Constructal Design of an Overtopping Wave Energy Converter Incorporated in a Breakwater

Abstract

A numerical study was performed in the present work about an Overtopping Device Wave Energy Converter (OTD-WEC) with one and two ramps incorporated in a real breakwater. The Constructal Design method was applied to evaluate the effects on the average dimensionless overtopping flow of the degrees of freedom of the device with one and two ramps. In addition, a comparison was carried out among the different geometry configurations of the OTD-WEC to determine which one presents the best hydrodynamic performance. The work used the JONSWAP spectrum and the multiphase Volume of Fluid model. It also solved the conservation equations for mass, momentum, and an equation for the transport of the volume fraction using the Finite Volume Method. Results showed that a device with a two-ramps configuration presented an average dimensionless overtopping flow 6.48% higher than those obtained for the one ramp. Present results obtained using Constructal Design theoretical recommendations about the influence of a complex configuration with four degrees of freedom over the performance of an OTD-WEC integrated into the east breakwater of the city of São José do Norte, State of Rio Grande do Sul (RS), Brazil.

1. Introduction

The increase in global energy demand, over 25% until 2040, has led to searching for alternatives to meet the growing requisition for electricity. Therefore, many renewable energy technologies have been studied, and their contribution could exceed 40% by 2040. Ocean wave energy stands out for its vast potential [1]. However, it is necessary to employ wave energy converters (WECs) to harness this energy from the sea. The WECs have different working principles, being one of them the overtopping. In the Overtopping Device Wave Energy Converter (OTD-WEC), the ocean water waves overtop an inclined ramp, facilitating water flow and entering a reservoir raised above sea level. The accumulated water returns to the sea, passing through a low head hydraulic turbine, generating electricity [2,3,4,5,6,7].

Some relevant works have been performed about OTD-WEC in the last decades using experimental and numerical approaches, some considering the integration of the OTD into a breakwater, others employing the Constructal Design method for the geometric evaluation of the equipment, in addition to devices with one, two or more ramps.

Vicinanza et al. [8] conducted a study on a 1:30 scale overtopping device incorporated into a breakwater (OBREC—Overtopping Breakwater for Energy Conversion) concerning experiments dedicated to breakwaters. The device ramp angle was kept at 34° and two ramp heights were tested. The JONSWAP spectrum was used through the software AwaSys, and the water pressure on the structure of the device was evaluated. The results showed that the OBREC does not guarantee the same level of protection as a traditional breakwater, both with similar dimensions.

Iuppa et al. [9] studied parameters such as type of ramp, flat and curved, ramp height, and reservoir length; of an OBREC with two ramps of different dimensions side by side to determine the best performance of the device. The JONSWAP spectrum was used through WaveLab software. For the flat ramp, the inclination was kept at 34°. The results showed a reduction of approximately 20% in overtopping for the curved ramp compared to the flat ramp.

Regarding numerical work and breakwaters, Margheritini et al. [10] conducted a study about the geometry optimization of the Sea wave Slot cone Generator (SSG) OTD-WEC as part of the feasibility study for implementing the device in the development plan of Hanstholm harbor in Denmark, aiming to present the expected power production, the overall performance and the price per kWh. The results showed that for the two ramps SSG, the expected power production was estimated to be 7.08 GWh/year, and for the three ramps SSG, it was 12.01 GWh/year.

Musa et al. [11] performed a 3D numerical OBREC evaluation study for the east coast of Malaysia, investigating geometric parameters, produced power, and economic aspects. The Flow3D software and the model of the overtopping device on a scale of 1:30 were used in the research. In addition, the JONSWAP irregular wave theory was used to generate the waves. The value of 34° was used for the slope of the ramp. The results showed that lower ramp heights than the average water level provide more significant overtopped water.

Lauro et al. [12] carried out a 2D numerical study on integrating OTD-WEC in a vertical breakwater (OBREC-V). They analyzed the performance of the equipment regarding wave reflection, the influence of geometry on wave overtopping, and wave loads exerted on the device on a scale of 1:30. The device analyzed has a single closed reservoir, i.e., no water outlet for the device. The ramp slope was maintained at 34°. Irregular waves were used according to the JONSWAP spectrum and the numerical model IH2VOF. The results showed that the integration of the overtopping device to the breakwater increases the stability of the structure. In addition, there is an approximate 40% reduction in reflection when using OBREC-V.

Numerical simulations of multi-level OWEC integrated with breakwaters were performed by Jungrungruengtaworn et al. [13] in a three-dimensional wave channel using the commercial Computational Fluid Dynamics (CFD) solver ANSYS FLUENT V17. The device has three ramps with an inclination of 26.56°, a distance between the ramps equal to 0.4 m, and a height between the mean water level and the height of the highest ramp of 2.0 m. The main parameter evaluated was the device width associated with the three-dimensional effects on hydraulic efficiency. The results showed that the device with the smallest width showed lower hydraulic efficiency, while the hydraulic efficiency has increased for wider devices.

Concerning the employment of Constructal Design and numerical simulation in the OTD-WEC, the following stand out: Machado et al. [14,15], Santos et al. [16], Goulart et al. [17], Martins et al. [18,19,20,21,22].

It is also worth mentioning some works carried out using the Constructal Design method in problems from other areas of engineering in order to show the versatility and applicability of the method. In mechanics of solids, for example, studies of Kucharski et al. [23] and Silveira et al. [24]. Concerning research in the area of heat transfer, mention is made of Bejan and Lorente [25], Kalbasi and Salimpour [26], Ziaei et al. [27], Moreira et al. [28], Ariyo and Ochende [29], Mustafa [30], Nan et al. [31], Zhang et al. [32], and Feijó et al. [33]. Regarding fluid mechanics problems, one can cite the works of Chen et al. [34], Izadpanahi et al. [35], Mardanpour et al. [36], Troncossi et al. [37], Kacimov et al. [38], and Miguel [39].

