Geometrical Evaluation of an Overtopping Wave Energy Converter Device Subject to Realistic Irregular Waves and Representative Regular Waves of the Sea State That Occurred in Rio Grande—RS

Abstract

Among the various potential renewable energy sources, sea waves offer significant potential, which can be harnessed using wave energy converter (WEC) devices such as overtopping converters. These devices operate by directing incident waves up a ramp into a reservoir. The water then passes through a turbine coupled with an electrical generator before returning to the ocean. Thus, the present study deals with the geometrical evaluation of an overtopping WEC, where the influence of the ratio between the height and length of the device ramp ($H_1/L_1$) on the amount of water mass (M) that enters the reservoir was investigated. Numerical simulations were performed using ANSYS-Fluent software, 22 R1 version, to generate and propagate realistic irregular (RI) waves and representative regular (RR) waves found in the coastal region of the municipality of Rio Grande, in the state of Rio Grande do Sul, southern Brazil. Consequently, through constructal design, the optimal WEC geometry for both wave approaches were identified as the same, where (${H_1/L_1)}_o=0.30$. Thus, considering the RI waves, $M=$ 200,820.77 kg was obtained, while, considering the RR waves, $M=$ 144,054.72 kg was obtained.

1. Introduction

Marine renewable energy (MRE) is a viable alternative for exploring renewable energy sources, given the various forms of energy available in the oceans, including wave energy [1]. Espindola and Araújo [2] estimate that the annual energy potential of waves in deep waters off Brazil is approximately 89.97 GW, with the country’s southern region having the highest average wave power. One of the possibilities for harnessing this energy is through wave energy converters (WECs) of the overtopping type. These devices operate on a partially submerged structure with a ramp that channels water into a reservoir through overtopping. The water then returns to the ocean, passing through low-head hydraulic turbines that drive coupled electrical generators [3].

Overtopping WECs have been studied in the experimental field, through real prototypes and laboratory devices, as well as in the numerical field, through computational modeling. An effective approach to investigate overtopping devices is through geometric evaluations, which can be conducted using the constructal design method. This method is based on constructal theory [4], according to which nature’s flow systems follow a physical principle for the generation of their configurations and geometric patterns. This principle is called constructal law [4,5,6], and is being applied to the evaluation of flow systems through the constructal design. This approach allows for the analysis of the influence of geometry on system performance.

Furthermore, concerning studies carried out in the numerical field, there are different approaches for wave generation, and, therefore, several numerical investigations regarding the propagation waves have been carried out. Regarding the conventional methodology commonly applied when addressing regular waves [7,8], there are several notable studies. For instance, Lisboa et al. [9] investigated the linear and quadratic damping coefficients considered for a numerical beach tool employed to suppress wave reflections at the end of a numerical wave channel, and Zabihi et al. [10] compared two software programs, ANSYS-Fluent and Flow-3D (https://www.flow3d.com/), for generating numerical waves.

In addition, there are alternative methods for wave generation that allow the reproduction of numerical irregular waves. Some examples can be seen in Higuera et al. [11,12], who developed [11] and validated [12] a numerical methodology for realistic wave generation associated with active wave absorption in OpenFOAM software (https://www.openfoam.com/), which was based on the linear shallow-water theory; Finnegan and Goggins [13], where the generation of irregular waves on a real scale was studied; and Machado et al. [14], who developed the WaveMIMO methodology, which was verified and validated by Maciel et al. [15] through numerical simulations in the ANSYS-Fluent software and laboratory experiments; this methodology can generate irregular waves based on realistic sea state data, making it possible to reproduce the phenomenon more reliably, akin to its occurrence in nature. The WaveMIMO methodology allows the reproduction of waves generated numerically or monitored by ocean buoys.

It should be emphasized that both the experimental and numerical approaches are important and complement each other in the search for advances in the MRE field, mainly in the wave energy realm. Combined, these investigation approaches allowed several analyses and advances regarding WECs, as can be seen in Tedd and Kofoed [16], who presented a study of overtopping flow series on the Wave Dragon prototype, a low-crested overtopping device, in an inland sea in Northern Denmark; Parmeggiani et al. [17], who conducted a laboratory experimental study on a 1.5 MW Wave Dragon, identifying its response to extreme conditions typical of the DanWEC test center; and Di Lauro et al. [18], who analyzed the stability of a breakwater with a coupled overtopping device, known as the overtopping breakwater for energy conversion (OBREC), combining laboratory-scale experiments and numerical simulations. Additionally, Palma et al. [19] carried out numerical studies to optimize the hydraulic and structural performance of the OBREC device by investigating the inclusion of a berm, the shape of the sloping plate, and the crown wall shape.

In the experimental field, Contestabile et al. [20] evaluated the effectiveness of a triangular parapet placed on top of the OBREC device in reducing wave overtopping when compared to a breakwater with a traditional crown wall. Di Lauro et al. [21] proposed a vertical OBREC (OBREC-V); the hydraulic performance and stability were analyzed under hydraulic loading, comparing it with a traditional breakwater, indicating that the non-conventional geometry enhances the structure’s stability. Musa et al. [22] investigated, through numerical and experimental approaches, the influence of the ramp shape parameters of an OBREC device under Malaysia’s wave condition.

Moreover, Koutrouveli et al. [23] presented conceptual proof of a hybrid WEC, which addresses the operational principles of two WECs, the overtopping and the oscillating water column (OWC); the results indicated that hybridization is an effective approach. Clemente et al. [24] conducted an experimental study to analyze the performance and stability of the armor layer and the toe berm of a model of the north breakwater extension project incorporating a hybrid WEC as proposed by Koutrouveli et al. [23].

Also, Martins et al. [25] conducted a numerical study of an overtopping device, using the constructal design method to evaluate the effects of the degrees of freedom on the average dimensionless overtopping flow for devices with one and two ramps; it was found that the two-ramp configuration performed better under the incidence of irregular waves generated through the JONSWAP wave spectrum. An et al. [26] analyzed different overtopping WEC models testing the length of the substructure, i.e., the water region under the ramp and reservoir of the device, inferring that the hydraulic efficiency decreases with the substructure length but does not affect the performance of the WEC. Barros et al. [27] applied the constructal design method to evaluate the influence of a trapezoidal berm, located at the bottom of the channel and connected to the ramp of the overtopping device, showing that this configuration leads to better results than traditional overtopping WEC under the incidence of regular waves. Goulart et al. [28] performed a geometric evaluation of the overtopping device, using constructal design, through experimental and numerical studies considering the incidence of regular waves on a laboratory scale; the results showed an agreement between the approaches, validating the numerical model proposed using the ANSYS-Fluent software.

It should be highlighted that the WaveMIMO methodology [14] has also been used in investigations regarding overtopping devices, as in Hübner et al. [29], who employed it to evaluate the overtopping WEC, where they compared the fluid dynamic behavior of the device under the incidence of realistic irregular (RI) waves and representative regular (RR) waves, considering realistic sea state data of three coastal regions along the Rio Grande do Sul (RS) state in southern Brazil. However, this methodology has not yet been applied to the geometric evaluation of overtopping devices under the incidence of RI waves occurring in a specific location.

As mentioned, several studies are addressing the overtopping WEC in the literature. More examples of these are found for the experimental approach in [30,31,32,33,34,35,36,37,38], including the ones addressing the hybrid WEC, which combines the overtopping and OWC operating principles [35,36,37,38], while for the numerical approach there are [39,40,41,42,43,44,45,46,47,48,49,50,51], including more studies where the constructal design was employed to evaluate the overtopping device’s geometry, such as [41,49,51]. As shown, despite the vast research on the overtopping device, few studies address realistic sea states when evaluating its geometry. This fact justifies the contribution of the present paper and is in line with what was stated by Romanowski et al. [52], who point out the lack of studies that use computational fluid dynamics (CFDs) to estimate and model irregular sea states through numerical simulations.

