Geometrical Analysis of an Oscillating Water Column Converter Device Considering Realistic Irregular Wave Generation with Bathymetry

Abstract

Given the increasing global energy demand, the present study aimed to analyze the influence of bathymetry on the generation and propagation of realistic irregular waves and to geometrically optimize a wave energy converter (WEC) device of the oscillating water column (OWC) type. In essence, the OWC WEC can be defined as a partially submerged structure that is open to the sea below the free water surface (hydropneumatic chamber) and connected to a duct that is open to the atmosphere (in which the turbine is installed); its operational principle is based on the compression and decompression of air inside the hydropneumatic chamber due to incident waves, which causes an alternating air flow that drives the turbine and enables electricity generation. The computational fluid dynamics software package Fluent was used to numerically reproduce the OWC WEC according to its operational principles, with a simplification that allowed its available power to be determined, i.e., without considering the turbine. The volume of fluid (VOF) multiphase model was employed to treat the interface between the phases. The WaveMIMO methodology was used to generate realistic irregular waves mimicking those that occur on the coast of Tramandaí, Rio Grande do Sul, Brazil. The constructal design method, along with an exhaustive search technique, was employed. The degree of freedom $H_1/L$ (the ratio between the height and length of the hydropneumatic chamber of the OWC) was varied to maximize the available power in the device. The results showed that realistic irregular waves were adequately generated within both wave channels, with and without bathymetry, and that wave propagation in both computational domains was not significantly influenced by the wave channel bathymetry. Regarding the geometric evaluation, the optimal geometry found, ${\left(H_1/L\right)}_o$ = 0.1985, which maximized the available hydropneumatic power, i.e., the one that yielded a power of 25.44 W, was 2.28 times more efficient than the worst case found, which had $H_1/L$ = 2.2789.

1. Introduction

As technological and scientific advances in human society progress, we are faced with an increase in the global energy demand that is a consequence of socio-economic development and the improvement in the population’s quality of life [1]. Since around 1850, fossil fuels such as coal, oil, and gas have dominated the energy supply, which has led to a significant increase in greenhouse gas emissions. As a result, the consumption of these non-renewable energy sources is responsible for most of the global anthropogenic emissions of these gases [2].

Since the turn of the century, due to the increase in investments in the renewable energy market, technological advances, and higher development rates in countries, the energy sector has undergone a major change [3]. As a consequence, there has been accelerated development of non-conventional energy sources and technological advances in the exploitation of all forms of energy. This fact has helped to reduce prices, decorrelating the concept of economic growth from the emission of gases that cause the greenhouse effect. Therefore, most countries have sought to diversify their energy matrix [1,3].

Among the renewable energy sources, it is possible to highlight the energy that comes from the oceans. The ocean can be considered a great reservoir of thermal and mechanical energy [4]. However, due to the low quality of heat and the high entropy within the marine environment, as well as the current level of technological advances, only mechanical energy can be used effectively. Ocean energy can be subdivided by origin into tides, waves, marine currents, temperature gradients, and salinity [5].

Wave energy is a concentrated form of solar energy: the sun produces temperature differences, causing winds to blow over the ocean surface. The waves generated are capable of traveling thousands of kilometers with almost no energy loss [6]. In Southern Brazil, the theoretical energy potential that can be obtained from ocean waves is 35 GW, which is higher than the potential found in other regions of the country [4].

The conversion of energy contained in ocean waves into electrical energy is carried out by means of devices such as an oscillating water column (OWC), which consists of a partially submerged hollow concrete structure. The OWC is composed of a hydropneumatic chamber and a duct, in which a turbine and an electric generator are coupled. The device chamber is open below the free water surface and the duct is open to the atmosphere; thus, the incidence of waves on the device makes the water column inside the chamber oscillate and generate a local pressure, expelling or sucking air through the duct. Consequently, this air flow drives the turbine, activating an electric generator [7].

Due to current and future needs in terms of using renewable energy sources, as well as the energy potential of the Brazilian coast, knowledge of how sea wave energy can be converted into electrical energy is essential, so that this type of technology can be used fully and sustainably. One of the methods to develop knowledge is through numerical simulation, which is a tool to study wave energy converter (WEC) devices, which in turn allows the construction of these devices (after testing), with the overall aim of their use in an optimized way with regard to energy conversion [8,9]. Regarding computational modeling as applied to OWC WEC devices, most studies approach regular waves (such as [10,11,12]). However, some recent research has been developed considering the incidence of irregular waves on OWC converters (such as [13,14,15]).

