocity boundary condition is assigned to the line segments (or velocity profile divisions) at the entrance of the channel, which requires the discretization of the number of subdivisions that provide the proper generation of numerical waves. Previous studies have adopted different numbers of line segments [22,51]; however, Maciel et al. [52] performed an investigation of the number of subdivisions that more adequately represent the generation of realistic irregular waves for the wave characteristics adopted in this study, in which the best results were obtained with 10 subdivisions at the entrance of the channel. Therefore, 10 velocity profile divisions were adopted for the boundary condition at the entrance of the wave channel in this study.

An additional aspect recently investigated regarding the generation of realistic irregular waves pertained to the process of spectral data conversion using the WaveMIMO methodology. During this process, the orbital wave velocity was calculated at the center of each line segment and then converted into horizontal and vertical wave velocity components (u and w). Nevertheless, some uncertainty remained regarding whether this location was suitable for accurate wave generation, or if alternative positions (such as the top or bottom of each line segment) could yield improved results [52]. Addressing this, an investigation was conducted to determine the most appropriate location for calculating the wave velocity components alongside each line segment. This analysis revealed that better results for the wave characteristics of the present investigation were achieved by computing the wave velocity components at the bottom of each subdivision, a method that was then adopted here.

Concerning the spatial discretization of the domain, four different meshes were evaluated to ensure that proper wave generation and propagation occurred. In each mesh, mesh elements with different horizontal and vertical sizes were analyzed for the region containing only water, the free surface elevation, and the region containing only air. In the free surface region, a refinement zone was applied along the entire length of the channel, based on the stretched mesh technique [54]. This refinement zone presented a height twice the value of the greatest free surface elevation during the 15 min of realistic sea data, H = $H_s$ = 0.66 m. The dimensions adopted in each mesh are shown in

Table 1. Length, height, and number of mesh elements used in the mesh sensitivity study.

	Lengths of Elements throughout the Channel	Number of Elements in the Region Containing Only Water	Heights of Elements Inside the Free Surface Refinement Zone	Number of Elements in the Region Containing Only Air
Mesh 1	$\lambda$/30	40	H/10	10
Mesh 2	$\lambda$/40	50	H/15	20
Mesh 3	$\lambda$/50	60	H/20	30
Mesh 4	$\lambda$/60	70	H/25	40

. As for the temporal discretization, Liu et al. [55] guaranteed numerical accuracy when $\mathsf{\Delta }t\le T$/50. Therefore, a time step of $\mathsf{\Delta }t$ = 0.05 s was adopted, as also performed in previous works [22,56].

Figure 10 illustrates the free surface elevation at x = 0 m over a simulation period of 5 min and the free surface elevation values derived from spectral data obtained via Tomawac conversion. In Figure 11, the same results are presented, but with a shorter time interval (160 s $\le t\le$ 220 s), highlighting the more pronounced differences in the free surface elevation.

As can be observed in Figure 10 and Figure 11, the free surface elevation at the crests and troughs varies considerably, a behavior that is characteristic of realistic irregular waves. It is also possible to notice that the numerical wave model adopted for this study adequately reproduced the free surface elevation values acquired from Tomawac by employing the WaveMIMO methodology. To assist with the analysis of the numerical model capability to represent the wave behavior, the RMSE values were calculated for each simulation and are presented in

Table 2. RMSE values for the mesh sensitivity study.

Mesh Number	RMSE (m)
1	0.0689
2	0.0687
3	0.0686
4	0.0686

. All analyzed meshes presented similar RMSE values, ranging from 0.0689 to 0.0686. Since meshes 3 and 4 presented the lowest values, mesh 3 was adopted for this study. This verification test ensured proper wave generation and propagation in the computational domain.

2.4. Numerical Simulations and Geometric Optimization

Once all parameters had been configured, numerical simulations were conducted to generate realistic irregular waves, which accurately reproduced the sea state conditions within the channel. Subsequently, the simulation results were compared with the time series data obtained from Tomawac. Following the tests to verify the accuracy and reliability of wave generation and propagation, the OWC WEC device was inserted into the wave channel.