The present work performs a two-dimensional (2D) numerical study about the geometry evaluation using the Constructal Design method [40] through the degrees of freedom and constraints of an OTD-WEC with one and two ramps incorporated into the east breakwater of the city of São José do Norte, State of Rio Grande do Sul (RS), Brazil, more precisely in 32°11′12.55″ S and 52°04′29.30″ W (Figure 1). Initially, the computational model verification of the OTD-WEC was carried out with one ramp with the ratio between its height and length of H₁/L₁ = 0.667 (33.69°). For this, the average dimensionless overtopping flow ($\overline{Q}$) was compared with the empirical average dimensionless overtopping flow ($\overline{Q}$_EurOtop) [41], and the spectral and analytical density energy of the wave was analyzed for four different meshes. In addition, three-time step values were evaluated, and a comparison was made between the simulation times of 100 and 400 s. It is worth mentioning that this case was also considered as the reference geometry since it has the same slope as the original breakwater. A mesh sensitivity analysis employing the Grid Convergence Index (GCI) was also performed; followed by a computational model validation.

Subsequently, a geometric evaluation for the one ramp OTD-WEC was performed using the Constructal Design method. More precisely, the objective was to determine the maximum magnitude of the average dimensionless overtopping flow, ($\overline{Q}$)_mm, varying the degree of freedom H₁/L₁ (0.508, 0.667, and 0.852) and comparing the results with the $\overline{Q}$_EurOtop. Finally, it was evaluated the geometry of the OTD-WEC with two ramps analyzing four degrees of freedom, the ratio between the heights and lengths of the ramps, H₁/L₁ and H₂/L₂, (0.508, 0.667, and 0.852), the vertical distance between the ramps, H_g (0.10 and 0.20 m), and the horizontal distance between the ramps, L_g (0.10, 0.20, 0.30 and 0.40 m); respecting the constraints: the area as a function of the wave parameters (A_wave), the areas of the ramps (A_r,i), and the maximum ramp height which was kept fixed as half the significant wave height (H_S/2).

The main contributions of this paper were the application of the Constructal Design method to evaluate the geometry (based on degrees of freedom and restrictions) of a one and two ramps OTD-WEC (with water stored in the reservoir of the device returning to the wave flume) over its performance. The device is incorporated to the actual and determined location, the east breakwater of the city of São José do Norte, Rio Grande do Sul, Brazil. It seeks the geometry configuration that presents the best hydrodynamic performance (the maximum magnitude of the average dimensionless overtopping flow ($\overline{Q}$)_mm) obtained numerically and compared it with the calculated by empirical equation recommended in a specific manual for wave overtopping on coastal defense structures.

2. Mathematical Modeling

Figure 2 and Figure 3 show the 2D models with dimensions, x (horizontal), z (vertical), and y (perpendicular to the plane), of the OTD-WECs in a real scale nearshore wave flume with one and two ramps integrated into the east breakwater of the city of São José do Norte, RS, Brazil.

Table 1. Used dimensions of the computational domains.

Total Length of the Flume	LT	3·Lmax + LBW
Maximum wavelength	Lmax	~118.47 m
Partial breakwater length	LBW	22.95 m
Total height of the flume	HT	18.00 m
Height between the MWL and maximum height of the breakwater	HMA	5.30 m
Height between the maximum height of the breakwater and Total height of the flume	HLV	2.70 m
Bottom/one ramp length	L1
Bottom/one ramp height	H1
Upper ramp length	L2
Upper ramp height	H2
One ramp reservoirs length	LR
Bottom reservoirs length	LR1
Upper reservoirs length	LR2
One ramp reservoirs height	HR
Bottom reservoirs height	HR1
Upper reservoirs height	HR2
Height between the base of the bottom reservoir to the MWL	HFR	0.20 m
Height between the MWL to the highest point of the one ramp	HC	0.75 m (value kept fixed and equal to HS/2)
Height between the MWL to the highest point of the bottom ramp	HC1
Height between the MWL to the highest point of the upper ramp	HC2	0.75 m (value kept fixed and equal to HS/2)
Height between the MWL and the lowest point of the bottom ramp	HD
Height between the lowest point of the bottom ramp and the reservoir water outlet	HSA	3.00 m
Length of the return of the reservoirs water to the flume	LTB	0.15 m
Ratio between the height and length of the bottom/one ramp	H1/L1 (α)	0.508 (26.95°), 0.667 (33.69°) and 0.852 (40.43°)
Ratio between the height and length of the upper ramp	H2/L2 (θ)	0.508 (26.95°), 0.667 (33.69°) and 0.852 (40.43°)
Breakwater slope	β	~33.69°
Horizontal distance between the bottom and upper ramps	Lg	0.10, 0.20, 0.30 and 0.40 m
Vertical distance the bottom and upper ramps	Hg	0.10 and 0.20 m
MWL	h	10 m
Wave length	L
Wave height	H

shows the dimensions of the wave flume and the OTD-WEC with one ramp (Figure 2) and two ramps (Figure 3).