There are also other ways to study the hydrodynamic behavior of WEC devices. Mavrakos et al. [16] addressed the hydrodynamics of bottomless cylindrical bodies with a vertical axis of symmetry floating in finite-depth water using two methods: an analytical approach and a CFD method. In this case, moonpool-type floaters were used, which have been employed as OWC-type devices. In turn, Zabala et al. [17] proposed experimental wave tank tests to validate a numerical model of wave tanks based on CFD that allowed, with limited resources, the necessary simulations for the design and optimization of marine WEC devices under various sea state conditions.

To maximize energy conversion, several studies have applied the constructal design method in the geometric evaluation of WEC devices [18,19,20,21]. Constructal design allows the device geometry to be subjected to global constraints and varied according to degrees of freedom [22]. To apply it, it is necessary to determine the performance indicator, the degrees of freedom, and geometric restrictions. The restrictions used in the present study were the volume of the hydropneumatic chamber and the total volume of the device; both were kept constant. The evaluated degree of freedom was the ratio between the height and the length of the hydropneumatic chamber. Finally, the performance indicator used was the OWC’s available power.

Therefore, the present study aims to develop an investigation regarding the influence (on the numerical wave generation and propagation) of the bathymetry of the municipality of Tramandaí, Brazil. Then, we analyzed and performed the geometric optimization of an OWC WEC device, through the constructal design method associated with an exhaustive search technique, while employing realistic irregular waves representing a sea state that occurred on 28 May 2018, at 10:14 a.m. For this purpose, numerical simulations were conducted considering two-dimensional computational modeling of the OWC WEC device.

2. Materials and Methods

The advancement of WECs is defined by the diversity of ideas and concepts on how to use resources from wave energy [23]. Therefore, WEC devices can be classified according to different criteria, such as the distance from the coast (onshore, nearshore, and offshore) or their conversion principle (OWC, oscillating bodies, and overtopping devices). Among these WECs, this study considered the OWC, as previously done in similar studies [24,25,26], which, as mentioned, is a partially submerged hollow structure, open to the sea below the free water surface. With the passage of the waves, there is an elevation and descent of the water column inside the chamber, which displaces the air through a turbine, thus generating electrical energy. Figure 1 illustrates this principle.

Models that describe the wave behavior and their incidence on an OWC WEC can be classified based on theories of regular and irregular waves. Regular wave theories allow waves with well-defined and constant behavior in each period of time or space, whereas, in irregular wave theories, waves are created from the sum of small regular waves with different heights and periods. Among the existing models, those that employ irregular waves are the ones that best represent realistic sea state conditions in oceanic environments [8]. Therefore, in this study, only realistic irregular waves were simulated. However, the characteristics of representative regular waves were used to discretize the computational model. For this reason, the characteristics of regular waves are shown in Figure 2.

In Figure 2, h is the water depth (m), H is the wave height (m), A is the wave amplitude (m), $\lambda$ is the wavelength, MWL is the mean water level, and $\eta$ is the free surface elevation (m). 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 [27].

2.1. Mathematical and Numerical Modeling

In this study, the computational fluid dynamics (CFD) software package Fluent was employed for the numerical simulations regarding the operating principle of the OWC WEC. Fluent is a computational program, based on the finite volume method (FVM), which allows the numerical simulation of fluid flow and heat transfer in complex geometries [28].

The volume of fluid (VOF) multiphase model, proposed by Hirt and Nichols [29], was used to treat the interface between the fluids. This model employs the technique of representing the phases (air and water) inside each computational cell as a volumetric fraction ($\alpha$), in which the sum of the phases in each cell is unitary; therefore, ${\alpha }_{water}+{\alpha }_{air}=1$.

In the VOF model, only one set of equations is used, consisting of the conservation equations for mass, volume fraction, and momentum [30]:

$$\frac{\partial \rho }{\partial t}+\nabla \left(\rho \overrightarrow{v}\right)=0$$

(1)

$$\frac{\partial \alpha }{\partial t}+\nabla ·\left(\alpha \overrightarrow{v}\right)=0$$

(2)

$$\rho \frac{\partial \overrightarrow{v}}{\partial t}+\rho (\nabla ·\overrightarrow{v})\overrightarrow{v}=-\nabla p+\nabla ·\overline{\overline{\tau }}-\rho \overrightarrow{g}+S$$

(3)

where $\rho$ is the density (kg/m${}^3$), t is the time (s), $\overrightarrow{v}$ is the velocity vector (m/s), p is the static pressure (Pa), $\overline{\overline{\tau }}$ is the strain rate tensor (N/m${}^2$), $\overrightarrow{g}$ is the gravity acceleration (m/s${}^2$), and S is the source term representing the energy dissipation when a numerical beach is implemented (N/m${}^3$).