The OWC WEC device used in this study was based on the onshore device located on the island of Pico in the Azores archipelago, Portugal. The report of the Non-Nuclear Energy Programme [57] presents a technical drawing of the OWC device at Pico island. The inner cross-sectional area of the hydropneumatic chamber is 12 m × 12 m at the bottom, the turbine axis is located 10 m from the mean water level, the turbine duct has a diameter of 2.8 m and a length of 11.3 m, and the draught of the front wall is 3.4 m from the mean water level, as shown in Figure 12.

In order to insert the OWC WEC device into the numerical wave channel shown in Figure 9, the geometry of the device was simplified, as a means to make the mesh generation easier, as performed by [49]. In the simplified geometry, the hydropneumatic chamber presented a rectangular shape, with the turbine duct placed vertically and the walls constructed with the minimal thickness. In the turbine duct, the dimensions of 11.3 m × 2.8 m were kept, but the duct was positioned vertically, above the hydropneumatic chamber. The width of the chamber was kept at 12 m, although in the real device, there is a narrowing towards the top, and, lastly, the distance from the mean water level to the entrance of the turbine duct was kept at 10 m. It is important to mention that, alongside this simplification, an adaptation regarding the device was made. The Pico island device is a fixed onshore device, where the rear wall of the hydropneumatic chamber is a solid barrier that completely blocks the passage of waves, and the one adopted for this study was a floating offshore device. The reason for this adaptation was to avoid wave reflection from the device and, thus, to allow a comparison with the input data.

In addition, it should be noted that energy losses due to the power take-off (PTO) system were not considered; hence, the investigation concerning the OWC performance was developed considering its available power. This simplification has been adopted in several studies regarding the OWC WEC device [21,22,23,58,59]. Therefore, considering the turbine was not the original goal of this study. Figure 13 shows the OWC WEC device inserted into the domain, and Figure 14 shows the domain, indicating the location of the device.

As can be seen in Figure 14, the OWC device was positioned at a distance of 1.2 wavelengths from the entrance of the wave channel, which corresponds to one wavelength, plus a mesh transition zone and a refined region before the device, with a length of $\lambda$/5. Regarding the spatial discretization inside the device, a refined discretization was adopted, which was the same as that used for the free surface region, i.e., 20 volumes per wave height. It is also important to mention that there is a region of transition from the wave channel before the device with rectangular volumes to the OWC region with square volumes, which was made using a hybrid mesh of triangular and quadrangular volumes (see Figure 13).

In order to measure physical quantities from the flow, numerical probes (NP) were inserted into the numerical wave channel located within the domain, as indicated in Figure 15. Probes NP1, NP2, and NP3 measured the free surface elevation, with NP1 located at x = 0 m, NP2 located at x = 50 m, and NP3 located at the center of the OWC device, as calculated by the relation x = 1.2$\lambda$ + ($L/$2). Positioned in the middle of the turbine duct, NP4 served as a measurement point for the static pressure and mass flow rate, which are essential variables for calculating the available hydropneumatic power, as indicated in Equation (10).

Upon defining the device geometry, the constructal design method could be applied. This draws its principles from the constructal theory established by Adrian Bejan [60]. This theory posits that the emergence of flow structures commonly observed in nature (such as river basins, lungs, atmospheric circulation, and vascularized tissues, among others) should be regarded as a physical phenomenon. Therefore, the principles of flow configuration evolution can be employed to obtain an optimization of the arrangement in a system, and this method has been widely used in various scientific fields.

In this study, the constructal design method was applied to explore alternative device geometries to analyze how varying the geometry of an OWC WEC device affects its performance under realistic sea state conditions and to estimate the hydrodynamic power available within the study area. The constructal design method requires some parameters to be applied, such as geometric constraints, degrees of freedom, and performance indicators. The procedural steps for implementing this method are outlined in Figure 16.

The degree of freedom $H_1/L$, which is the ratio of the height and length of the hydropneumatic chamber, was varied in this case. Meanwhile, the degrees of freedom $H_2/l$ (ratio of the height and length of the turbine duct) and $H_3$ (submergence depth of hydropneumatic chamber) were kept constant. Figure 17 illustrates these dimensions.