The imposition of a velocity profile at the left inferior surface of the wave flume (Prescribed Velocity—blue line, Figure 2 and Figure 3) generates irregular waves through the Joint North Sea Wave Project (JONSWAP) spectral representation model used in the present study. Furthermore, the numerical results obtained for the average dimensionless overtopping flow, $\overline{Q}$, were compared with the average overtopping dimensionless flow equation ($\overline{Q}$_EurOtop) [41] for inclined coastal structures (between 1:2 to 1:4/3) covered with large rocks or tetrapod, given by:

$${\overline{Q}}_{\mathrm{EurOtop}}=\frac{\overline{q}}{\sqrt{{g H}_{ \mathrm{S}}^{ 3}}}=0.09 \exp \left[-{\left(1.5\frac{H_{\mathrm{C}}}{H_{\mathrm{S}}}\right)}^{1.3}\right]$$

(1)

where $\overline{q}$ is the average overtopping flow (m³/s/m)—value obtained experimentally or numerically, g is the gravitational acceleration (m/s²), H_S is the significant wave height, H_C is the height from the mean water level (MWL) to the highest point of the one ramp.

More specifically, the values obtained for $\overline{Q}$ from the numerical simulations referring to the one ramp OTD-WEC with a slope of 33.69° (defined as reference geometry) were compared with those from Equation (1), allowing the analysis of the consistency of the results.

2.1. Constructal Design Applied of the OTD-WEC

According to the Constructal Design method [40], the studied domain is subjected to three constraints, i.e., the distance between the MWL and the maximum ramp height (kept fixed as half of the significant wave height, H_S/2), the area as a function of the wave parameters (A_wave), and the areas of the ramps (A_r,i):

$$A_{\mathrm{wave}}=\frac{H_{\mathrm{S}}{ L}_{\mathrm{P}}}{2}$$

(2)

$$A_{\mathrm{r},\mathrm{i}}=\frac{H_{\mathrm{i} }{L}_{\mathrm{i}}}{2}$$

(3)

where L_P is the peak wavelength and i = 1 and 2.

The ramps area fractions of the device are given by the ratio between the areas of the ramps and the area of the wave:

$${\varphi }_{\mathrm{i}}=\frac{A_{\mathrm{r},\mathrm{i}}}{A_{\mathrm{wave}}}$$

(4)

Moreover, the area fractions of the ramps (ϕ_i) were used as follows: ϕ₁ = 0.034 when only one ramp was considered and ϕ₁ = 0.022 (bottom ramp) and ϕ₂ = 0.002 (upper ramp), for the analysis with two ramps. Notice that these values aim to maintain the proportions of the device geometry with one ramp. Figure 4 shows a diagram with details of the application of the Constructal Design method.

Regarding the simulation process, it was divided as illustrated in

Table 2. Geometric evaluation of OTD-WEC with one ramp and one degree of freedom.

Case	H1/L1
1	0.5085
2	0.6667
3	0.8519

and Figure 5, with three (3) simulations for the OTD-WEC with one ramp varying the degree of freedom H₁/L₁ ratio in search of the maximum magnitude of the average overtopping dimensionless flow ($\overline{Q}$)_mm through the Equation (1) using Equation (5), as follows, to obtain the numerical value of $\overline{Q}$.

$${\overline{q}}_{\mathrm{n}}=\frac{\overline{\dot{m}}}{{\rho }_{\mathrm{water}}}$$

(5)

where ${\overline{q}}_{\mathrm{n}}$ is the average overtopping flow (m³/s/m) obtained numerically by simulations, $\overline{\dot{m}}$ is the average mass flow rate and ${\rho }_{\mathrm{water}}$ is the water density.

Analogously, for the simulations of the device with two ramps (see

Table 3. Geometric evaluation of OTD-WEC with two ramps and four degrees of freedom.

Case	H1/L1	H2/L2	Hg	Lg	Case	H1/L1	H2/L2	Hg	Lg	Case	H1/L1	H2/L2	Hg	Lg
4	0.508	0.508	0.10	0.10	28	0.667	0.508	0.10	0.10	52	0.852	0.508	0.10	0.10
5	"	"	"	0.20	29	"	"	"	0.20	53	"	"	"	0.20
6	"	"	"	0.30	30	"	"	"	0.30	54	"	"	"	0.30
7	"	"	"	0.40	31	"	"	"	0.40	55	"	"	"	0.40
8	0.508	0.508	0.20	0.10	32	0.667	0.508	0.20	0.10	56	0.852	0.508	0.20	0.10
9	"	"	"	0.20	33	"	"	"	0.20	57	"	"	"	0.20
10	"	"	"	0.30	34	"	"	"	0.30	58	"	"	"	0.30
11	"	"	"	0.40	35	"	"	"	0.40	59	"	"	"	0.40
12	0.508	0.667	0.10	0.10	36	0.667	0.667	0.10	0.10	60	0.852	0.667	0.10	0.10
13	"	"	"	0.20	37	"	"	"	0.20	61	"	"	"	0.20
14	"	"	"	0.30	38	"	"	"	0.30	62	"	"	"	0.30
15	"	"	"	0.40	39	"	"	"	0.40	63	"	"	"	0.40
16	0.508	0.667	0.20	0.10	40	0.667	0.667	0.20	0.10	64	0.852	0.667	0.20	0.10
17	"	"	"	0.20	41	"	"	"	0.20	65	"	"	"	0.20
18	"	"	"	0.30	42	"	"	"	0.30	66	"	"	"	0.30
19	"	"	"	0.40	43	"	"	"	0.40	67	"	"	"	0.40
20	0.508	0.852	0.10	0.10	44	0.667	0.852	0.10	0.10	68	0.852	0.852	0.10	0.10
21	"	"	"	0.20	45	"	"	"	0.20	69	"	"	"	0.20
22	"	"	"	0.30	46	"	"	"	0.30	70	"	"	"	0.30
23	"	"	"	0.40	47	"	"	"	0.40	71	"	"	"	0.40
24	0.508	0.852	0.20	0.10	48	0.667	0.852	0.20	0.10	72	0.852	0.852	0.20	0.10
25	"	"	"	0.20	49	"	"	"	0.20	73	"	"	"	0.20
26	"	"	"	0.30	50	"	"	"	0.30	74	"	"	"	0.30
27	"	"	"	0.40	51	"	"	"	0.40	75	"	"	"	0.40