It is worth mentioning that the continuity and momentum equations are solved for the mixture, therefore, the density ($\rho$) and viscosity ($\mu$) of the mixture can be described respectively, by [31]:

$$\rho ={\alpha }_{water}{\rho }_{water}+{\alpha }_{air}{\rho }_{air}$$

(4)

$$\mu ={\alpha }_{water}{\mu }_{water}+{\alpha }_{air}{\mu }_{air}$$

(5)

Lastly, some numerical parameters were adopted to control the solution of the numerical simulations. The first-order upwind was used for the discretization of the momentum equation, assuming that the value of the variable in the center of the control volume could be considered as the average value of the entire volume; the PRESTO (PRessure STaggering Option) was used for pressure discretization; the geo-reconstruct scheme was employed to tackle the volumetric fraction; and PISO (Pressure-Implicit Splitting of Operators) was used for the pressure–velocity coupling, based on the highest degree of approximation between the corrections for pressure and velocity. The algorithm presented two additional correlations, neighbor correction and skewness correction, which increased the calculation efficiency. This numerical computational model has already been used in previous works (see [21,32]).

2.2. WaveMIMO Methodology

Machado et al. [33] proposed a numerical technique for the simulation of irregular waves, which may represent a realistic sea state in a given region; they named this technique the WaveMIMO methodology, which has been verified and validated for regular wave generation in Maciel et al. [34]. For this, in the present study, a wave spectrum was obtained from the spectral model TOMAWAC (Telemac-based Operational Model Addressing Wave Action Computation). This spectrum was transformed into a time series of free surface elevations and then transformed into orbital velocity profiles. With this, the discrete data obtained could be imposed as boundary conditions of wave velocity in a wave channel in the Fluent software, thus enabling the numerical simulation of realistic irregular waves. In summary, the WaveMIMO methodology was applied in the present work by means of the following steps.

• Location and time of study: a place and a time interval were chosen, and, thus, a simulation of the sea state was performed with these conditions.
• TOMAWAC: the software was used for the numerical simulation of the sea state.
• Wave spectrum simulation: a wave spectrum was extracted from the sea state simulation at the chosen location and time interval.
• Spectral data conversion: the wave spectrum was transformed into a free surface elevation time series statistically identical to the initial wave spectrum, thus representing the sea state [33].
• Dispersion relation: $\eta$ was decomposed into horizontal (u) and vertical (w) velocity components using the method presented by Machado et al. [33] and Oleinik et al. [35].
• Orbital velocities u and w: velocity profiles were imposed as boundary conditions in Fluent to generate a realistic sea state.

For the present work, a sea state that occurred in Tramandaí, in the state of Rio Grande do Sul, Brazil, was considered. This sea state condition happened on 28 May 2018, at 10:14 am, at a study area located 2094.33 m from the coast, with geographic coordinates 50${}^{\circ }$06${}^{\prime }$18${}^{″}$ W and 29${}^{\circ }$59${}^{\prime }$52${}^{″}$ S (see Figure 3).

In order to determine the most frequent sea state, realistic data of the significant height $\left(H_s\right)$ and mean period $\left(T_m\right)$ referring to the selected study area were analyzed. For this, a wave recurrence bivariate histogram relating $T_m$ and $H_s$ was designed, which can be seen in Figure 4.

Next, the wavelength ($\lambda$) was calculated using the dispersion relation equation [27]:

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

(6)

in which the angular wave frequency $\sigma$ is related to the wave period, in this case $T_m$, by $\sigma =2\pi /T_m$ (Hz); the angular wave number k is related to $\lambda$ by k = $2\pi /L$ (1/m); and g is the gravitational acceleration (m/s${}^2$).

Thus, the characteristics of the regular waves representative of the sea state, which were used for the spatial discretization of the computational domain, were as follows: significant height, $H_s$ = 0.90 m; length, $\lambda$ = 45.9915 m; mean period, $T_m$ = 5.70 s; and depth, h = 10.976 m.