The hydropneumatic chamber volume ($V_{HC}$) and the total volume of the OWC WEC device ($V_T$) are the geometric constraints of the problem and, as mentioned, they were kept constant. The volumes $V_{HC}$ and $V_T$ are described by:

$$\begin{array}{c}{V}_{HC}=H_{1}L{L}_1\end{array}$$

(13)

$$\begin{array}{c}{V}_T=V_{HC}+H_{2}l{L}_1\end{array}$$

(14)

in which the third dimension, $L_1$, was kept constant and possessed a value equal to one, so the problem was regarded as two-dimensional. The equations for the length and height of the hydropneumatic chamber can be derived from Equation (13), which changed accordingly to the degree of freedom $H_1/L$:

$$L={\left[{\displaystyle \frac{{\displaystyle V_{HC}}}{{\displaystyle \left(\frac{H_1}{L}\right)L_1}}}\right]}^{1/2}$$

(15)

$$H_1=L\left(\frac{H_1}{L}\right)$$

(16)

Likewise, using Equation (14), the equations for both the length and height of the turbine duct could be obtained:

$$l={\left[{\displaystyle \frac{{\displaystyle V_T-V_{HC}}}{{\displaystyle \left(\frac{H_2}{l}\right)L_1}}}\right]}^{1/2}$$

(17)

$$H_2=l\left(\frac{H_2}{l}\right)$$

(18)

Nevertheless, the ratio $H_2/l$ was kept constant for all simulations, with $H_2$ being equal to 11.3 m and l being equal to 2.8 m, as previously mentioned. Next, the values for the ratio $H_1/L$ were defined, which allowed the definition of values for both the lengths and heights of the hydropneumatic chambers to be simulated. Therefore, to define the volume restriction for the device chamber, dimensions akin to those of the OWC WEC device on Pico island, Azores, Portugal, were adopted, where the length of the hydropneumatic chamber is 12 m and its height is 13.4 m, as shown in Figure 12. This represents an $H_1/L$ value of 1.1167. For the minimum value of $H_1/L$ investigated, the adopted height of the hydropneumatic chamber was half the height of the turbine duct, which led to L being equal to 28.46 m, $H_1$ being equal to 5.65 m, and $H_1/L$ being equal to 0.1985. For the maximum value of $H_1/L$ considered, the adopted hydropneumatic chamber length was three times the turbine duct length, which led to an L value equal to 8.40 m, an $H_1$ value equal to 19.14 m, and an $H_1/L$ value equal to 2.2789. All values of $H_1/L$ considered in this study are presented in

Table 3. Device geometries used in the numerical simulations.

Case	${\mathit{H}}_1/\mathit{L}$	L (m)	${\mathit{H}}_1$ (m)	l (m)	${\mathit{H}}_2$ (m)	${\mathit{H}}_3$ (m)
1	0.1985	28.46	5.65	2.80	11.30	6.12
2	0.4297	19.34	8.31	2.80	11.30	6.12
3	0.6608	15.60	10.31	2.80	11.30	6.12
4	0.8920	13.43	11.98	2.80	11.30	6.12
5	1.1167	12.00	13.40	2.80	11.30	6.12
6	1.3543	10.90	14.76	2.80	11.30	6.12
7	1.5854	10.07	15.97	2.80	11.30	6.12
8	1.8166	9.41	17.09	2.80	11.30	6.12
9	2.0478	8.86	18.15	2.80	11.30	6.12
10	2.2789	8.40	19.14	2.80	11.30	6.12

.

Throughout this study, the performance metric analyzed was the root mean square (RMS) of the available hydropneumatic power generated by the OWC WEC device, and the objective was to maximize this indicator.

Upon obtaining the optimal geometry for the ratios analyzed, this device configuration was subjected to regular waves with wave parameters representative of that sea state, i.e., H of 0.66 m and T of 6.30 s (see Figure 7 and Figure 8). This analysis was performed in order to evaluate how the available hydropneumatic power changed in comparison to the use of realistic irregular waves.