and Figure 6) seventy-two (72) simulations were performed to determine the configuration that presents the ($\overline{Q}$)_mm, varying four degrees of freedom, H₁/L₁, H₂/L₂, L_g, and H_g where the distance H_g was varied keeping the remaining parameters fixed, in this case, L_g, H₂/L₂, and H₁/L₁. Then, the same process was repeated for L_g values, H₂/L₂, and H₁/L₁.

2.2. The Multiphase Volume of Fluid (VOF) Model

The conservation equation of mass and momentum for the mixture of air and water in an isothermal, laminar and incompressible flow is given by [42]:

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

(6)

$$\frac{\partial }{\partial t}(\rho \overrightarrow{v})+\Delta ·(\rho \overrightarrow{v}\overrightarrow{v})=-\nabla p+\nabla ·(\mu \stackrel{=}{\tau })+\rho \overrightarrow{g}+\overrightarrow{F}$$

(7)

where ρ is the density of the mixture (kg/m³), $\overrightarrow{v}$ is the flow velocity vector (m/s), p is the pressure (N/m²), $\mu$ is the dynamic viscosity (Pa·s), $\stackrel{=}{\tau }$ is the tensor of deformation rate (N/m²), and $\rho$ and $\overrightarrow{F}$ are, respectively, the buoyancy and external body forces (N/m³).

The Volume of Fluid (VOF) method has been employed to tackle the mixture of air and water and its device interaction. The VOF is used to model multiphase fluid flows with two or more phases. In this model, the phases are immiscible, i.e., the volume of one phase may not be occupied by another phase [43,44].

In the numerical simulations of this study, it was considered two different phases: air and water. Values of air and water properties were respectively: ρ_air = 1.225 kg/m³, ρ_water = 1025 kg/m³, μ_air = 1.789 × 10⁻⁵ kg/(ms), and μ_water = 1.077 × 10⁻³ kg/(ms). Therefore, the volume fraction concept (α_q) represents the two phases in the control volume. In this model, the sum of the volume fractions within a control volume must be unitary (0 ≤ α_q ≤ 1). Consequently, if α_water = 0, the control volume is empty of water and full of air (α_air = 1). If the fluid is a mixture of air and water, one phase complements the other, α_air = 1 − α_water. Thus, only one additional transport equation for the volume fractions is required [43,44]:

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

(8)

It is noteworthy that the conservation equations of mass and momentum are solved for the mixture. Therefore, it is necessary to obtain density and dynamic viscosity values for the mixture, which can be written, respectively, by:

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

(9)

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

(10)

2.3. Wave Spectrum

The irregular waves were generated by imposing a velocity profile at the left surface of the channel (see Figure 2 and Figure 3), based on the JONSWAP spectrum [45,46]:

$${\mathrm{S}}_{\mathrm{JS}}\left(\omega \right)=\frac{5}{16}\frac{H_{ \mathrm{S}}^{ 2}{ \omega }_{\mathrm{P}}^4}{{\omega }^5}\exp \left[-\frac{5}{4}{\left(\frac{{\omega }_{\mathrm{P}}}{\omega }\right)}^4\right]A_{\gamma }{\gamma }^{\exp \left\{-\left[\frac{1}{2}{\left(\frac{\omega -{\omega }_{\mathrm{P}}}{{\sigma \omega }_{\mathrm{P}}}\right)}^2\right]\right\}}$$

(11)

where S_JS(ω) is the energy spectral density (J·s), H_S is the significant height (m), ω is the angular frequency (rad/s), ω_P is the peak angular frequency (rad/s), γ is the spectrum peak parameter, σ spectrum shape parameter and Aγ = 1 − 0.287ln(γ).

Equation (11) considered the wave characteristics for the region of interest according to Almeida et al. [47], Cuchiara et al. [48], and Lisboa et al. [49] (

Table 4. Wave characteristics adopted.

Frequency (ω)	Period (T)	Wavelength (L)
ωmin = 0.503 rad/s	Tmax = 12.50 s	Lmax = 118.47 m
ωP = 0.838 rad/s	TP = 7.50 s	LP = 65.40 m
ωmax = 1.396 rad/s	Tmin = 4.50 s	Lmin = 30.59 m

). The values chosen for significant wave height and the peak period are, respectively, H_S = 1.50 m and T_P = 7.50 s.

The frequency range used in the JONSWAP spectrum was determined between T_min = 4.50 s (minimal period) and T_max = 12.50 s (maximal period) and was divided into 12 regular waves to adequately represent the spectral density of energy for the JONSWAP spectrum.