Bathymetry Influence Study and Numerical Model Verification

This study was subdivided into two moments. In the first, a bathymetry study was performed with the objective of evaluating the bathymetry influence on the generation and propagation of realistic irregular waves, and, in the second, the verification of the generation of waves by means of the WaveMIMO methodology was performed. For this purpose, two computational domains were considered for the two-dimensional numerical wave channel. The first one (Figure 5a) presented a horizontal bottom (i.e., without bathymetry) and the second one (Figure 5b) presented the bathymetry found in the studied place, which was obtained from the nautical charts of the Directorate of Hydrography and Navigation of the Brazilian Navy [36] and catalogued by Cardoso [37]. For both cases, the wave channel presented a length of $L_c=229.56$ m, which corresponded to $5\lambda$, following the recommendation of Gomes et al. [38], and a height of $H_c=15$ m. In addition, a constant water depth of $h=10.976$ m for the domain without bathymetry was adopted, and a variable depth was employed for the domain with bathymetry, where the left side of the wave channel measured $h=10.976$ m and the right side measured $h_1=10.520$ m.

As for the boundary conditions, in both cases, the following were used: atmospheric pressure (green line); wall (continuous black line), where the velocities were considered null; hydrostatic profile (blue line); and velocity inlet (red line), which was subdivided into 14 segments of size h/14, as recommended by Machado et al. [33]. It is important to mention that a numerical beach was used in the computational domain, in which a damping sink term was added to Equation (3), being defined as [28,39]

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

(7)

in which $C_1$ is the linear damping coefficient (s${}^{-1}$), $C_2$ is the quadratic damping coefficient (m${}^{-1}$), V is the fluid velocity at a given point (m/s), and $z_{fs}$, $z_b$, $x_e$, and $x_s$ are the free surface, bottom, starting, and ending points of the numerical beach domain (m), respectively.

The numerical beach tool can be used to absorb wave energy and reduce the effects of reflection, which is caused by the incidence of waves on the right wall of the channel. The damping coefficients, following recommendations from previous studies [39], were defined as $C_1=20$ s${}^{-1}$ and $C_2=0$ m${}^{-1}$, and the numerical beach length was defined as $2\lambda$, i.e., $L_{nb}=91.82$ m.

Regarding the spatial discretization, a stretched mesh was used [40]. For this, the computational domain was subdivided into 3 vertical regions with the following discretizations: 60 computational cells in the region that contained only water; 40 cells in the free surface region (air/water interface); and 20 cells in the region containing only air. Horizontally, the computational domain was subdivided into 250 computational cells, i.e., 50 cells per wavelength. Regarding temporal discretization, a time step of $\Delta t=0.05$ s was considered, as recommended by Machado et al. [33]. The total simulation time considered was 200 s of wave generation and propagation in the channel.

2.3. Geometric Optimization of OWC WEC

Following the bathymetry influence study, the OWC WEC device was inserted into the wave channel and its geometry was analyzed. To do so, the constructal design method was employed, which is based on the constructal theory. The constructal theory, developed by Adrian Bejan [22], explains how the generation of structures subjected to flow, such as those commonly found in nature, can be based on the principle of access to flow over time.

There are several studies that have used the constructal design in the geometric optimization of OWC devices under the incidence of regular waves (for instance, [41,42,43]), allowing the evaluation and improvement of the conversion of the energy contained in sea waves into electrical energy. In this study, the constructal design method, which was used to define the search space, was employed with an exhaustive search technique to obtain the optimized geometric shape of an OWC WEC device subjected to the sea state found in the city of Tramandaí, in the state of Rio Grande do Sul, Brazil.

For the constructal design application, it was necessary to define a performance indicator, the degrees of freedom, and restrictions. In this case, the performance indicator was the available hydropneumatic power, which must be maximized. The varied degree of freedom was the ratio $H_1/L$, being $H_1$ the height and L the length of the device’s hydropneumatic chamber. Two degrees of freedom were kept constant, $H_2/l=4.03$, which defines the ratio between the height and length of the turbine duct, and the submersion depth of the chamber $H_3=6.12$ m. These dimensions can be seen in Figure 6.

In this study, the geometric constraints were the volume of the hydropneumatic chamber, $V_{ch}$, and the total volume of the OWC, $V_t$, which were kept constant and are given by:

$$V_{ch}=H_{1}L{L}_1$$

(8)

$$V_t=V_{ch}+H_{2}l{L}_1$$

(9)

The third dimension $L_1$ had a unitary value, as the computational modeling adopted was two-dimensional. From Equation (8), it was possible to find formulations for the length, L, and height, $H_1$, of the hydropneumatic chamber:

$$L=\sqrt{{\displaystyle \frac{V_{ch}}{\left({\displaystyle \frac{H_1}{L}}\right)L_1}}}$$

(10)

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

(11)

The present study was based on Maciel [15], where the geometry of the OWC WEC was optimized considering the sea state found in the city of Rio Grande, in the state of Rio Grande do Sul, Brazil. Therefore, the same values of the degree of freedom $H_1/L$ evaluated in that study were considered here. It is worth noting that the main difference between the studies was the location of the device and, consequently, the time series of irregular waves to which the OWC WEC was subjected. Therefore, similar to Maciel [15], the device used was based on the dimensions of the OWC located in Pico Island, Azores, Portugal, with the following dimensions: L = 12 m; $H_1$ = 13.4 m; l = 2.8 m; H = 11.3 m; and $H_3$ = 6.12 m (see Figure 6).