In this context, the present paper conducts a geometric evaluation of an overtopping device subject to the incidence of RI waves and RR waves from the sea state occurring near the Molhes da Barra breakwater, located on the coast of Rio Grande—RS, in southern Brazil. Therefore, the numerical models employed for the RI and RR waves generation were verified, comparing the numerical results obtained to realistic sea state data (RI waves) and the analytical solution (RR waves), proving that the waves were adequately reproduced. Then, the constructal design method was used to evaluate the influence of the ratio between the height and length of the ramp ($H_1/L_1$) on the amount of water mass ($M$) entering the device’s reservoir and determine the geometry that maximizes converter performance.

It is important to note that the $H_1/L_1$ parameter has been evaluated using constructal design through both numerical [27,41,51] and experimental [28] approaches under the incidence of regular waves. Additionally, it has been evaluated using the JONSWAP wave spectrum for irregular waves [25,49]. However, in this paper, the parameter is investigated based on realistic sea state data rather than theoretical or random waves. This consideration is crucial, given that the performance of WEC devices varies according to the wave climate they are subjected to. Furthermore, a comparison was made between the results obtained, highlighting the differences and similarities in the fluid dynamic behavior of the device depending on the wave approach used. It is important to emphasize that the geometric evaluation of an overtopping WEC, employing the constructal design, under the incidence of RI waves generated using the WaveMIMO methodology is an original scientific contribution of the present paper, as is the comparison of its performance with the one that occurred due to the incidence of the RR waves from the same sea state.

2. Mathematical and Numerical Modeling

The numerical simulations carried out in the present study used the ANSYS-Fluent software [53], 22 R1 version, a CFD package that is based on the finite volume method (FVM) [54]. It should be mentioned that numerical simulations of wave channels in this software have already been validated with laboratory experiments [15,28].

The multiphase volume of fluid (VoF) model [55] was employed in the treatment of the interface between the phases considered, the water and the air. In this model, the concept of volumetric fraction ($\alpha$) is used, considering that the sum of the phases contained in each control volume must always equal unity. Thus, when a computational cell contains only water or only air, is the following applies, respectively:

$${\alpha }_{water}=1,$$

(1)

$${\alpha }_{air}=1.$$

(2)

Additionally, since the VoF model is used for immiscible fluids, if a cell contains the interface between the two phases, it holds that:

$${\alpha }_{water}=1-{\alpha }_{air}.$$

(3)

Along with using the VoF model, a set of three equations is solved, which is composed of the mass conservation equation, given as follows [54,56]:

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

(4)

where t is the time (s); $\overrightarrow{V}$ is the velocity vector (m/s); and $\rho$ is the fluid density (kg/m³), which is calculated for the mixture of phases (water and air) as [57]:

$$\rho ={\alpha }_{water}{\rho }_{water}+\left(1-{\alpha }_{water}\right){\rho }_{air}.$$

(5)

The model also includes the conservation of the volumetric fraction, given by [54]:

$${\displaystyle \frac{\partial \left(\alpha \right)}{\partial t}}+\nabla \left(\alpha \overrightarrow{V}\right)=0.$$

(6)

Finally, the last equation that composes the VoF model is the conservation of momentum, defined as [53,56]:

$${\displaystyle \frac{\partial }{\partial t}}\left(\rho \overrightarrow{v}\right)+\rho \left(\nabla \overrightarrow{v}\right)\overrightarrow{v}=-\nabla p+\nabla \stackrel{̿}{\tau }-\rho \overrightarrow{g}+S,$$

(7)

where p is the static pressure (Pa); $\overrightarrow{g}$ is the gravity acceleration vector (m/s²); and $\stackrel{̿}{\tau }$ is the strain rate tensor (N/m²). Moreover, the term S is a sink, referring to the numerical beach tool, which attributes a dissipative profile to the region of the wave channel where it is applied, avoiding the reflection effect of waves that reach the end wall of the channel, given as follows [58,59]:

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

(8)

where $C_1$ and $C_2$ are, respectively, the linear (s^–1) and quadratic (m⁻¹) damping coefficients; $V$ is the velocity along the z direction (m/s); $z_{fs}$ and $z_b$ are the vertical positions of the free surface (FS) and the channel bottom (m); and $x_s$ and $x_e$ are the starting and ending positions of the numerical beach (m). Finally, it should be noted that the damping coefficients, $C_1$ and $C_2$, are defined, respectively, as $20$ s^–1 and $0$ m^–1, following the indications of Lisboa et al. [9].

To perform the numerical simulations in the ANSYS-Fluent software, configurations are selected to solve the Equations (4), (6), and (7). Thus, the pressure-implicit with splitting of operators (PISO) scheme, a non-iterative transient calculation procedure based on temporal precision [60], was used to solve the pressure–velocity coupling. Furthermore, it is emphasized that when the ANSYS-Fluent software is used, the PISO scheme is the most stable option for simulation problems that use the multiphase VoF model [52]. Regarding the spatial discretization for the pressure equation, the pressure staggering option (PRESTO) scheme [53] was employed.

For the treatment of the advective terms, the discretization method adopted was the first-order upwind, as it generally leads to better convergence [53]. According to Patankar [61], with this scheme, the quantities on all faces of the volumes are determined assuming that the center of the volume for a variable field represents an average value throughout the entire volume.

Moreover, to determine the surface occupied by water, the geo-reconstruction method is used, which, in ANSYS-Fluent, presents the best accuracy [53]. In this approach, ANSYS-Fluent’s standard interpolation schemes are used to obtain the flow at the faces whenever a volume is filled with one phase, water or air. However, when a volume contains a mixture of phases, the geo-reconstruction scheme is employed, which represents the interface between the fluids using a piecewise-linear approximation.

The aforementioned numerical methods, along with the remaining configuration sets, are summarized in

Table 1. Methods and parameters used in the present numerical simulations.

Parameters		Numerical Inputs
Solver		Pressure-Based
Pressure–Velocity Coupling		PISO
Spatial Discretization	Gradient Evaluation	Green–Gauss-Cell-Based
	Pressure	PRESTO
	Momentum	First Order Upwind
	Volume Fraction	Geo-Reconstruct
Temporal Differencing Scheme		First Order Implicit
Under-Relaxation Factors	Pressure	0.3
	Momentum	0.7
Residual	Continuity	10–3
	x-velocity
	z-velocity
Regime Flow		Laminar

. It is worth noting that these methods are based on methodologies used in previous studies, including numerical ones [14,25,27,41,49,51], as well as validation studies where the results were compared to laboratory experiments [15,28].

Additionally, the flow was considered under a laminar regime without addressing a turbulence model. This simplification about the real problem can be adopted without significant loss of accuracy, as indicated in Gomes et al. [8], who obtained a good agreement between the numerical model, under a laminar regime, and experimental results. When considering laminar flow, in addition to reducing processing time, the convergence of the numerical solution is facilitated, which justifies the adoption of this simplification given the total time of each simulation in the present study.