2.5. Statistical Measures

As previously mentioned, the instantaneous available hydropneumatic power for each OWC WEC geometry was calculated. Thus, to obtain the mean available power values, the root mean square (RMS) was employed, a variable that is commonly used for transient flow problems.

For an instantaneous hydropneumatic power indicated by $P_i\left(t\right)$ varying with time, the equivalent RMS value $P_{RMS}$ over the number of observations n was calculated by the following relation [30,61]:

$$P_{RMS}=\sqrt{{\displaystyle \frac{1}{n}}\sum _{i=1}^n{P}_i^2}$$

(19)

To compare the results of the numerical simulations, the root mean squared error (RMSE) was employed, as given by [62]:

$$\mathrm{RMSE}=\sqrt{{\displaystyle \frac{1}{n}}\sum _{i=1}^n{(S_i-O_i)}^2}$$

(20)

in which $S_i$ stands for the values obtained from the simulations, $O_i$ represents the observation values, which can be either the analytical solution or the experimental results, and n indicates the number of observations available for analysis.

3. Results and Discussion

In this study, discrete wave velocity data obtained using the WaveMIMO methodology were used as boundary conditions in a numerical wave channel that was 258 m long, which is five times the characteristic wavelength. As previously mentioned, 10 subdivisions were adopted for the fractioning of the region at the entrance to the channel. In addition, the wave velocity components were obtained by calculating the values at the bottom of each subdivision during the spectral data conversion using the WaveMIMO methodology.

3.1. Application of the Constructal Design Method

After inserting the OWC WEC device into the numerical wave channel, it was subjected to numerical simulations under the influence of realistic irregular waves, taking into account various device geometries (see Table 3).

3.1.1. Preliminary Analysis

First, a preliminary analysis was conducted between the free surface elevation values obtained by numerical probes positioned at the entrance of the wave channel and the free surface elevation values obtained through the conversion of spectral data from Tomawac utilizing the WaveMIMO methodology. Figure 18 illustrates the free surface elevation at x = 0 m during simulations of 15 min for each geometry corresponding to values of $H_1/L$, and Figure 19 presents the same dataset but with a reduced time interval (500 s $\le t\le$ 600 s), making the differences between the free surface elevations more pronounced.

Notably, a subtle difference between the wave crests and troughs is noticeable, despite all simulations employing identical velocity inlet boundary conditions. This discrepancy can likely be attributed to the wave reflection caused by the walls of the OWC WEC device. Additionally, as each simulated device features a distinct geometry, the reflected waves behaved distinctively in each simulation. The respective RMSE values at x = 0 m, calculated in relation to the free surface elevation derived from the conversion of spectral data from Tomawac, are presented in

Table 4. RMSE values for different values of the degree of freedom $H_1/L$.

${\mathit{H}}_1/\mathit{L}$ (–)	RMSE (m)
0.1985	0.0721
0.4297	0.0725
0.6608	0.0731
0.8920	0.0740
1.1167	0.0739
1.3543	0.0748
1.5854	0.0742
1.8166	0.0752
2.0478	0.0762
2.2789	0.0764

for each $H_1/L$ value analyzed.

It is crucial to emphasize that directly comparing elevation time series from different positions within the wave channel does not yield favorable results. This can be explained by the fact that the wave propagation velocity changes with the wave channel length. Since the time series comprises the sum of numerous sinusoidal components with varying wavelengths, each one propagates with a different velocity, and this causes the shape of the time series to evolve over time [49]. In Figure 20, it is possible to see the time series of the free surface elevation measured at the wave channel entrance overlaid with the elevation at the center of the OWC WEC device (with ratio $H_1/L=1.1167$). The values are visibly displaced with respect to each other due to the propagation of waves along the channel and to the presence of the device. In the context of irregular waves, meaningful comparisons are only feasible when the time series is recorded at the same location.