Table 1. Configurations of the geometries used in the study.

${\mathit{H}}_1/\mathit{L}$	${\mathit{H}}_1$ (m)	L (m)
0.1985	5.65	28.46
0.4297	8.31	19.34
0.6608	10.31	15.16
0.8920	11.98	13.43
1.1167	13.40	12.00
1.3543	14.76	10.90
1.5854	15.97	10.07
1.8166	17.09	9.41
2.0478	18.15	8.86
2.2789	19.14	8.40

shows the cases defined by means of the constructal design application (variation of the ratio $H_1/L$) and its respective dimensions. As already stated, the value of $H_3$ remained constant, equal to $6.12$ m. It is important to mention that ratio $H_1/L$ = 0.1985 was the optimal case found in Maciel [15] and ratio $H_1/L$ = 1.1167 was the reference case with the dimensions of the Pico OWC device.

Regarding the computational domain, the OWC WEC was inserted in the wave channel in which the bottom of the channel presented bathymetry (see Figure 5b), according to the results from the bathymetry influence study. Furthermore, as recommended by De Lima [9], the OWC WEC was placed at 1.5$\lambda$ from the beginning of the wave channel. Regarding the boundary conditions, they were the atmospheric pressure (green line), wall (continuous black line), hydrostatic profile (blue line), and velocity inlet (red line), which, as in the bathymetry influence study, it was subdivided into 14 segments of size h/14, as recommended by Machado et al. [33]. An illustration of the computational domain considering the OWC WEC is presented in Figure 7. Again, the numerical beach was used (Equation (7)).

Regarding the spatial discretization, a stretched mesh was used for the regions both before and after the device [40], as performed in the bathymetry influence study. For the discretization of the device, square-shaped cells with a size of 0.1 m were used and the thickness of its wall was 0.1 m, as recommended by De Lima [9]. As for temporal discretization, the time step $\Delta t=0.05$ s was used [33]. In this study, the total simulation time was expanded to 900 s of wave generation in the channel in order to compare the results obtained with those of Maciel [15].

2.4. Error Metrics

To analyze the quality of the results, the mean absolute error, MAE, and root mean square error, RMSE, metrics were considered, respectively, given by [44]:

$$\mathrm{MAE}=\frac{1}{n}\sum _{i=1}^n|O_i-R_i|$$

(12)

$$\mathrm{RMSE}=\sqrt{\frac{1}{n}\sum _{i=1}^n{|O_i-R_i|}^2}$$

(13)

in which $O_i$ represents the value found numerically, $R_i$ represents the value used as a reference, and n represents the total number of data.

To monitor the free water surface elevation, vertical numerical probes were used, in which the integral was calculated by [28]:

$$\int \varphi dA=\sum _{i=1}^n{\varphi }_i\left|A_i\right|$$

(14)

in which ${\varphi }_i$ is a variable field and $A_i$ is the area of each computational cell volume $\left({\mathrm{m}}^2\right)$. In the study of the influence of bathymetry, numerical probes were placed at $x=0.000000$, 10.455425, 20.910851, 31.366276, 41.821702, 50.186042, 60.641468, 71.096893, 81.552318, 92.007744, and 102.463169 m, and in the geometric optimization study of the OWC WEC, the numerical probe was positioned at the center of the device, i.e., x = $(1.5\lambda$ + $L/2)$ m.

To monitor the static pressure and air mass flow rate, a horizontal numerical probe was used at the center of the OWC turbine duct, i.e., $x\in [1.5\lambda +L/2-1.4,1.5\lambda +L/2-1.4],$ in which the integrals were calculated, respectively, by [28]:

$$\frac{1}{A}\int \varphi dA=\frac{1}{A}\sum _{i=1}^n{\varphi }_i\left|A_i\right|$$

(15)

$$\int {\rho }_{air}\overrightarrow{v}·d\overrightarrow{A}=\sum _{i=1}^n{\rho }_{air}{\overrightarrow{v}}_i·{\overrightarrow{A}}_i$$

(16)

in which $\overrightarrow{v}$ represents the velocity vector on the face (m/s) and ${\rho }_{air}$ the air density $(\mathrm{kg}/{\mathrm{m}}^3)$.

