1. Introduction
The global energy sector is undergoing a critical transition as societies confront the dual challenges of climate change and the depletion of fossil fuel reserves. Growing environmental concerns about greenhouse gas emissions and ecological degradation, coupled with the finite nature of fossil resources, have driven nations to explore and integrate renewable alternatives into their energy matrices [
1]. Among these, oceans represent an underexploited renewable resource, with wave energy standing out as particularly promising. Numerous global studies have assessed and quantified this potential across different countries, including Mexico [
2], Algeria [
3], France [
4], Indonesia [
5], India [
6], Argentina, Brazil, Chile, Colombia, Uruguay [
7], Australia [
8], China [
9], among others.
The energy contained in ocean waves can be extracted through converter devices that transform kinetic energy into electricity, such as oscillating water column (OWC), oscillating bodies, overtopping devices, and submerged horizontal plates (SHP) [
10,
11], along with other proposed technologies. These devices have been investigated through both experimental and numerical approaches. According to the International Energy Agency [
12], beyond laboratory testing, multiple prototypes are currently under development or testing worldwide, including in China, South Korea, Singapore, Australia, New Zealand, the United States, Ireland, France, and Portugal.
Numerical modeling offers a cost-effective alternative for studying wave energy converters, requiring fewer financial and human resources. Consequently, numerous numerical studies have investigated the performance of wave energy conversion devices. Examples include OWC [
13,
14,
15,
16,
17,
18,
19,
20,
21], overtopping [
22,
23,
24,
25,
26,
27,
28,
29,
30], SHP [
11,
31,
32,
33], and even hybrid devices combining multiple conversion principles [
34,
35,
36]. However, to enable reliable numerical investigations, it is crucial to accurately understand and reproduce wave behavior, particularly irregular waves, which represent real ocean conditions. Thus, developing robust numerical modeling methodologies for this phenomenon remains essential.
Numerical wave generation has evolved through diverse approaches, each addressing specific challenges in wave propagation modeling. For regular waves, conventional methodologies have been rigorously tested, such as Lisboa et al. [
37], who performed numerical simulations using ANSYS-Fluent software to quantify the effects of linear and quadratic damping coefficients in numerical beaches, demonstrating significantly reduced wave reflection. Zabihi et al. [
38] conducted a benchmark study between ANSYS-Fluent and Flow-3D, showing both models are powerful tools for wave generation. Windt et al. [
39] presented a validation of a numerical wave tank model demonstrating strong agreement between simulated and experimentally monitored free surface elevations. Jiang et al. [
40] studied regular wave propagation on a sloping bed under steady wind action, analyzing nearshore wave-wind interaction mechanisms; the results indicated that wind accelerates wave breaking, increases the breaker index, and enhances turbulence.
Regarding irregular wave generation, Oh et al. [
41] developed a nonlinear wave simulation code using a higher-order spectral (HOS) method, employing fourth-order Runge–Kutta time integration and anti-aliasing zero-padding for stability; the results confirmed its efficiency in predicting nonlinear wave interactions. Choi et al. [
42] presented a methodology for simulating nonlinear irregular waves using the HOS method, validated against extreme wave events; the results showed strong agreement between OpenFOAM, HOS, and experimental data in predicting wave breaking and elevations. Kim et al. [
43] introduced a unified nonlinear wave generation and validation procedure for numerical and experimental wave tanks; a comparative analysis of mild and extreme breaking waves demonstrated strong agreement across stochastic and deterministic metrics between HOS simulations, OpenFOAM, and experimental results.
Moreover, Kim et al. [
44] presented a real-time wave forecasting system for directional seas using Lagrangian-based models, enabling efficient processing of sparse optical measurement data; experimental validation confirmed that directional wave components must be considered to achieve accuracy comparable to unidirectional forecasts. Canard et al. [
45] developed a wave generation method controlling both spectrum and crest statistics, enabling precise irregular wave field reproduction beyond spectral methods, particularly for unidirectional non-breaking conditions. Kim and Ducrozet [
46] advanced the complementary improved choppy wave model (CICWM) to address directional wave field limitations by incorporating essential nonlinear terms. Experimental validation demonstrated CICWM reduces linear model prediction errors in directional seas while maintaining real-time forecasting capability through simplified assimilation.
Another example is the WaveMIMO methodology, proposed and verified by Machado et al. [
47], which is the focus of the present study. This methodology uses realistic sea state data to generate irregular waves, enabling reproduction of natural wave phenomena. To do so, spectral data are processed to obtain discretized wave propagation velocity profiles, imposed as prescribed velocity boundary conditions. Notably, the methodology’s capability for regular wave generation was validated by Maciel et al. [
48] through numerical–experimental comparisons. Several studies have employed WaveMIMO to analyze wave energy converters fluid–dynamic behavior under realistic irregular waves, including geometric evaluations of the OWC [
16,
17], overtopping [
25,
30], and SHP [
33] devices. Furthermore, specific aspects of the WaveMIMO methodology have been investigated in dedicated studies: Mocellin et al. [
17] examined the influence of numerical wave channel bathymetry on realistic irregular wave generation, while Paiva et al. [
49] and Maciel et al. [
50] presented case studies evaluating key application parameters.
In this sense, the present study aims to establish general theoretical recommendations regarding the application of the WaveMIMO methodology [
47]. The importance of refining the WaveMIMO methodology lies in its potential to generate realistic irregular waves, thereby achieving a more accurate representation of natural wave phenomena. Therefore, studies were carried out to simulate realistic irregular waves based on the sea states found on the coast of the Rio Grande do Sul (RS), southern Brazil. Locations with different characteristics of significant height, mean period, and water depth were analyzed, enabling an evaluation of the methodology’s performance under different wave climates. As a final contribution, this study validates the refined methodology by comparing numerical results with experimental data from the literature, a novel aspect of this paper.