2.4. Boundary Conditions

Regarding the wave generation, a velocity profile based on the JONSWAP spectrum was imposed at the left side surface of the wave channel (blue lines in Figure 2 and Figure 3), simulating the behavior of a wavemaker with the velocity components in the horizontal (u) and vertical (w) directions given, respectively, by [50,51,52,53]:

$${u=\mathsf{\zeta }}_{{\mathrm{a}}_{\mathrm{n}}}{g k}_{\mathrm{n}}\frac{{\cos h(k}_{\mathrm{n}}{\mathrm{z}+k}_{\mathrm{n}}h)}{{\omega }_{\mathrm{n}}{\cos h(k}_{\mathrm{n}}h)}{\cos (k}_{\mathrm{n}}{\mathrm{x} - \omega }_{\mathrm{n}}\mathrm{t})$$

(12)

$${w=\mathsf{\zeta }}_{{\mathrm{a}}_{\mathrm{n}}}{g k}_{\mathrm{n}}\frac{{\sin h(k}_{\mathrm{n}}{\mathrm{z}+k}_{\mathrm{n}}h)}{{\omega }_{\mathrm{n}}{\sin h(k}_{\mathrm{n}}h)}{\sin (k}_{\mathrm{n}}{\mathrm{x} - \omega }_{\mathrm{n}}\mathrm{t})$$

(13)

where ${\mathsf{\zeta }}_{{\mathrm{a}}_{\mathrm{n}}}$ is the amplitude of a component (12 waves) of the spectrum (m), g is the acceleration of the gravity (m/s²), k_n is the wave number (rad/m), ω_n is the frequency of a component (rad/s), z is the position between the free surface and the bottom of the channel (m), x is the horizontal position (m), and t is the time (s).

The superior region of the left side and the superior surface were considered at atmospheric pressure P_abs = 101.3 kPa (green lines in Figure 2 and Figure 3) for the other boundary conditions. In the lower and right surfaces and the overtopping device surfaces (red lines in Figure 2 and Figure 3), it was imposed a non-slip condition and impermeability (u = w = 0 m/s). Regarding the initial conditions, it was considered that the fluid was wavy, and the water depth with h = 10.0 m.

3. Numerical Modeling

The present work employed the Finite Volume Method (FVM) [54,55] for the numerical solution of conservation equations of mass, momentum, and the transport equation of volume fraction, i.e., Equations (6)–(8). It was also used the CFD commercial package Fluent^TM. The multiphase Volume of Fluid (VOF) model [43,44] was applied for the interaction among air, water, and device. The numerical parameters adopted in the commercial software ANSYS FLUENT V.16 are shown in

Table 5. Methods adopted in 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
Flow Regime		Laminar
Under-Relaxation Factors	Pressure	0.3
	Momentum	0.7
Residual	Continuity	10−6
	x-velocity
	z-velocity
Open Channel Initialization Method		Wavy

.

Finally, the simulations were processed on a computer with an AMD Quad-Core 3.3 GHz processor and 16 GB of RAM, and the Message Passing Interface (MPI) was used for parallelization. The processing time for each simulation was approximately 1.26 × 10⁵ s (35 h).

4. Results and Discussion

4.1. Mesh Sensitivity Analysis, Numerical Model Verification and Validation

Initially, structured stretched meshes were developed, following recommendations found in the literature [56], to analyze the mesh sensitivity and verification of the numerical model. For that, it was used the reference geometry with one ramp, ratio H₁/L₁ = 0.6667 and ϕ₁ = 0.0344, allowing a comparison between $\overline{Q}$ and $\overline{Q}$_EurOtop = 0.045 (Equation (1)) with analytical (Equation (11)) and numerical solutions of the energy spectral density.

Figure 7 illustrates the gauges used in the one ramp study. The gauges S1, S2, S3, and S4, respectively, 50, 100, 150, and 200 m from the left line of the domain (responsible for the measurement of the water free surface elevation) and the gauge Sor monitors the mass flow rate of water ($\dot{m}$). Thus, using the Fast Fourier Transform (FFT) on the temporal series of the water free surface elevation during the 100 s of simulation, from the values obtained from gauges S1, S2, S3, and S4, it was possible to compare with the analytical solution of the spectral density energy.

Figure 8 indicates the gauges Sb and Su to measure the mass flow rate of water, respectively, from the bottom ramp and the upper ramp. Therefore, with the instantaneous values of $\dot{m}$ measured, the averages were calculated for the 100 s of simulation and the values used in Equation (5) to determine the average overtopping flow $\overline{Q}$ (Equation (1)).

Furthermore, the relative error between the values of $\overline{Q}$ and $\overline{Q}$_EurOtop, as well as, between the successive time steps, was calculated for the generated meshes by:

$$\overline{R_E}=\left|\frac{{\overline{x}}_{\mathrm{num}}-\overline{x}}{\overline{x}}\right|·100\%$$

(14)

Moreover, for the comparison between the spectral densities, analytical and numerical, three indicators were used: l_∞ norm, Mean Absolute Error (MAE), and Root Mean Square Error (RMSE) [57], respectively, defined as:

$${\left|\right|l\left|\right|}_{\infty }=\underset{\mathrm{j}}{\max }\left|l_{\mathrm{j}}\right|$$

(15)

$$\mathrm{MAE}=\frac{1}{\mathrm{n}}\sum _{\mathrm{i}=1}^{\mathrm{n}}|{\mathrm{x}}_{\mathrm{num},\mathrm{i}}-{\mathrm{x}}_{\mathrm{ana},\mathrm{i}}|$$

(16)

$$\mathrm{RMSE}=\sqrt{\frac{{\sum }_{\mathrm{i}=1}^{\mathrm{n}}{\left({\mathrm{x}}_{\mathrm{num},\mathrm{i}}-{ \mathrm{x}}_{\mathrm{ana},\mathrm{i}}\right)}^2}{\mathrm{n}}}$$

(17)

The results for $\overline{Q}$ from the mesh sensitivity analysis are presented in Figure 9a, for the four meshes analyzed (1, 2, 3, and 4) in comparison with $\overline{Q}$_EurOtop (dashed green line). The relative errors between numerical and analytical solutions for each considered mesh are: $\overline{R_{E1}}$ = 31.09%, $\overline{R_{E2}}$ = 12.34%, $\overline{R_{E3}}$ = 9.64%, and $\overline{R_{E4}}$ = 8.10%. Figure 9b, as expected, presents that the increase of the mesh refinement also significantly increases the processing time.