It is important to mention that the position of the numerical probes inside the device changed according to the geometric configurations. With this, it was possible to calculate the available hydropneumatic power, which is given by [45]:

$$P_H=\left(p+\frac{1}{2}{\rho }_{air}{v}_{air}^2\right)\frac{\dot{m}}{{\rho }_{air}}$$

(17)

in which p is the static pressure in the duct (Pa), $v_{air}$ is the air velocity in the duct $(\mathrm{m}/\mathrm{s})$, and $\dot{m}$ is air mass flow rate inside the turbine duct (kg/s).

Finally, to quantitatively evaluate the available power of the geometries studied and determine the ratio $H_1/L$ that maximized its value, the root mean square, RMS, was used [46]:

$$P_{\mathrm{RMS}}=\sqrt{\frac{1}{n}\sum _{k=1}^n{P}_k^2}$$

(18)

in which $P_k$ indicates the instantaneous available power (W) calculated in each time step.

3. Results and Discussion

3.1. Study of Bathymetry Influence in the Generation and Propagation of Realistic Irregular Waves and Numerical Model Verification

As earlier mentioned, the study of bathymetry considering realistic irregular waves was divided into two cases. In the first case, the channel presented a horizontal bottom, i.e., without bathymetry, and the second case presented a sloped bottom, reproducing the bathymetry found in the study area (see Figure 5). Aiming to evaluate the difference in wave propagation in channels with and without bathymetry,

Table 2. MAE and RMSE results considering the bathymetry cases subjected to realistic irregular waves.

Probe Position (m)	MAE (m)	RMSE (m)
0.000000	0.004264	0.006133
10.455425	0.019815	0.020653
20.910851	0.019803	0.020501
31.366276	0.019803	0.020468
41.821702	0.019835	0.020556
50.186042	0.019836	0.020555
60.641468	0.019802	0.020494
71.096893	0.019757	0.020514
81.552318	0.019746	0.020618
92.007744	0.019743	0.020814
102.463169	0.019682	0.021028

presents the metrics MAE and RMSE monitored at 11 points of the bathymetry. The results obtained in the numerical simulation that considered the wave channel bathymetry were taken as a reference, as it represented the computational domain that most closely resembled the simulation performed in TOMAWAC, which originated the boundary conditions imposed in Fluent, being more similar to the reality of the study area.

As expected, the smallest differences were found in the wave generation zone ($x=0$ m), as it was the location in the wave channel that was least influenced by the bathymetry. Next, it was observed that the largest differences in MAE and RMSE, respectively, occurred at positions $x=50.186042$ m and 102.463169 m. When comparing them with the significant wave height of the regular waves representative of this sea state, differences of approximately 2.2% were found in both metrics, which confirmed that, in the analyzed situation, the use of bathymetry did not have a significant influence on the generation and propagation of realistic irregular waves.

Figure 8a and Figure 8b present, respectively, the comparison between the free surface elevation of realistic irregular waves at positions $x=0$ m and $71.096893$ m. It is worth noting that these positions were chosen for evaluation due to the fact that the generation of realistic irregular waves occurred at $x=0$ m, whereas, at $x=71.096893$ m, the OWC WEC was inserted into the wave channel, as recommended by De Lima [9].

Figure 8a shows the waves numerically generated (using Fluent) in the wave channels with and without bathymetry, along with the free surface elevation from TOMAWAC, allowing the verification of the wave generation. As can be observed, the waves generated in the wave channels with and without bathymetry adequately represented the sea state from TOMAWAC. To confirm the observed qualitative results, the free surface elevations obtained with Fluent were compared with those from TOMAWAC using the MAE and RMSE metrics. For the case of the wave channel with bathymetry, the differences were found to be 0.0999 m and 0.1285 m, respectively, while, for the wave channel without bathymetry, the differences were 0.1003 m and 0.1288 m, respectively. These results are in agreement with those found by Machado et al. [33] and confirm the appropriate generation of realistic irregular waves in both wave channels, with and without bathymetry. Furthermore, as observed in Figure 8a, the free surface elevations monitored in the generation zone for the channels with and without bathymetry were similar. When comparing the MAE and RMSE metrics presented in Table 2 (for $x=0$ m) with the significant wave height of the regular waves representative of this sea state, differences of 0.47% and 0.68%, respectively, were found.