2.1. Generation of Realistic Irregular Waves

To generate the RI waves addressed in the present study, the WaveMIMO methodology was used. According to Machado et al. [14], this methodology consists of imposing discrete data of the wave propagation velocity in the horizontal ($u_{IR}$) and vertical ($w_{IR}$) directions as a prescribed velocity boundary condition (BC) in a numerical wave channel. Therefore, it was necessary that realistic sea state data be processed using the Spec2Wave software, 1.2.1 version. In this software, the procedure proposed in Oleinik et al. [62] is used to transform the wave spectrum into a time series of FS elevations. This procedure approximates the spectral data through a finite sum of monochromatic waves, individually described by Airy’s Linear Wave Theory [63]:

$${\eta }_1={\displaystyle \frac{H}{2}}\mathit{cos}\left(kx-\omega t\right),$$

(9)

which allows the irregular FS elevation series to be decomposed into orbital velocity profiles $u_1$ and $w_1$ that vary with depth. Thus, the RI waves can be reproduced by applying these velocity profiles as BC in the numerical wave channel. The configurations for the computational modeling needed to employ the WaveMIMO methodology are presented in Section 3.1. More information regarding the treatment method of the spectral data can be found in Oleinik et al. [62].

The realistic data addressed in the present study comes from a database of the realistic sea state occurred in the coastal region of RS state in the year 2018. This database was generated through the TELEMAC-based operational model addressing wave action computation (TOMAWAC) spectral model. According to Awk [64], in this model, sea state data are obtained from an equation that represents the general behavior of wave propagation in an unsteady and inhomogeneous medium, given by:

$$Q\left(t,x,y,k_i,k_j\right)={\displaystyle \frac{\partial N}{\partial t}}+{\displaystyle \frac{\partial \left(\dot{x}N\right)}{\partial x}}+{\displaystyle \frac{\partial \left(\dot{y}N\right)}{\partial y}}+{\displaystyle \frac{\partial \left({\dot{k}}_{i}N\right)}{\partial k_i}}+{\displaystyle \frac{\partial \left({\dot{k}}_{j}N\right)}{\partial k_j}},$$

(10)

where Q is the source term (m²/rad); N represents the directional spectrum of wave action density (m²/Hz/rad); $k_i$ the component x of the wave number vector (m^–1); and $k_j$ the component z of the wave number vector (m^–1).

For the present study, the realistic sea state data considered refer to a point with geographic coordinates 32°11′24″ S, 52°04′45″ W, located 171.06 m away from the Molhes da Barra breakwater in the municipality of Rio Grande in southern Brazil. It is important to mention that this data is from 11 September, from 7:15 a.m. to 7:30 a.m., with more details about it presented in the next section.

2.2. Generation of Representative Regular Waves

Although a realistic sea state is composed of irregular waves, it can be represented through regular waves using statistical parameters. One of the possible metrics to address is the mean period ($T_m$) of the irregular waves, which essentially consists of an arithmetic average of the periods of the waves that make up the spectrum, weighted by their contribution to the spectrum’s energy [64]. Another statistical parameter to be considered is the significant height ($H_s$), which, according to Holthuijsen [65], is commonly used to describe a sea state, as it represents the waves causing the most significant surface agitation.

Thus, the statistical parameters of $H_s$ and $T_m$ referring to the selected local were analyzed to establish the regular waves related to this realistic sea state. Thereby, Figure 1 presents the bivariate histogram showing the occurrence of combinations between $H_s$ and $T_m$ and points out the most frequent one.

Using these characteristics and the depth ($h$) of the analyzed location, it is possible to establish the characteristics of the RR waves of this sea state. To do so, it is determined the wavelength, which is calculated using the dispersion relation [66]:

$$Q\left(t,x,y,k_i,k_j\right)={\displaystyle \frac{\partial N}{\partial t}}+{\displaystyle \frac{\partial \left(\dot{x}N\right)}{\partial x}}+{\displaystyle \frac{\partial \left(\dot{y}N\right)}{\partial y}}+{\displaystyle \frac{\partial \left({\dot{k}}_{i}N\right)}{\partial k_i}}+{\displaystyle \frac{\partial \left({\dot{k}}_{j}N\right)}{\partial k_j}},$$

(11)

where $\omega$ is the angular frequency (Hz); and $k$ is the wave number (m⁻¹), given, respectively, by:

$$\omega ={\displaystyle \frac{2\pi }{T}},$$

(12)

$$k={\displaystyle \frac{2\pi }{\lambda }},$$

(13)

where $\lambda$ is the wave length (m); and T is the wave period (s).

Thus, the characteristics of the RR waves generated in the present study are shown in

Table 2. Characteristics of the RR waves.

Characteristic	Nomenclature	Magnitude
Height	$H$ (m)	1.14
Length	$\lambda$ (m)	31.50
Period	$T$ (s)	4.50
Depth	$h$ (m)	13.29

, where $H$ stands for $H_s$ and $T$ for $T_m$. It should be emphasized that these characteristics are used in both the spatial and temporal discretization of the computational domain. In addition, these regular wave characteristics are similar to those observed in other parts of the world, such as China [67] and Spain [68].

According to Chakrabarti [69], the regular waves of Table 2 are defined by the 2nd order Stokes theory. Therefore, the FS elevations ($\eta$) caused by the RR waves addressed in the present study are analytically described as [70]:

$${\eta }_2={\displaystyle \frac{H}{2}}\mathit{cos}\left(kx-\omega t\right)+{\displaystyle \frac{H2kcosh\left(kh\right)}{16{sinh}^3\left(kh\right)}}\left[2+cosh\left(2kh\right)\right]\mathit{cos}\left[2(kx-\omega t)\right],$$

(14)

while the associated velocity potential ($\Phi$) is given by:

$$\Phi =-{\displaystyle \frac{H}{2}}{\displaystyle \frac{g}{\omega }}{\displaystyle \frac{\mathit{cosh}\left(kh+kz\right)}{\mathit{cosh}kh}}sin\left(kx-\omega t\right)-{\displaystyle \frac{3H^2\omega \mathit{cosh}[2k(h+z)]}{32{sinh}^4\left(kh\right)}}\sin [2\left(kx-\omega t\right)].$$

(15)

Furthermore, calculating the partial derivatives of $\Phi$ in the $x$ and $z$ directions, there are the regular wave propagation velocities, which are imposed as BC of velocity inlet in the numerical wave channel for the generation of RR waves as in [7,8,27]. Thus, these horizontal ($u$) and vertical ($w$) propagation velocities are given, respectively, by [70]:

$$u_2={\displaystyle \frac{H}{2}}gk{\displaystyle \frac{cosh(kz+kh)}{\omega cosh\left(kh\right)}}\mathit{cos}\left(kx-\omega t\right)+{\displaystyle \frac{H^2}{4}}\omega k{\displaystyle \frac{\mathit{cosh}[2k(h+z)]}{{sin}^4\left(kh\right)}}\mathit{cos}[2\left(kx-\omega t\right)],$$

(16)

$$w_2={\displaystyle \frac{H}{2}}gk{\displaystyle \frac{sinh(kz+kh)}{\omega sinh\left(kh\right)}}\mathit{sin}\left(kx-\omega t\right)+{\displaystyle \frac{H^2}{4}}\omega k{\displaystyle \frac{\mathit{sinh}[2k(h+z)]}{{cos}^4\left(kh\right)}}\mathit{sin}[2\left(kx-\omega t\right)].$$

(17)

2.3. Model Verification

To verify the numerical models employed for the generation and propagation of the RI waves and the RR waves inside a numerical wave channel, the FS elevation was monitored using probes at: $x=0$ m, in the wave generation zone, for the RI waves; and $x=29.37$ m, approximately one wavelength away from the beginning of the channel, for the RR waves. Then, the results obtained, i.e., the FS elevation data from each case, were compared with data of FS elevation coming from the TOMAWAC model in the case of RI waves, or with the results of the analytical solution (Equation (14)) in the case of RR waves. As for the quantitative analysis, two metrics were considered, the mean absolute error (MAE) and root mean square error (RMSE), which are given, respectively, as follows [71]:

$$\mathrm{M}\mathrm{A}\mathrm{E}={\displaystyle \frac{{\sum }_{i=1}^n{|O}_i-P_i|}{n}},$$

(18)

$$\mathrm{R}\mathrm{M}\mathrm{S}\mathrm{E}=\sqrt{{\displaystyle \frac{{\sum }_{i=1}^n{\left(O_i-P_i\right)}^2}{n}}},$$

(19)

where O_i represents the numerical value (m), obtained in the ANSYS-Fluent software; P_i is the reference value (m), which comes from TOMAWAC or Equation (14); and $n$ represents the total amount of data.