Merely comparing the RMSE values for different locations in the wave channel was not an effective assessment. Therefore, in order to assess the capability of the hydrodynamic model to replicate the sea state accurately, the free surface elevation spectral density was computed. This calculation was performed by following the same procedure as that used by Oleinik et al. [42]. The spectral density was calculated for three different locations, at x = 0 m, x = 50 m, and x = 67.57 m (center of the OWC device) for $H_1/L$ = 1.1167, as shown in Figure 21.

From observing the results presented in Figure 21, one can notice that for waves with periods between approximately 15 s and 5 s, there is a good agreement between the results. This represents the majority of the waves present within the time series employed. However, for wave periods greater than 15 s and less than 5 s, there is divergence. As previously mentioned, this is a limitation regarding the WaveMIMO methodology, in which high-frequency waves are not fully represented. However, as low- and medium-frequency waves are the most relevant for wave energy conversion, this limitation does not significantly impact the numerical simulation of the fluid-dynamic behavior of the OWC WEC device, as indicated by Machado et al. [22].

3.1.2. Geometric Investigation of the OWC WEC Device

With the spectral density analysis, it was possible to ensure that, despite a drop in the spectral density along the wave channel, the numerical model was able to reproduce the waves encompassed in the time series adopted for this study. In addition to measuring the free surface elevation, data on the static pressure and mass flow rate were also recorded through numerical probes positioned within the numerical wave channel (see Figure 15). Figure 22 displays the static pressure within the turbine duct over 15 min simulation periods for various geometries equivalent to different values of $H_1/L$, while Figure 23 presents the same results but for an interval between 150 s and 250 s, underlining the existing differences. As previously mentioned, the coordinates of the center of the OWC WEC device change according to the device’s geometry.

As can be observed in Figure 22, the results for all $H_1/L$ ratios analyzed exhibited similar behaviors. However, it is possible to observe that there were differences in the phase between the geometries analyzed. This discrepancy is likely attributed to the distance between the front wall and the center of the device, which was shortened as the $H_1/L$ values increased, thus causing the waves to reach the center of the device faster in low $H_1/L$ value geometries.

Comparing all ratios analyzed, the lowest ratio, $H_1/L$ = 0.1985, presented higher pressure values. This can be justified by the fact that, for this ratio, L is larger, causing greater compression and decompression of the air inside the hydropneumatic chamber. It is important to highlight that this fluid dynamic behavior was also observed in previous works, such as Mocellin et al. [23] and Gomes et al. [20]. As shown in Figure 23, the highest static pressure peak occurred during the decompression of the hydropneumatic chamber, at 573.55 s, and corresponded to −35.70 Pa. The highest compression pressure peak was 31.21 Pa and this occurred at 243.85 s.

Figure 24 illustrates the mass flow rate within the turbine duct during simulation periods of 15 min for different geometries corresponding to various values of $H_1/L$. Meanwhile, Figure 25 presents the same dataset, but with a narrower time interval (500 s $\le t\le$ 600 s), highlighting the more pronounced differences in the mass flow rate values. It should be noted that the same difference in phase between the geometries can also be observed when analyzing this variable. As shown in Figure 25, the highest mass flow rate peak occurred during the compression of the hydropneumatic chamber at 571.50 s and corresponded to 10.12 kg/s. The highest decompression mass flow rate peak was −8.16 kg/s and this occurred at 295.40 s.

Figure 26 depicts the instantaneous hydropneumatic available power for each value of $H_1/L$, calculated using Equation (10). It can be observed that the highest peak in the hydropneumatic available power was $P_{hyd}=$ 165.18 W, which occurred during the compression of the hydropneumatic chamber at $t=$ 570.55 s for $H_1/L=$ 0.1985. The highest hydropneumatic available power during decompression also occurred with $H_1/L=$ 0.1985 and was $P_{hyd}=$ 145.22 W at $t=$ 572.80 s. Then, as a means to obtain the mean available power values, the RMS (Equation (19)) was calculated for each geometry corresponding to values of $H_1/L$, as shown in Figure 27 and

Table 5. Available hydropneumatic power values for the OWC WEC device geometries simulated.