Figure 8b, on the other hand, shows the monitored free surface elevation at the probe located at $x=71.096893$ m in both wave channels, with and without bathymetry. Qualitatively, it is evident that the propagation of realistic irregular waves was not significantly influenced by the wave channel bathymetry. Quantitatively, when comparing the MAE and RMSE metrics with the significant wave height of the regular waves representative of this sea state, differences of approximately 2.2% were found again. Based on the results obtained, the bathymetry was taken into consideration for the simulations of the OWC WEC device.

Lastly, aiming to compare the results obtained for the OWC WEC in the present study with those found in Maciel [15], Figure 8c presents the verification of 900 s of wave generation in the channel with bathymetry. For this purpose, the monitored free surface elevation in the wave generation zone ($x=0$ m) was compared to the elevation from TOMAWAC at the same position. Here, MAE and RMSE values of 0.0814 m and 0.1044 m, respectively, were found. It should be noted that realistic irregular waves were properly generated using the WaveMIMO methodology [33], as evidenced by the low error metric values obtained.

3.2. Study of Geometric Optimization of OWC WEC Subjected to Realistic Irregular Waves

The OWC WEC device was introduced in the wave channel with bathymetry (see Figure 7), in which it was possible to carry out the geometric optimization of the device and to maximize the available hydropneumatic power, considering the 10 configurations listed in Table 1, under the influence of realistic irregular waves, as can be seen in Figure 9 and Figure 10.

Figure 9a shows the variation in static pressure that occurred during the 900 s of simulation for all $H_1/L$ ratios. Meanwhile, in Figure 9a, it is possible to observe the variation in static pressure that occurred at $t\le 900$ s for the ratio that qualitatively presented the best result, $H_1/L$ = 0.1985, and for the ratio that qualitatively presented the worst result, $H_1/L$ = 2.2789. Next, Figure 9c presents the same cases as Figure 9b, in a specific time interval of 100 s $\le t\le 300$ s, which was when the second highest pressure peak occurred, among the 900 s of simulation. As can be observed in Figure 9a, the results for all $H_1/L$ ratios analyzed exhibited similar behavior, except for the case $H_1/L$ = 0.1985, which presented higher static pressure peaks compared to the others. This can be justified by the fact that, for this ratio, L is larger (see Table 1), causing the 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 De Lima et al. [12] and Gomes et al. [41]. As seen in Figure 9b, for $H_1/L$ = 0.1985, the highest static pressure peak occurred during the decompression of the hydropneumatic chamber, at $t=668.55$ s, and corresponded to −50.481 Pa. Meanwhile, for $H_1/L$ = 2.2789, the highest static pressure peak was 17.2 Pa and occurred during the compression, at $t=597.85$ s. In Figure 9c, the second highest static pressure peak can be observed at $t=171.85$ s and 191 s for $H_1/L$ = 0.1985 and 2.2789, respectively. There is a difference of 61% between these static pressure peaks. Additionally, it can be observed that the smallest $H_1/L$ ratios resulted in the highest static pressure peaks, either during compression or decompression of the hydropneumatic chamber of the OWC WEC, and these peaks stabilized as the $H_1/L$ values increased.

Figure 10a shows the variation in the available hydropneumatic power for all $H_1/L$ ratios during the 900 s of simulation. Meanwhile, in Figure 10b, it is possible to observe the variation in hydropneumatic power that occurred at $t\le 900$ s for the ratio that presented the best qualitative results, $H_1/L$ = 0.1985, and for $H_1/L$ = 2.2789, which presented the worst qualitative results. Next, Figure 10c shows the ratios $H_1/L$ = 0.1985 and 2.2789 for 100 s $\le t\le 300$ s, the interval during which one of the highest peaks in hydropneumatic power occurred among the 900 s of simulation.

As can be observed in Figure 10a,b, the highest peak in hydropneumatic power found, $P_H$ = 279.98 W, occurred during the compression of the hydropneumatic chamber, for the ratio $H_1/L$ = 0.1985, at t = 668.95 s. Additionally, the second highest peak in hydropneumatic power was also found for $H_1/L$ = 0.1985, which occurred during the decompression of the hydropneumatic chamber (t = 667 s) and was equal to −236.61 W. On the other hand, the highest peak in hydropneumatic power for $H_1/L$ = 2.2789 occurred at $t=598.6$ s and was 75.5% and 71.1% lower than the highest and second highest peaks in hydropneumatic power found for $H_1/L$ = 0.1985, respectively. In Figure 10c, it is confirmed that the peaks in hydropneumatic power for $H_1/L$ = 0.1985 were prominent compared to those found for $H_1/L$ = 2.2789, confirming that this case presented the best results.