Furthermore, the uncertainly analyses Grid Convergence Index (GCI) [58,59] was used in meshes 2, 3, and 4. The refinement ratio estimate was calculated by:

$$r_{\mathrm{j},\mathrm{i}}={\left(\frac{M_{\mathrm{i}}}{M_{\mathrm{j}}}\right)}^{\raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$D$}\right.}$$

(18)

where M_i is the coarse mesh, M_j is the fine mesh and D is the number of dimensions of the problem.

The relative error is given by:

$${\in }_{\mathrm{i},\mathrm{j}}=\frac{f_{\mathrm{j}}-f_{\mathrm{i}}}{f_{\mathrm{i}}}$$

(19)

where f_i is the result obtained for the coarse mesh and f_j is the result obtained for the fine mesh.

And the convergence order was calculated by:

$$p=\left[\frac{1}{ln\left(r_{21}\right)}\right]\left[ln\left|\frac{{\in }_{32}}{{\in }_{21}}\right|+ln\left(\frac{r_{21}^p-s}{r_{32}^p-s}\right)\right]$$

(20)

$$s=1.sign\left(\frac{{\in }_{32}}{{\in }_{21}}\right)$$

(21)

The GCI was defined by:

$$GC{I}_{\mathrm{i},\mathrm{j}}=\frac{F_{\mathrm{S}}.{\in }_{21}}{r_{21}^p-1}$$

(22)

where F_S is the factor of safety equal to 1.25.

Finally, the asymptotic range of convergence, where the value should be approximately 1, can be written by:

$$\chi =\frac{GC{I}_{32}}{r_{21}^p.GC{I}_{21}}$$

(23)

Therefore,

Table 6. Values calculated using GCI.

p	ri,j	εi,j	GCIi,j	χ
4.48	1.15	0.02	2.43%	1.03
-	1.14	0.03	4.70%	-

presents de results of the uncertain analyses through the GCI, where the GCI values were below 5%, as recommended by the literature, and the asymptotic range of convergence remained approximately 1.

Figure 10 presents the spectral density graphs for gauges S1 = 50 m, S2 = 100 m, S3 = 150 m, and S4 = 200 m for the four meshes analyzed. Thus, it is possible to observe a good agreement between the measured and analytical-numerical spectral densities for each case, showing that the proposed numerical model adequately represents the JONSWAP spectrum for the imposed sea conditions. It is also observed that there is an increase in the error with the increase of the distance of the gauge in relation to the wave generator, a fact that may be linked to the reflection of the waves after interacting with the structure of the breakwater.

Table 7. MAE, RMSE and l_∞ norm for the spectral densities for the gauges S1, S2, S3 and S4.

S1 = 50 m				S2 = 100 m
Mesh	MAE (m2s)	RMSE (m2s)	l∞ (m2s)	Mesh	MAE (m2s)	RMSE (m2s)	l∞ (m2s)
1	0.045	0.057	0.051	1	0.051	0.062	0.078
2	0.041	0.053	0.026	2	0.044	0.056	0.063
3	0.043	0.054	0.030	3	0.038	0.055	0.048
4	0.041	0.053	0.013	4	0.037	0.053	0.038
S3 = 150 m				S4 = 200 m
Mesh	MAE (m2s)	RMSE (m2s)	l∞ (m2s)	Mesh	MAE (m2s)	RMSE (m2s)	l∞ (m2s)
1	0.060	0.077	0.199	1	0.052	0.081	0.243
2	0.048	0.056	0.082	2	0.036	0.048	0.126
3	0.045	0.053	0.071	3	0.035	0.047	0.116
4	0.045	0.053	0.072	4	0.035	0.044	0.088

presents the values obtained for the MAE, RMSE, and l_∞ norm for the spectral densities of the meshes developed.

Therefore, based on the relative errors ($\overline{R_E}$) between $\overline{Q}$ and $\overline{Q}$_EurOtop and of the MAE, RMSE, l_∞ norm for the spectral densities and the GCI analyses, the spatial discretization adopted for the study was the mesh 3 with approximately 182,000 finite quadrilateral volumes, since it presents results with good accuracy (since it is in an asymptotic region of the influence of the number of volumes over the $\overline{Q}$) generated with an adequate processing time.

Figure 11 shows the partial discretization of the domain in the ramp region of the overtopping device for the independent mesh with nearly 182,000 volumes.

Finally, the time step adopted in the simulations, Δt = 2.0 × 10⁻² s, was previously used for overtopping devices studies in works found in the literature [17,18,19,20,21,22].

Regarding the computational model validation procedure, Figure 12 presents the illustrations of the wave channel (Figure 12a) used in the experimental analyses and the computational domain (Figure 12b) with the dimensions: L_T = 34.2 m, H_T = 1.0 m, S = 0.15 m, L_R = 0.5 m, h = 0.862 m, h₁ = 0.392 m, and H₁/L₁ = 0.39 (21.3°). In addition, the same numerical methods used in the present work were used in Goulart [60]. The difference was the initialization method, flat in Goulart [60] and wavy in the present paper.