It is important to highlight that for RI waves, the verification was conducted solely through a comparison at the channel inlet (in $x=0$ m), which may seem somewhat unusual. However, based on the previous verification and validation results for the WaveMIMO methodology presented in Machado et al. [14], where the comparison for the FS elevation was made at $x=0$ m, as well as at the end of the wave channel through the spectral density. In the present study, only the comparison at the channel entrance was made; more details on the verification and validation of WaveMIMO can be found in Maciel et al. [15] and in Oleinik et al. [72]. Therefore, one can affirm that if the velocity profile was correctly imposed at the channel inlet, the waves are generated and propagated in an adequate way. Hence, it is possible to state that the numerical procedure was verified.

3. Problem Description

In this section, the studies conducted to achieve the objectives of this paper are described. In this sense, the first study presented is the one referring to the verification of the numerical models used for the generation of RI waves and RR waves (Section 3.1). Subsequently, the study of the geometric evaluation of the overtopping WEC device is presented (Section 3.2 and Section 3.3). Figure 2 illustrates the flowchart of this study.

3.1. Computational Domain for Model Verification

Two computational domains were considered for the studies carried out in the present paper. The first one, which can be observed in Figure 3, was employed to verify the accuracy of the numerical model employed. Thus, the generation of RI waves and RR waves were simulated in the wave channel without the presence of the overtopping WEC to analyze if the waves are being adequately reproduced. Moreover, Figure 3 highlights the main dimensions of the wave channel and the BC applied to the computational domain, in addition to the water level at rest (WLR).

Regarding the applied BCs, there are: in the lower portion of the left wall (red line), the velocity inlet; in the upper part of the left wall and at the top of the channel (blue dashed line), the pressure outlet as atmospheric pressure; at the bottom of the channel (black line), the non-slip and impermeability; on the right wall, also the pressure outlet, characterizing an open flow channel, where, however, a hydrostatic profile is imposed in order to prevent the wave channel from emptying. Thus, it was possible to employ the numerical beach tool (gray region), which was used to reduce the energy of the waves, avoiding their reflection when they reach the end of the channel.

Furthermore, it is worth noting that to use the WaveMIMO methodology [14], it is necessary to subdivide the region where the velocity inlet BC is imposed in smaller segments, where the discretized profiles of orbital velocities propagation of waves ($u_1$ and $w_1$) are applied. It should be mentioned that the calculation of the velocity vectors takes place in the Spec2Wave software, where their positions are defined. In accordance with the recommendations obtained in Paiva [73], the vectors were considered in the upper part of each segment of the prescribed velocity BC imposition region.

Therefore, in accordance with the recommendations of Paiva [73] for the generation of RI waves through the WaveMIMO methodology, the BC imposition region was subdivided into 15 segments. In this approach the first two segments, those closest to the WLR, have a size of $h/28$, while the remaining 13 segments, which go till the bottom of the channel, are sized $h/14$, as illustrated in Figure 4. It is worth emphasizing that in the present study this methodology is employed solely for the RI waves generation. For the RR waves generation, the BC imposition region consists of a straight segment (see Figure 3).

Regarding the dimensions of the wave channel, the length is $L_c=171.06$ m, which corresponds to the distance between the location selected for analysis of the realistic sea state and the Molhes da Barra breakwater; while the numerical beach region length was $L_B=2\lambda$, following the indication by Lisboa et al. [9]. Moreover, the height is $H_c=21.29$ m, while the depth varies from $h=13.29$ m to $h_f=10.54$ m, reproducing the bathymetry found in the Rio Grande coast at the bottom of the wave channel. It should be noted that for this purpose, the bathymetric data were obtained from digitized nautical charts from the Hydrography and Navigation Directorate of the Brazilian Navy [74].

As for the spatial discretization adopted for the first computational domain (see Figure 3), a stretched mesh was used, which has been employed in Machado et al. [14] as well as in studies considering different WECs, such as Gomes et al. [8], for the OWC device, and Hübner et al. [28], for the overtopping device. Therefore, the wave channel was subdivided vertically into three regions, namely: R1, which contains only air, discretized into 20 computational cells; R2, which contains the interface between air and water, discretized into 40 cells; and R3, which contains only water, discretized into 60 cells. Also, horizontally there is the R4 region, discretized into 50 computational cells per $\lambda$. In this sense, the regions described are illustrated in Figure 5.

However, it is worth highlighting that the FS region is discretized differently in the case of RI waves (see Figure 3 and Figure 4). Following the recommendation of Paiva [73], the R2 region (see Figure 5) is composed of 2 segments below the WLR, of size $h/28$, and 2 segments above the WLR, of the same size. In this approach, the segments closest to the WLR, those in the center of the FS region, are discretized into 20 cells, while the others are discretized into 10, totaling 60 computational cells in the region, as illustrated in Figure 6.

As for the temporal discretization, a time step of $\Delta t$ = 0.01 s was used. This value corresponds to $\Delta t=T/450$, being in agreement with the indication of Barreiro [75], which recommends $\Delta t$ between $T/300$ and $T/600$ for regular waves. Thus, the same $\Delta t$ was applied for both the RI and RR waves, since it is a more refined temporal discretization than needed for the WaveMIMO methodology, which is $\Delta t=0.05$ s according to Machado et al. [14]. Furthermore, in both studies, a total simulation time of 900 s was considered for the generation and propagation of RI waves and RR waves.

3.2. Computational Domain for Geometric Evaluation of the Overtopping Device

Regarding the second computational domain considered, Figure 7 illustrates the one employed in the geometric evaluation study, where the overtopping device is coupled to the end of the channel. In this case, the numerical beach region is absent, since the purpose of the simulations is that the waves reach the end of the numerical channel without major energy dissipations.

As before, the main dimensions and BCs applied to the wave channel are presented; the geometric characteristics of the WEC, in turn, are presented at a later moment. Moreover, the pressure outlet BC was applied at the bottom of the device’s reservoir (green line) in order to keep it empty, as in Koutrouveli et al. [23], and prevent overflow, which occurred in Hübner et al. [29]. In this last study, some of the evaluated cases overflowed before the simulation ended, making it necessary to reduce the total time considered.

Regarding the spatial discretization of the domain presented in Figure 7, the same previous configurations were used. However, the region above the ramp and the converter reservoir were discretized horizontally with mesh elements of size $\Delta x=0.05$ m, while, inside the overtopping device reservoir, a regular mesh with quadrilaterals of size $\Delta x=0.05$ m was used, as in Martins et al. [25]. Figure 8 presents in detail the mesh adopted for the present study in the device region. As before, regarding the temporal discretization, a time step of $\Delta t$ = 0.01 s was considered for the 900 s of simulation of RI waves and RR waves.