Case	H${}_1/$L	L (m)	H${}_1$ (m)	l (m)	H${}_2$ (m)	H${}_3$ (m)	P${}_{\mathbf{hyd}}$ (W)
1	0.1985	28.46	5.65	2.80	11.30	6.12	29.63
2	0.4297	19.34	8.31	2.80	11.30	6.12	21.19
3	0.6608	15.60	10.31	2.80	11.30	6.12	16.79
4	0.8920	13.43	11.98	2.80	11.30	6.12	13.84
5	1.1167	12.00	13.40	2.80	11.30	6.12	11.89
6	1.3543	10.90	14.76	2.80	11.30	6.12	10.42
7	1.5854	10.07	15.97	2.80	11.30	6.12	9.24
8	1.8166	9.41	17.09	2.80	11.30	6.12	8.28
9	2.0478	8.86	18.15	2.80	11.30	6.12	7.26
10	2.2789	8.40	19.14	2.80	11.30	6.12	6.83

.

From Figure 27 and Table 5, one can observe that lower values of $H_1/L$ result in higher available hydropneumatic power values. The best geometry performance, which occurred with geometry $H_1/L=$ 0.1985, presented a hydropneumatic available power of $P_{hyd}=$ 29.63 W. This is 4.34 times higher than the power achieved with the least efficient geometry, which was 6.83 W, corresponding to the degree of freedom $H_1/L=$ 2.2789, and 2.49 times higher than the power obtained by the device with the same dimensions as those of the one on Pico island ($H_1/L=$ 1.1167), $P_{hyd}=$ 11.89 W. As it can be seen in Table 5, the length of the hydropneumatic chamber for the optimal geometry was 28.46 m, which is 0.55$\lambda$. This is in accordance with a relation found in the literature, in which, for an L value approximately equal to $\lambda /$2, higher hydropneumatic power values were obtained with 0.04 $<H_1/L<$ 0.23 [20].

One of the geometries studied in the constructal design assessment conducted by Gomes et al. [20] employed wave parameters akin to those utilized in this study, reproducing a wave channel under the action of regular waves with a wave period of T = 6 s, water depth of h = 10 m, wave height of H = 1 m, and wavelength of L = 48.5 m. However, the device geometries were somewhat different, with submergence depth $H_3$ values ranging from 9 m to 10.25 m, while in this study, $H_3$ was kept constant at 6.12 m. In the mentioned study, $P_{hyd}$ ranged from 15 W to ≈140 W. Perhaps, the hydropneumatic power achieved by the WEC device in this study might be improved by evaluating distinct $H_3$ values. Gomes et al. [59] also performed a similar application of the constructal design methodology by varying $H_1/L$ and employing it to perform a geometric evaluation of an OWC device, for which the authors found an optimal ratio of $H_1/L$ = 0.2152, a value similar to that obtained in the present study.

Mocellin et al. [23] investigated the same $H_1/L$ values and device geometries as the ones used in the present study. The authors, however, focused on employing the WaveMIMO methodology to evaluate the hydrodynamic performance of the OWC WEC device in another possible installation site with a different wave climate. The geometric evaluation presented similar curves, and the optimal geometry was the same in both studies ($H_1/L=$ 0.1985); however, the available hydropneumatic in this study was 16.47% higher.

Lastly, to better visualize the motion of the waves through the OWC WEC device, Figure 28 and Figure 29 display the computational domain solution for certain instances of the simulation and the matching velocity field, respectively. In Figure 28, the water phase is shown in red, while the water phase is shown in blue, and it is possible to see the passage of the waves. At the first moment, the free water surface on the left of the device is at a lower level than on its right. Then, as the waves propagate, the water level on the left of the device becomes lower.

The velocity field in the device can be seen in Figure 29, in which higher velocity magnitudes are shown in red, while lower velocity magnitudes are shown in blue. It is possible to note that greater velocity values occurred near the edges of both the front and rear walls, as well as in the turbine duct. This is due to the fact the incidence of waves on the walls of the device results in a region of vortex and turbulence. A moment of compression can also be noted, in which airflow with higher velocities occurred at the lower end of the turbine duct, indicating an upward flow of air.