Figure 13 shows a free surface elevation (η) of the MWL numerically obtained with ANSYS FLUENT, v.18 and experimentally for t = 50 s and with measurements carried out in Gauge (see Figure 12a). Numerically with the 2nd order Stokes theory, for regular waves, with period and height of, respectively, T = 7.5 s and H = 1.0 m for an MWL height h = 10.0 m. However, the analysis was performed on a laboratory scale in a geometric reduction of 1:15 by the Froude criterion due to the dimensions of the physical channel. Thus, values of wave characteristic were T = 1.94 s, H = 0.067, L_w = 3.54 m, and L_w1 = 3.54 m [60].

Therefore, one can observe the agreement between the waves generated experimentally and numerically, with values obtained for the MAE, RMSE, and l_∞ norm presented in

Table 8. MAE, RMSE and l_∞ norm for the free surface elevation (η) for the Gauge.

MAE (m)	RMSE (m)	l∞ (m)
0.007	0.010	0.012

. In addition, it was noted from t = 30 s greater disturbance of the waves due to the reflection caused by the structure of the ramp.

Thus, the mathematical model implemented in the computational code is suitable for representing the physics of the problem. Moreover, the wave spectrum is a variation of the boundary condition imposed on the model, a superposition of several linear waves, and the spectrum used in this paper presents energy similar to the regular wave used in Goulart [60]. From this, it was possible to consider the computational model validated.

4.2. Analysis of Simulation of OTD-WEC Considering a Time of 400 s

The simulation time was analyzed for the reference geometry (one ramp, ratio H₁/L₁ = 0.6667 and ϕ₁ = 0.0344). Thus, a time of 400 s was simulated to determine the influence of a time four times longer than that initially used (100 s) on the results of $\overline{Q}$ (

Table 9. $\overline{Q}$ for simulation times of 100 and 400 s.

Proc. Time (h)	Sim. Time (s)	Qnum	QEurOtop	ER (%)
148	400	0.045	0.045	1.06%
35	100	0.041	"	9.64%

) and MAE, RMSE, and l_∞ norm of the spectral density (

Table 10. MAE, RMSE and l_∞ norm for the spectral densities for the gauges S1, S2, S3 and S4 and simulation times of 100 and 400 s.

S1 = 50 m				S2 = 100 m
Sim. time (s)	MAE (m2s)	RMSE (m2s)	l∞ (m2s)	Sim. time (s)	MAE (m2s)	RMSE (m2s)	l∞ (m2s)
100	0.043	0.054	0.030	100 s	0.038	0.055	0.048
400	0.042	0.053	0.032	400 s	0.038	0.053	0.043
S3 = 150 m				S4 = 200 m
Sim. time (s)	MAE (m2s)	RMSE (m2s)	l∞ (m2s)	Sim. time (s)	MAE (m2s)	RMSE (m2s)	l∞ (m2s)
100	0.045	0.053	0.071	100 s	0.035	0.047	0.116
400	0.045	0.053	0.072	400 s	0.037	0.045	0.175

).

Therefore, analyzing the data from Table 9 and Table 10, it is inferred that the simulations carried out with duration of 100 s adequately represent the physical phenomenon. The simulations present a difference of only 8.59% between the relative errors of $\overline{Q}$ and $\overline{Q}$_EurOtop. They also present values close to the means MAE and RMSE, and for the l_∞ norm for spectral densities measured in the gauges S1, S2, S3, and S4.

4.3. Results for One Ramp OTD-WEC (One Degree of Freedom: H₁/L₁)

Three geometric configurations were evaluated for the OTD-WEC with one ramp incorporated in the east breakwater of São José do Norte. Case 2 (Table 2) was defined with the reference geometry, and the ratio between height and length of the ramp (H₁/L₁) used was based on the real slope of the original breakwater, which is equal to 1:1.5 (H₁/L₁ = 0.667~33.69°). The other two cases (1 and 3) had the H₁/L₁ ratios defined as 20% of the ratio of case 2. Thus, case 1 has H₁/L₁ = 0.508 (26.95°) and case 3 has H₁/L₁ = 0.852 (40.43°). In these three cases it was fixed a ramp area fraction of ϕ₁ = 0.0344, a value defined to take advantage of and avoid exceeding the limits of the existing geometry of the east breakwater, respecting the constraint of the ramp height (H_S/2 = 0.75 m). Figure 14 presents the geometries of cases 1, 2, and 3.

Figure 15 shows the results obtained for the numerical average dimensionless overtopping flow, $\overline{Q}$, for the three values of the degree of freedom H₁/L₁, i.e., cases 1, 2, and 3. The geometric evaluation indicates that the smaller magnitudes of H₁/L₁ led to the highest values of $\overline{Q}$ for the conditions imposed for the problem. Therefore, case 1 showed the maximum magnitude for the average dimensionless overtopping flow, ($\overline{Q}$_mm = 0.041, with a $\overline{R_E}$ = 1.59% higher if compared to case 2 and $\overline{R_E}$ = 9.60% when compared with case 3, i.e., case 1 was the geometry that presented the lowest resistance to water flow from the incident waves on the OTD-WEC with one ramp, providing the highest average dimensionless overtopping flow.

Figure 16 shows the hydrodynamic behavior for the instants t = 92–100 s for the best (Figure 16a) and worst (Figure 16b) ramp geometries. Therefore, it can be noticed in Figure 14 for the instants t = 94 and 100 s that the volume of water overtopped was greater for the best geometry (case 1—Figure 16a), a fact that can be confirmed by analyzing the instants t = 92, 96, and 98 s due to the higher amount of water inside the reservoir. It is also observed at instant t = 98 s, for both cases, a water volume run-down over the ramp and returning to the flume, due to the water movement inside the reservoir, with a greater water volume for the best case (case 1).