3.3. Geometric Evaluation of the Overtopping Device Through the Constructal Design

When applying the constructal design method, the geometry is deduced by a principle of maximizing performance while being subjected to constraints, varying parameters according to the defined degrees of freedom [4,5,6]. Therefore, according to Dos Santos et al. [76], to employ the constructal design in the geometric study of a physical system, it is necessary to define:

• Performance indicator: a quantity to be maximized or minimized, which, in the present study, is the amount of water mass that entered in the converter reservoir;
• Geometric restrictions: the parameters that shall be kept constant, which are here the areas of the wave channel and the device ramp;
• Degrees of freedom: the geometric parameter that will be varied, in this case, is the ratio between height and length ($H_1/L_1$) of the overtopping device ramp.

It should be mentioned that the total water mass ($M$) obtained was selected as the performance indicator since it directly correlates with the potential energy generation. According to the physical operating principle of the overtopping device, the water from the incident waves that enters the reservoir subsequently returns to the ocean, thereby activating low-head turbines and generating electricity. Thus, a higher $M$ means a greater amount of energy converted from the sea waves.

As for the overtopping device, the object of study in the present paper, Figure 9 presents its configuration, highlighting the geometric characteristics. Furthermore, the monitoring probe used to analyze the performance of the WEC is also represented (pink line), which remains aligned with the WLR in all cases evaluated, as in Barros et al. [27].

Regarding the dimensions adopted, the WEC studied by Barros et al. [27] was taken as a reference, which has a reservoir height $H_r=6.50$ m and length $L_r=20.00$ m, characteristics that are kept fixed. Concerning the submersion ($H_2$) of the device, it was established at $H_2=4.00$ m and kept fixed, being 0.50 m greater than that adopted in Barros et al. [27], due to the variation of the same magnitude in the depth of the location where the converter is considered. This geometry was used as a basis because it underwent a geometric evaluation [27], where it was subjected to regular waves with different characteristics from the RR waves employed in the present study.

As mentioned, the degree of freedom investigated was the ratio between the height and length of the device ramp, i.e., $H_1/L_1$. Therefore, two geometric area restrictions were considered. The channel area ${(A}_c)$, defined as:

$$A_c=H_c{L}_c,$$

(20)

which, in the present study, is $A_c=2736.96$ m²; and the device ramp area ($A_1$), given by:

$$A_1={\displaystyle \frac{H_1{L}_1}{2}},$$

(21)

which, in the present study, is $A_1=78.4455$ m², as in Barros et al. [27]. Then, it was possible to define $H_1/L_1$ by dividing both sides of Equation (20) by ${L_1}^2$ and isolating the degree of freedom, so it is obtained:

$$\left({\displaystyle \frac{H_1}{L_1}}\right)={\displaystyle \frac{2A_1}{{L_1}^2}},$$

(22)

allowing to determine the values of $H_1$ and $L_1$ for each ratio of $H_1/L_1$ considered. Also, a dimensionless area fraction can be defined by:

$$\varphi ={\displaystyle \frac{A_1}{A_c}},$$

(23)

being in the present study $\varphi =0.03$.

Additionally, restrictions were defined regarding the height of the device’s ramp, as illustrated in Figure 10. It was established that the part of the ramp above the WLR should be lower than $H_{max}$, as in Figure 10a, and higher than $H_s/4$, as in Figure 10b; being $H_{max}$ the module of the greatest crest or trough found in the series of FS elevations from TOMAWAC, which in the present study corresponds to $H_{max}=1.64$ m.

Thus, the maximum and minimum height of the overtopping device ramp were defined as follows:

$${\left(H_1\right)}_{max}<h_f+H_{max}-H_2,$$

(24)

$${\left(H_1\right)}_{min}>h_f+{\displaystyle \frac{H_s}{4}}-H_2.$$

(25)

Once the upper and lower limits of the height of the overtopping device ramp were defined, 13 ratios were established for the degree of freedom $H_1/L_1$, ranging from 0.30 to 0.42, which are presented in

Table 3. Values considered for the degree of freedom $H_1/L_1$investigated.

${\mathit{H}}_1/{\mathit{L}}_1$	${\mathit{H}}_1$ (m)	${\mathit{L}}_1$ (m)
0.30	6.8606	22.8685
0.31	6.9740	22.4967
0.32	7.0856	22.1424
0.33	7.1954	21.8043
0.34	7.3036	21.4813
0.35	7.4103	21.1726
0.36	7.5154	20.8760
0.37	7.6190	20.5920
0.38	7.7213	20.3192
0.39	7.8222	20.0570
0.40	7.9219	19.8047
0.41	8.0203	19.5617
0.42	8.1175	19.3274

, along with the dimensions adopted for $H_1$ and $L_1$ in each geometry. It is important to emphasize that a ratio higher than 0.42 or lower than 0.30 would infringe the maximum and minimum ramp height restrictions. Since the submersion of the device was kept fixed, lower values for $H_1/L_1$ would leave the device completely submerged, while higher values would turn the device into a breakwater. In both cases, the device would cease to function as a WEC.

It should be mentioned that the importance of evaluating the degree of freedom $H_1/L_1$ lies in the fact that how waves break on a structure, such as an overtopping device, depends on the wave conditions and the slope of the ramp. Thus, Figure 11 illustrates the effect of varying the degree of freedom investigated on the design of the device geometry, where it is possible to observe changes in the slope of the WEC ramp when considering both extremes and an intermediate geometry.

Thereby, with variations in the degree of freedom $H_1/L_1$, it was possible to analyze the influence of the device geometry over the performance of the converter subject to RI and RR waves occurring in the municipality of Rio Grande, in addition to determining the best geometric configuration. As mentioned, the performance of the overtopping WEC geometries is evaluated by the amount of water mass that entered the device’s reservoir. Therefore, a mass flow monitoring probe was used at the reservoir inlet (see the pink line in Figure 9), which made it possible to calculate the total water mass, given as follows:

$$M={\displaystyle \frac{1}{t_f}}{\int }_0^{t_f}\dot{M}dt,$$

(26)

where, $M$ is the total mass of water that entered in the reservoir (kg); $\dot{M}$ is the instantaneous mass flow rate (kg/s); and $t_f$ is the time interval (s) between the first overtopping registered and the end of the simulation.

4. Results and Discussions

In this section, the results of the conducted studies are presented and discussed. To do so, it is divided into four sections. The first results presented refer to the verification of the numerical models employed (Section 4.1), followed by the geometric evaluation of the overtopping device (Section 4.2). Then, there is a more detailed analysis of the results monitored for selected geometries (Section 4.3). Finally, a graphical representation of the physical phenomenon is shown (Section 4.4).

4.1. Wave Generation Models Verification

As mentioned, the initial results presented are from studies verifying the numerical models used for generating and propagating RI waves and RR waves, which were simulated in channels without the presence of the overtopping WEC device. In this context, Figure 12a shows the qualitative comparison between the irregular FS elevation monitored in ANSYS-Fluent and the irregular FS elevation data from the TOMAWAC spectral model over the 900 s simulated, while Figure 12b focuses on the first 300 s of the simulation in order to provide a clearer visualization of the results.

As might be seen in Figure 12, the RI waves generated through the WaveMIMO methodology adequately reproduced the sea state found off Rio Grande. However, one can note that the waves generated in the ANSYS-Fluent software do not reach some of the crests and troughs present in the FS elevation from the TOMAWAC model, a phenomenon also observed in [14,73]. Furthermore, the effects of the initial inertia of the fluids were not observed in the generation of RI waves due to the fact that the position of the monitoring probe is in the wave generation zone, i.e., at $x=0$ m. As for the quantitative evaluation of the results, metrics $\mathrm{M}\mathrm{A}\mathrm{E}=0.101$ m and $\mathrm{R}\mathrm{M}\mathrm{S}\mathrm{E}=0.131$ m were obtained, indicating good results when compared to those present in the literature.