2. Mathematical and Numerical Modeling
To carry out this study, numerical simulations of the generation and propagation of realistic irregular waves in a wave channel were conducted using the ANSYS-Fluent 2024 R2 software [
51], based on the finite volume method (FVM). To treat the interface between the phases (water and air), the multiphase volume of fluid (VOF) [
52] model was used. The representation of the phases in the computational model requires the concept of volumetric fraction (α), where the sum of the fractions in each control volume must be equal to 1. Thus, each computational cell can assume three distinct states, that is, containing only water, where [
53]
containing only air, where [
53]
or, containing the interface between both phases, which are immiscible fluids, where [
53]
Furthermore, it is important to note that, when employing the VOF model, a single set of equations is solved, comprising the conservation equations for mass, volumetric fraction, and momentum. These equations are provided, respectively, by [
54]
where
is the fluid density (kg/m
3), calculated for the mixture of phases as [
53]
while
t is the time (s);
is the velocity vector (m/s);
p is the static pressure (Pa);
is the gravity acceleration vector (m/s
2); and
is the strain rate tensor (N/m
2). Additionally, it is important to note that the sink term (
S) refers to the numerical beach tool, which applies damping to minimize reflections, and is given by [
55,
56]
where
and
are the linear (s
−1) and quadratic (m
−1) damping coefficients, respectively;
is the velocity along the
z direction (m/s);
and
are the vertical positions of the free surface (FS) and the channel bottom (m); and
and
are the starting and ending positions of the numerical beach (m). Lastly, it is important to highlight that the damping coefficients,
and
, are set to
s
−1 and
m
−1, respectively, as recommended by [
37].
In addition to the governing equations, the following boundary conditions (detailed later in
Section 4.1) were applied to model the numerical wave channel:
Prescribed velocity: where discretized orbital velocity profiles are imposed (WaveMIMO methodology);
Pressure outlet: atmospheric pressure was considered, i.e., kPa;
No-slip and impermeability: where null velocities were considered, m/s.
Numerical Methods Applied
To conduct numerical simulations in the ANSYS-Fluent software, specific set of configurations were defined to solve the governing equations of the problem. The pressure-implicit with splitting of operators (PISO) scheme employed for the pressure–velocity coupling, while the pressure staggering option (PRESTO) scheme was used for pressure spatial discretization. These approaches are summarized in
Table 1, along with other numerical methods used in this study.
It is highlighted that the flow was considered under a laminar regime, which, according to Gomes et al. [
57], is a simplification that does not cause significant accuracy loss while reducing computational processing time. The simulation was initialized with inertia-dominated flow conditions. Additionally, the numerical procedures adopted were adapted from methodologies established in previous studies, such as those by [
17,
47,
48,
49].
6. Conclusions
The present study systematically evaluated the application parameters of the WaveMIMO methodology [
47] through four sequential investigations of realistic irregular wave generation. Numerical simulations were performed using wave climate data from coastal areas of the Brazilian state of Rio Grande do Sul near the cities of Rio Grande and Tramandaí, with comparative analysis incorporating previous results from the coastal region of city of Mostardas [
49]. These assessments enabled the development of theoretical recommendations for implementing the WaveMIMO methodology in the realistic irregular wave generation. Final validation was achieved through qualitative and quantitative comparison with laboratory experimental data from Koutrouveli et al. [
34], confirming the methodology’s effectiveness for both wave generation and propagation of these waves.
Among these studies, there are those that focus on the spatial discretization of the computational domain employed, such as the investigation into the discretization of the prescribed velocity BC imposition region. Based on the evaluations and analyses conducted, it was determined that the best results were achieved in Case 1, where it was subdivided into 14 segments. In another study, the mesh sensitivity in the FS region was analyzed. It was possible to establish Case 5 as a theoretical recommendation, where the central segments of the FS region were discretized with greater refinement, totaling 60 computational cells. Additionally, the temporal discretization was investigated, where the time step was related to the mean period. In this evaluation the processing time required was considered to determine the most suitable relation. Thus, the recommendation of using , referred to as Case 9, was established. The final study addressed the evaluation of the velocity vector’s location. As a result, Case 11 was determined as the theoretical recommendation, which consists of placing the vector at the upper part of each segment of the prescribed velocity BC imposition region.
In summary, when applying the WaveMIMO methodology for realistic irregular wave generation, the prescribed velocity BC region should comprise 15 segments: the first two, located just below the WLR, with a size of , while the remaining 13 segments must have a size of . Thus, the FS region on the stretched mesh consists of the first two segments below the WLR and two equally sized segments above it. For spatial discretization, segments near the WLR should contain 20 computational cells, whereas the remaining segments should use 10 cells. The temporal discretization requires a time step of and the velocity vector should be considered at the top of each segment in the prescribed velocity BC region.
At last, it is worth highlighting that, when comparing the optimized configuration (Case 12) to recommendations from the previous literature (Case 1) [
47], the metrics were reduced by 7.58% (NMAE) and 8.36% (NRMSE) for Rio Grande, while Tramandaí’s sea state showed more modest improvements of 2.36% (NMAE) and 2.44% (NRMSE). Thus, after performing the investigations, it was possible to validate the WaveMIMO methodology against experimental data from Koutrouveli et al. [
34]. One can highlight that, in a general way, an excellent agreement between the irregular waves generated with WaveMIMO methodology and the experimental data was achieved, since most employed indicators were below 10%. For future studies, it is suggested to implement a prescribed velocity BC positioned in the upper section of the left wall of the channel. This approach could emulate the effects of wind on the FS of the channel, thereby improving the adjustment of missed crests and troughs, enhancing wave generation accuracy.