Figure 11 relates the values of $P_{\mathrm{RMS}}$ obtained in the present study with those found in Maciel [15]. In both studies, the case that optimized the available hydropneumatic power was ${(H_1/L)}_o$ = 0.1985. It is noteworthy that the RMS hydropneumatic power found in this study was 25.44 W, representing a 14.14% loss in device efficiency compared to the power found by Maciel [15].

It is worth noting that the case with $H_1/L$ ratio = 1.1167 represented the dimensions of the OWC WEC installed at Pico Island. For this case, a hydropneumatic RMS power of 14.88 W was obtained, a value 20.09% larger than that found by Maciel [15], which was 11.89 W. As for the worst efficiency, it was found for $H_1/L$ = 2.2789 in both studies. However, in this study, the minimum RMS hydropneumatic power was found to be 38.74% higher than the value found in Maciel [15], which was 6.83 W.

Finally, the optimal case found in this study, ${(H_1/L)}_o$ = 0.1985, was 2.28 times more efficient than the worst case, $H_1/L$ = 2.2789. Furthermore, as $H_1/L$ increased, the RMS hydropneumatic power decreased, which was also observed in Maciel [15]. Additionally, it could be concluded that even with the OWC WEC subjected to different wave elevation series, the geometric evaluations showed similar curves, which can be justified by the significant similarity between the wave climate behavior in Rio Grande and Tramandaí [47].

Figure 12 depicts the phase topology, in which water and air are represented in red and blue colors, respectively. In Figure 12a–d, the case of optimal geometric configuration ${(H_1/L)}_o$ = 0.1985 is shown, while, in Figure 12e–h, the worst case, $H_1/L$ = 2.2789, is presented. In Figure 12a,e, the initial instant ($t=0$ s) of the simulation is shown, where the flow was at rest and, therefore, there was no incidence of irregular waves into the hydropneumatic chamber of the OWC WEC. Next, Figure 12b,f shows the flow behavior at $t=300$ s, a moment where a larger volume of water could be observed inside the device chamber for the optimal case, in contrast to the worst case (Figure 12f), in which there was less oscillatory movement. Figure 12c,g depicts $t=600$ s, a moment that, similar to the previous $t=300$ s, shows a higher volume of water in the hydropneumatic chamber for the case ${(H_1/L)}_o$ = 0.1985 compared to $H_1/L$ = 2.2789, confirming the results presented in Figure 11. Finally, Figure 12d,h shows the final instant of the simulation, $t=900$ s, where once again it can be observed that the optimal case presented a larger volume of water in the hydropneumatic chamber compared to the case with $H_1/L$ = 2.2789. It is important to highlight that, during compression, wave crests could be observed inside the device, while, during decompression, wave troughs were visible. The alternating motion of the waves inside the hydropneumatic chamber of the device led to alternating air flow in the turbine duct, where the turbine was installed, converting kinetic energy into electrical energy.

4. Conclusions

Regarding the study of bathymetry influence, concerning the generation of realistic irregular waves, the best results were achieved with the bathymetric channel. However, in the propagation of realistic irregular waves, there was no significant difference in the free water surface elevation whether the bathymetry was considered or not. Therefore, the local bathymetry was used to verify the generation of a realistic sea state found in Tramandaí.

Next, a geometric optimization study of an OWC WEC device was carried out, taking into account the wave climate and local bathymetry. The optimal geometry that maximized the available hydropneumatic power in the converter presented a ratio of ${\left(H_1/L\right)}_o$ = 0.1985, in which case $H_1$ = 5.65 m and L = 28.46 m. The RMS hydropneumatic power found when considering this geometric configuration was 25.44 W, which was 2.28 times more efficient than the worst-case scenario, $H_1/L$ = 2.2789. Furthermore, by analyzing the 10 investigated geometries, it was observed that the degree of freedom $H_1/L$ and the RMS hydropneumatic powers found were inversely proportional. In other words, as the $H_1/L$ ratio increased, the $P_{\mathrm{RMS}}$ decreased, which was a behavior trend analogous to those observed in previous works, as in Maciel [15].

For future studies, an investigation regarding the influence of bathymetry on the representative regular waves generated through the WaveMIMO methodology can be suggested. Additionally, one could reproduce the geometric optimization carried out in this study considering representative regular waves of this sea state, as a means to investigate whether the optimal geometry would be influenced by the wave climate.