Continuing the verification of the numerical models, regarding the RR waves, Figure 13a presents the qualitative comparison between the FS elevation monitored in ANSYS-Fluent and the results obtained analytically (Equation (14)) over the 900 s of simulation; as done before, the first 300 s of the simulation are highlighted in Figure 13b to provide a better visualization of the results found.

Analogously to the previous case, it is possible to observe in Figure 13 that the generation of RR waves of the sea state that occurred in Rio Grande accurately reproduces the analytical waves; however, a difference is noted in the first 10 s of simulation. This is due to the fact that the FS elevation was monitored at $x=29.37$ m and, due to the initial condition of fluid inertia, the flow starts from rest, which causes damping in the first waves generated in the channel. Thus, in order to verify this numerical model, the results monitored between $10\le t\le 900$ s were considered, achieving metrics of $\mathrm{M}\mathrm{A}\mathrm{E}=0.062$ m and $\mathrm{R}\mathrm{M}\mathrm{S}\mathrm{E}=0.085$ m, indicating, once again, good quantitative results when compared to those present in the literature.

After that, for the visualization of the physical phenomenon reproduced numerically, in Figure 14 the volumetric fraction field is presented, where the water phase is represented in blue, while the air phase is in red. In this sense, Figure 14a shows the initial instant at $t=0$ s where the fluids are at rest, i.e., there are no FS elevations. This fluid-dynamic behavior occurs due to the initial condition of inertia applied for both wave approaches. Furthermore, the final moments of generation and propagation of the waves, at $t=900$ s, are depicted in Figure 14b and Figure 14c, respectively, for the RI and RR waves.

Therefore, it is possible to observe in Figure 14 that at the end of each simulation, at $t=900$ s, the difference between the wave approaches addressed is evident. In Figure 14b there is irregular behavior in the FS elevations, reproducing the realistic sea state that occurred in Rio Grande in 2018. However, in Figure 14c a regular oscillatory behavior is observed, which, in turn, represents the same sea state through the combination of the most frequent $H_s$ and $T_m$ characteristics. Moreover, it can be seen in Figure 14b,c the operation of the numerical beach tool at the end of the numerical channel, where there are no FS elevations, since the waves reaching this region are damped.

Thereby, based on the results of MAE and RMSE, as well as from Figure 12, Figure 13 and Figure 14, one can infer that the computational models for the generation of RI and RR waves were properly verified. Regarding the computational modeling of the overtopping device operation, it is important to emphasize that the numerical model used in the present study is based on the model validated by Goulart et al. [28], where a geometric evaluation was carried out through experimental and numerical studies. To do so, a numerical wave channel, with the device inserted in it, was simulated considering regular waves with the same parameters as laboratory experiments. The numerical model proposed was validated regarding the wave propagation, instantaneous mass flow rates, and the effect of $H_1/L_1$ on the device performance.

4.2. Geometric Evaluation of the Overtopping Wave Energy Converter Device

Once the numerical models employed were verified, it was possible to carry out the geometric investigation of the overtopping WEC. First of all, it is worth highlighting that the constructal design method is a method for geometric evaluation. In this study, it allowed to analyze the influence of an overtopping device’s geometry on its performance. Thus, a qualitative analysis of the results was performed to evaluate their tendencies.

In this context, Figure 15 shows the influence of the degree of freedom $H_1/L_1$ on the total mass of water ($M$) entering the device’s reservoir when subjected to RI waves and RR waves from the sea state occurring near the Molhes da Barra breakwater in Rio Grande—RS. Highlighting that as this is a two-dimensional domain, the third dimension is unitary, i.e., it was considered 1 m in the y direction.

As can be seen in Figure 15, regarding the geometric evaluation of the overtopping WEC device, there is an inversely proportional behavior between the total water mass and the values considered for the degree of freedom. This trend is explained since a greater ratio for $H_1/L_1$ means that a greater portion of the ramp is above the WLR, making it more difficult for the waves to overtop it. This kind of behavior has been previously seen in numerical studies, such as [25,27,41] as well as in [28], which is a numerical and experimental study. However, it remains valid to investigate the ratio $H_1/L_1$ when considering a different wave climate, as the amount of water entering the device’s reservoir is associated with the way the wave breaks on the WEC ramp. This, in turn, depends on both the wave characteristics and the structure’s geometry. For instance, in da Silva [51], two geometric evaluations of the overtopping device were carried out through the constructal design. In each evaluation, a different wave climate and device geometries were considered; in both cases, the presence of local minima was found for the lowest values of $H_1/L_1$.

In other words, the curve that illustrates the influence of the degree of freedom $H_1/L_1$ on $M$ has the tendency of a descending function. Thus, for both the RI waves and RR waves, the best geometry is that with (${H_1/L_1)}_o=0.30$, while the worst one has $H_1/L_1=0.42$. Moreover, when observing the curves illustrated in Figure 15, it is noted that they are, essentially, parallel. Other than showing a decreasing trend, for instance, it is possible to observe a change in their inclinations recorded for the same geometry, being the one with $H_1/L_1=0.35$. The parallel behavior, however, does not occur for the higher values of $H_1/L_1$ (from 0.39), when the device performance shows more stability under the incidence of RR waves than for RI waves.

Additionally, as noted in Figure 15, all geometries evaluated obtained higher $M$ when subjected to RI waves than to RR waves. However, it is important to mention that this behavior is the opposite of what was found in Hübner et al. [29], where a single geometry of the overtopping device was subjected to different wave climates found in three coastal cities of the RS state. To better understand this phenomenon, the instantaneous mass flow rates of the water entering the device’s reservoir are analyzed later, for selected geometries. Despite the differences in the magnitude of $M$ obtained for each wave approach, the curves approximate for geometries with the higher values investigated, i.e., $0.40\le H_1/L_1\le 0.42$. As previously mentioned, the higher the $H_1/L_1$ ratio is, the greater the portion of the ramp above the WLR, making wave overtopping more difficult, which explains the tendency of proximity between the results obtained with the RI and RR waves approaches.

To conduct a quantitative evaluation of these results,

Table 4. Quantitative comparison of the influence of $H_1/L_1$ on the $M$ that entered in the overtopping device’s reservoir.

${\mathit{H}}_1/{\mathit{L}}_1$	$\mathit{M}—$RI Waves (kg)	${\mathit{D}}_{\mathit{R}}$ (%)	$\mathit{M}—$RR Waves (kg)	${\mathit{D}}_{\mathit{R}}$ (%)	${\mathit{D}}_{\mathit{W}}$ (%)
0.30	200,820.77	-	144,054.72	-	–28.26
0.31	182,093.85	–9.32	128,128.68	–11.05	–29.63
0.32	161,409.70	–11.35	109,880.09	–14.24	–31.92
0.33	144,289.64	–10.60	94,068.96	–14.38	–34.80
0.34	128,352.21	–11.04	79,899.03	–15.06	–37.75
0.35	120,552.95	–6.07	72,727.34	–8.97	–39.67
0.36	106,433.13	–11.71	60,828.26	–16.36	–42.84
0.37	96,307.13	–9.51	50,905.02	–16.31	–47.14
0.38	85,269.83	–11.46	41,566.58	–18.34	–51.25
0.39	76,661.55	–10.09	37,428.83	–9.95	–51.18
0.40	63,339.88	–17.38	30,986.83	–17.21	–51.07
0.41	57,760.40	–8.80	29,009.56	–6.38	–49.77
0.42	49,925.58	–13.56	26,215.31	–9.63	–47.49

presents the $M$ values achieved for each geometry of the overtopping device over the 900 s of generation and propagation of RI waves and RR waves that occurred in Rio Grande—RS. In this context, Table 4 also presents relative differences ($D_R$) found between the results obtained, which indicate the variation of $M$ between the geometries evaluated. Therefore, the performance of each geometry is always compared with that located on the line immediately above in the table. This quantification elucidates the effect of the investigated degree of freedom on the converter’s performance. Furthermore, the difference regarding the wave approaches ($D_W$) is also found in Table 4, where the values of $M$ obtained in cases with the same $H_1/L_1$ are compared, taking the RI waves approach as the reference.