3.2. Regular Waves

Although the focus of this study is the use of realistic irregular waves, an analysis with regular waves representative of the sea state was also performed in order to evaluate the hydrodynamic performance of the OWC WEC device in comparison to the realistic irregular waves. For this, the optimal geometry ($H_1/L=$ 0.1985) was subjected to these regular waves in 15 min simulations. The wave parameters for these regular waves were obtained from the statistical analysis performed previously, in which $H=$ 0.66 m and $T=$ 6.30 s. As a means to visualize the main differences between regular and irregular waves, they were both plotted together and are shown in Figure 30 and Figure 31.

As can be seen in Figure 30a,b, the hydrodynamic behaviors for these scenarios are different. Both free surface elevation and static pressure presented uniform behavior when regular waves were employed, on the other hand, they presented unsteady behavior when irregular waves were employed. However, their magnitude remained approximately the same.

The same tendency occurred with the mass flow rate and available hydrodynamic power, as can be seen in Figure 31a,b. The RMS averaged available hydropneumatic power for the device under regular waves was also calculated, resulting in $P_{hyd}=$ 30.50 W, a value similar to those of the realistic irregular waves. This indicates that, for $H_1/L=$ 0.1985, regular waves reproduced the behavior of the realistic irregular waves in a satisfactory manner, which might not be the case for different values of $H_1/L$.

4. Conclusions

This paper conducted a two-dimensional numerical analysis of an OWC WEC device within a wave channel. To achieve this, the WaveMIMO methodology was utilized to generate random waves of an existing sea state. This method converts spectral sea data into time series of the free surface elevation, which are then processed and converted into the orbital velocity components of the water particles. For this, a verification of the numerical wave model was conducted, in which the numerical wave channel was simulated without the OWC WEC device, and an RMSE value of 0.0686 m was obtained.

Next, the OWC device was introduced and simulated as a case study. By employing the constructal design method, a geometric evaluation of this device was conducted. This evaluation allowed an examination of the available hydropneumatic power generated by the device when subjected to the wave conditions of Rio Grande, Rio Grande do Sul, Brazil. Numerical simulations of 15 min were subsequently conducted with $H_1/L$ values ranging from 0.1985 to 2.2789. The degrees of freedom $H_2/l$ and $H_3$ remained constant: 4.0357 and 6.12 m, respectively. For the physical constraints of the methodology, the volume of the hydropneumatic chamber $V_{HC}$ and the total volume $V_T$ also remained constant, with values of 160.80 m${}^2$ and 192.44 m${}^2$, respectively. During this analysis, the free surface elevation values at the wave channel entrance were continuously monitored and compared to the free surface elevation values obtained through the conversion of spectral data from Tomawac to ensure proper wave generation and propagation.

The optimal geometry that maximized the available hydropneumatic power in the OWC WEC occurred with an $H_1/L$ value of 0.1985, which is in accordance with the relation found in the literature, and an available hydropneumatic power $P_{hyd}$ of 29.63 W was found. This is 4.34 times higher than the power achieved with the least efficient geometry, which was 6.83 W, corresponding to a degree of freedom $H_1/L$ value of 2.2789. In addition, the available hydropneumatic power obtained with the optimal geometry is 2.49 times higher than the power obtained by the device with the same dimensions as those from the one on Pico island—11.89 W. As an additional analysis, the optimal geometry was also subjected to regular waves representative of the coastal region of Rio Grande, Brazil for the year 2014 with $H=$ 0.66 m, and $T=$ 6.30 s. This analysis indicated that the available hydropneumatic power of the device was $P_{hyd}=$ 30.50 W, a value similar to that obtained when subjected to realistic irregular waves. Despite the similarity of the available power for the device employed in this study, it is possible to encounter differences in the hydrodynamic performance when employing regular or irregular waves. Therefore, other geometrical parameters should be investigated for this scenario.

As for future studies, the WaveMIMO methodology may be further employed to obtain sea state data from other regions of the country where it might be possible to have device installation sites. Other OWC WEC device geometries may also be investigated, as well as other degrees of freedom regarding the constructal design method.