Therefore, observing the variations of $D_R$ in Table 4, it is possible to identify the tendencies previously observed for the curves that describe the influence of the device geometry on $M$ (see Figure 15). It is noted that, when considering the RI waves, the variations in the total water mass that entered the device reservoir remain around 11% for most of the geometries evaluated. An exception occurs for $H_1/L_1=0.35$, where there is a smaller drop, corresponding to the change observed in the slope of the blue curve in Figure 15. Other examples occur for the geometries with $H_1/L_1=0.40$ and $H_1/L_1=0.42$, where there are sharper drops in the values obtained for $M$.

Regarding the incidence of RR waves, the variations are more unstable and generally higher than those obtained for the RI waves; however, it is possible to say that they were around 15% for most geometries. In this case, the exceptions refer to lower variations than the rest, such as for the geometry with $H_1/L_1=0.35$, where a change in the slope of the red curve illustrated in Figure 15 was previously highlighted. Additionally, low variations were recorded for some geometries with higher $H_1/L_1$ ratios, where the device showed stability in its performance, such as for $H_1/L_1=0.39$, $H_1/L_1=0.41$, and $H_1/L_1=0.42$.

In turn, observing the variations in $D_W$ in Table 4, one can note that the differences caused by the wave approach are accentuated as the height of the ramp and its portion above the WLR increase. This occurs until it becomes more difficult for the waves to overtop the ramp and enter the device reservoir. Thus, $D_W$ varies between approximately 28% for $H_1/L_1=0.30$ and 51% for $H_1/L_1=0.38$. Then, from $H_1/L_1=0.39$, it reverses the growth trend and gradually decreases to 47% at $H_1/L_1=0.42$, which is when the curves approximate in Figure 15. Despite the proximity illustrated in Figure 15 for the higher values of the degree of freedom ($0.40\le H_1/L_1\le 0.42$), the quantitative evaluation shows that the difference in the overtopping device performance caused by waves approached is relatively smaller considering the lowest ratios ($0.30\le H_1/L_1\le 0.32$). This highlights the importance of performing both qualitative and quantitative analyses when applying the constructal design method.

Therefore, the constructal design method allowed finding the best geometry for the device, that is, the one that maximizes the $M$ that entered in the overtopping WEC reservoir. Accordingly, it is highlighted that, for both wave approaches, the best geometry was the one with (${H_1/L_1)}_o=0.30$ ($H_1=6.8606$ m and $L_1=22.8685$ m); while, the worst geometry was the one with $H_1/L_1=0.42$. Thus, considering the incidence of RI waves, $M=\mathrm{200,820.77}$ kg entered the converter’s reservoir. This amount of water is 4.02 times superior to the total obtained for the worst geometry evaluated. On the other hand, when RR waves are considered, $M=\mathrm{144,054.72}$ kg was obtained in the device’s reservoir, showing an even greater difference, corresponding to an amount 5.49 times of what was obtained by the worst geometry evaluated.

4.3. Detailed Analysis of the Monitored Results

In the present section, a detailed analysis of the results monitored by the probe at the entrance of the device’s reservoir is carried out. Thus, the overtopping device’s performance was analyzed considering the best geometry, (${H_1/L_1)}_o=0.30$; a geometry with intermediate performance, $H_1/L_1=0.35$; and the worst-performing geometry, $H_1/L_1=0.42$ (see Figure 11). In this context, Figure 16 shows the instantaneous mass flow rates ($\dot{M}$) entering the device’s reservoir, considering (a) RI waves and (b) RR waves.

The first noticeable aspect in Figure 16 is the difference in the scale of the $\dot{M}$ peaks, which are higher considering the incidence of RI waves. Additionally, the frequency of the flows also differs; in this case, it is higher with the incidence of RR waves. Thus, it can be observed that the hydrodynamic behavior of the overtopping device varies depending on the wave approach. There are flows of variable magnitude, with higher peaks but less frequent occurrences for RI waves. In contrast, there are lower magnitude flows, practically constant, occurring more frequently with RR waves. It can be partially explained by observing Figure 12 and Figure 13, in Section 4.2. As seen in Figure 12, there are moments where the FS elevation reached for the RI waves exceeds what is achieved for the RR waves, shown in Figure 13; these waves cause the highest $\dot{M}$ peaks. Furthermore, there are also times when the elevations shown in Figure 12 (RI waves) are lower than those in Figure 13 (RR waves); these waves are not able to overtop the ramp, thus causing the difference in flow frequency.

Another point that should be highlighted is that when considering RR waves, the peaks of mass flow rates decreased over time for geometries with lower $H_1/L_1$ ratios, as seen for (${H_1/L_1)}_o=0.30$ in Figure 16b; while they maintained a more constant behavior for geometries with higher $H_1/L_1$ ratios, as seen for $H_1/L_1=0.42$ in Figure 16b. Thus, one can infer that those geometries with lower ramps suffered more from the effect of wave reflection. Since a greater amount of water mass was frequently able to overtop the ramp, the subsequent waves faced reflection. On the other hand, this behavior does not occur when considering the incidence of RI waves, even for geometries with lower $H_1/L_1$ ratios, as observed in Figure 16a. Since water mass overtopping is less frequent due to waves that are unable to overtop the ramp, the subsequent waves do not face the same effects of reflection. Thus, large peaks of water mass entering the device’s reservoir can be observed throughout the simulation.

In a quantitative analysis of the results, considering the RI waves (Figure 16a), the first overtopping, or water mass flow rates, occurred close to: $t=20$ s for $({H_1/L_1)}_o=0.30$; $t=40$ s for $H_1/L_1=0.35$; and $t=55$ s for $H_1/L_1=0.42$. Besides that, the largest mass flow peaks occurred in different instants for the three geometries considered, which correspond to: $\dot{M}=3295.17$ kg/s at $t=434.34$ s, for (${H_1/L_1)}_o=0.30$; $\dot{M}=2748.39$ kg/s at $t=418.80$ s, for $H_1/L_1=0.35$; and $\dot{M}=2174.47$ kg/s at $t=261.14$ s, for $H_1/L_1=0.42$. Thus, it was noted that for (${H_1/L_1)}_o=0.30$ and $H_1/L_1=0.35$ these peaks occurred close to 50% of the total simulation time, while for $H_1/L_1=0.42$ it was close to 30%. Furthermore, the difference in the magnitude of the peaks was practically the same when comparing one geometry with the other, being around $\dot{M}=550$ kg/s.

Regarding the incidence of RR waves (Figure 16b), the first overtopping also occurs first for the best geometry, (${H_1/L_1)}_o=0.30$. However, the range for the other geometries is smaller, it started close to: $t=36$ s for (${H_1/L_1)}_o=0.30$; $t=41$ s for $H_1/L_1=0.35$; and $t=50$ s for $H_1/L_1=0.42$. Moreover, the largest mass flow peaks were monitored close to $t=50$ s for the three geometries covered in this analysis. These peaks correspond to: $\dot{M}=1258.65$ kg/s at $t=50.02$ s, for (${H_1/L_1)}_o=0.30$; $\dot{M}=934.81$ kg/s at $t=54.25$ s, for $H_1/L_1=0.35$; and $\dot{M}=948.83$ kg/s at $t=50.01$ s, for $H_1/L_1=0.42$. Thus, it is important to mention that the worst geometry, $H_1/L_1=0.42$, recorded a higher $\dot{M}$ peak than the intermediate geometry, $H_1/L_1=0.35$.

Continuing with the analysis of the overtopping WEC performance, Figure 17 depicts the evolution of the amount of water mass ($M$) that entered in the overtopping device’s reservoir over the 900 s simulation for both wave approaches considered. The same geometries were considered for this analysis. Moreover, the fitted curves for the growth of $M$ (dotted black lines) are also included in Figure 17.

As expected, the curves illustrated in Figure 17 differ depending on the wave approach used. For the RI waves (Figure 17a), the curves resemble steps, with plateau intervals where $M$ remains constant between growth intervals. For instance, some examples of intervals where it occurs are: 485.89 s$\le t\le$505.29 s, for (${H_1/L_1)}_o=0.30$; 377.90 s $\le t\le$ 387.31 s, for $H_1/L_1=0.35$; and 162.53 s$\le t\le$186.62 s, for $H_1/L_1=0.42$. On the other hand, the amount of $M$ increases steadily over time when considering the RR waves (Figure 17b). This is a direct consequence of what was observed in Figure 16,b, where it was noted that the frequency of wave overtopping depends on the wave approach. For the same reason, this behavior was observed in Hübner et al. [30]. Additionally, it is possible to observe that RI waves led to better results, as previously presented (see Figure 15 and Table 4 for full results). Thus, it is safe to state that the higher frequency of overtopping considering the incidence of RR waves was not enough to compensate for the higher mass flow rates $\left(\dot{M}\right)$ obtained when the overtopping device was subjected to RI waves.

To draw the fitted curves shown in Figure 17a,b, a first-degree polynomial was defined, relating the amount of water entering the reservoir with the time elapsed, which characterizes a linear correlation between the variables [77]. Thus, it was determined the correlation coefficient ($R^2$) for each polynomial. In this context,

Table 5. Statistical parameters of fitting curves.

Geometry	Wave Approach	First-Degree Polynomial	${\mathit{R}}^2$
(${H_1/L_1)}_o=0.30$	RI	$M_p=233.27t-4161.20$	0.9979
	RR	$M_p=165.44t+708.64$	0.9952
$H_1/L_1=0.35$	RI	$M_p=141.17t-5629.30$	0.9964
	RR	$M_p=82.17t-951.74$	0.9992
$H_1/L_1=0.42$	RI	$M_p=57.96t-2639.10$	0.9925
	RR	$M_p=29.79t-23.856$	0.9956

presents the equations for the polynomial approximation of the water mass ($M_p$), as well as the correlation coefficient for the highlighted geometries.

The results from Table 5 indicates that the doted black curves presented in Figure 17 are an adequate approximation to the numerical results found in the geometric evaluation. Since $R^2\ge 0.99$, one can infer that all the fitted curves represent a strong linear relationship between $M_p$ and t [77]. Thus, it allows the use of the polynomial approximation for the water mass that enters the reservoir in future studies considering longer activity times of the overtopping device.

4.4. Visualization of Physical Phenomenon

Finally, Figure 18 shows the behavior of wave flow over the best geometry evaluated for the overtopping device, for both wave approaches, at the start and end of the simulations. As before, the water phase is represented in blue while the air phase is represented in red. Additionally, the overtopping WEC ramp and the bottom of the reservoir are represented in gray.

Thus, there are the volumetric fraction field for (${H_1/L_1)}_o=0.30$ subjected to RI waves and RR waves, respectively, at: $t=0$ s, in Figure 18a,b; and $t=900$ s, in Figure 18c,d. It is possible to observe in Figure 18a,b the initial instant of the simulations, where no waves were impinging on the device due to the initial condition of fluids at rest in the channel. On the other hand, Figure 18c shows a wave reaching the overtopping WEC device at the final instant of the simulations, while Figure 18d shows a wave returning down the ramp after its overtopping. As seen in Figure 16b, in Section 4.3, there was a mass flow entering the device’s reservoir in the final moment of the simulation of the RR waves.

Additionally, it should be noted that due to the pressure outlet BC imposed at the bottom of the reservoir, there is no accumulation of water. Thus, it is recommended that in future studies, the overtopping device reservoir should be reduced in order to decrease the total processing time of the simulations. Since it is a very refined region of the computational domain, discretized with quadrilateral computational cells of size $\Delta x=0.05$ m (see Figure 8), reducing this area will result in a significant reduction in the total number of computational cells employed.

5. Conclusions

The present study aimed to investigate the influence of the geometry of the overtopping WEC device on its performance when subjected to RI waves and RR waves from the sea state occurring near the Molhes da Barra breakwater in Rio Grande—RS, southern Brazil, in 2018. Thus, in both studies, numerical simulations were conducted to generate and propagate waves impinging on the overtopping WEC in a channel. To do so, the ANSYS-Fluent software was employed, which is based on the FVM and applies the multiphase VoF model to treat the interface between the air and water.

The numerical models used were verified quantitatively through the MAE and RMSE metrics, and qualitatively through a visual comparison. Both the RI waves and the RR waves generated numerically provided an accurate representation of the physical phenomenon addressed. It was found that the geometries that maximize the overtopping device’s performance are the same for both wave approaches. Additionally, it was noted that using RR waves underestimates the device’s performance.

Regarding geometric evaluation, constructal design was applied in order to investigate the influence of the degree of freedom $H_1/L_1$ on $M$, that is, the influence of the ratio between the height and length of the ramp on the total mass of water that entered the overtopping device reservoir. Thus, it was possible to infer that the best geometry evaluated was (${H_1/L_1)}_o=0.30$, which has $H_1=6.8606$ m and $L_1=22.8685$ m; while the worst was the one with $H_1/L_1=0.42$. That is, it was observed an inversely proportional behavior between the total water mass and the values considered for the degree of freedom, as seen in [25,27,28,41]. As for the total mass of water that entered the reservoir throughout the 900 s of simulation, $M=\mathrm{200,820.77}$ kg was obtained for the RI waves, an amount 4.02 times higher than the worst geometry analyzed. $M=\mathrm{144,054.72}$ kg was obtained for the RR waves, an amount 5.49 times greater than the worst geometry analyzed.

As noted, all the geometries analyzed had their performance underestimated when subjected to RR waves, that is, all obtained higher M when subjected to RI waves. For instance, considering the optimized geometric configuration of the overtopping WEC, the RI waves achieved a total mass of water in the reservoir approximately 40% higher than the water amount reached by the RR approach. However, as mentioned, this behavior is the opposite of what was found in Hübner et al. [29]. This highlights the importance of geometric evaluation studies for WECs being carried out considering the realistic sea state corresponding to the location where they will be installed.

Finally, regarding future studies, it is suggested to investigate the influence of the device’s submersion ($H_2$) preserving the maximum and minimum height restrictions ($H_1$) of the ramp that were established based on the characteristics of the realistic sea state addressed. Thus, different values for the area of the converter ramp ($A_1$) can be evaluated, leading to different values for the area fraction ($\varphi$). Additionally, it is suggested to carry out analyses considering the sea state in other regions of Rio Grande do Sul, with the aim of identifying whether there is an ideal geometry to be reproduced along its coast.