4.4. Results for Two Ramps OTD-WEC (Four Degrees of Freedom: H₁/L₁; H₂/L₂; H_g; and L_g)

For the overtopping device with two ramps, it was considered ramp area fractions of ϕ₁ = 0.0218 for the bottom ramp and ϕ₂ = 0.0025 for the upper ramp. These area fractions were defined to maintain a total height between the two ramps equal to 1.50 m when H₁/L₁ = H₂/L₂ = 0.6667, as well as set for a reference geometry (case 2). Figure 17 shows the geometry for case 40, illustrating one of the used configurations, where HFRu was the distance between MWL and the base of the upper reservoir.

Figure 18, Figure 19, Figure 20, Figure 21, Figure 22, Figure 23, Figure 24, Figure 25 and Figure 26 show the graphics of the behavior of $\overline{Q}$ × L_g for all studied cases of the OTD-WEC with two ramps. In a general way, the fluid-dynamic behavior was similar, i.e., for the gauge Sb, for example, when H_g = 0.10 m, an increase in $\overline{Q}$ was observed for greater distances between the ramps (L_g), while for H_g = 0.20 m the trend of the curve was similar but less accentuated. For the Su probe, the behavior was opposite to that of the Sb probe. There was a decrease of $\overline{Q}$ with the increase of L_g for most of the studied cases submitted to the sea conditions for the analyzed region. In addition, in general, for cases considering H_g = 0.20 m, i.e., bottom ramp elevated by 0.10 m in relation to H_g = 0.10 m, there was a decrease in the $\overline{Q}$ value, except in some specific cases. Moreover, for all cases the curves referring to the sum of the $\overline{Q}$ values of the Sb and Su probes (Sb + Su) are similar to the curves of the Su probes.

Therefore, analyzing the cases globally and according to the Constructal Design method, for the bottom ramp, it was observed that for higher L_g and lower H_g values, the flow of water from the sea waves over the bottom ramp suffer the lowest resistance, providing better results for $\overline{Q}$. On the other hand, for the upper ramp lower values of L_g and H_g provided lower resistance to water flow, reaching higher $\overline{Q}$ values.

The graphs were divided into blocks (Figure 18, Figure 19, Figure 20, Figure 21, Figure 22, Figure 23, Figure 24, Figure 25 and Figure 26) to facilitate the analysis of the influence of degrees of freedom H₁/L₁ and H₂/L₂ on the value of $\overline{Q}$. Thus, for the block of Figure 18a, Figure 19a and Figure 20a, it is noted that for the fixed H₁/L₁ ratio and with the increase of the H₂/L₂ ratio, there was an overall decrease in the values of $\overline{Q}$ measured a in the gauge Sb. However, for the gauge Su, there was a small increase in the magnitude of $\overline{Q}$ with the increase in the H₂/L₂ ratio, being the most noticeable in case 12, and a decrease presented in case 23.

In its turn, for the block of Figure 18b, Figure 19b and Figure 20b, the behavior was similar to that presented previously for the gauge Sb. For the gauge Su, the value of $\overline{Q}$ decreases with increasing from H₂/L₂ = 0.508 to 0.667. However, for H₂/L₂ = 0.852, cases 24 and 25 were like cases 8 and 9, while cases 26 and 27 had an increase in the value of $\overline{Q}$ concerning the H₂/L₂ ratios = 0.508 and 0.667.

The block of Figure 21, Figure 22 and Figure 23, presents for the gauge Sb similar behavior to the block of Figure 18, Figure 19 and Figure 20. The increase in the H₂/L₂ ratio from 0.508 to 0.667 led to an increase in $\overline{Q}$ for all distances L_g, highlighting the local maximum at L_g = 0.30 m, for the gauge Su (Figure 21a, Figure 22a and Figure 23a). However, as the H₂/L₂ ratio increased from 0.667 to 0.852, the $\overline{Q}$ values were close to those obtained for H₂/L₂ = 0.508; except for L_g = 0.40 m, where there was a reduction in $\overline{Q}$. For the block of Figure 21b, Figure 22b and Figure 23b, the gauge Sb also presents similar behavior to the block of Figure 18, Figure 19 and Figure 20 and the increase in the H₂/L₂ ratio from 0.508 to 0.667 led to a global increase in $\overline{Q}$ and showed a local maximum at L_g = 0.30 m for the gauge Su. Moreover, as the H₂/L₂ ratio increased from 0.667 to 0.852, the curve behavior was similar to the obtained with H₂/L₂ = 0.508 but with higher $\overline{Q}$ magnitudes.

The gauge Sb for the block of Figure 24, Figure 25 and Figure 26 showed similar behavior to the previous blocks. However, from Figure 24a, Figure 25a and Figure 26a the gauge Su achieved close values for $\overline{Q}$ when L_g = 0.10 m for the three slopes of the upper ramp; but for the other L_g distances, the increase of the H₂/L₂ ratio led to the reduction of $\overline{Q}$ values. Moreover, for the block of Figure 24b, Figure 25b and Figure 26b the gauge Su led to increased $\overline{Q}$ values with an increasing H₂/L₂ ratio.

Figure 27 shows the maximum magnitude graphs found among all proposed ramp configurations over the incidence of the sea conditions at the site of interest. Analyzing Figure 27, similar behavior and magnitudes of ($\overline{Q}$)_mm values can be observed as a function of H₁/L₁ and H₂/L₂ for the gauge Sb. However, case 38 (Figure 27b) presented a higher value of ($\overline{Q}$)_mm when compared with cases 12 and 63. Furthermore, for gauge Su, the curves had the same behavior for Figure 27a,b, with local maximums for H₂/L₂ = 0.667; whereas in Figure 27c a different curve trend was noticed, presenting a local minimum for H₂/L₂ = 0.667.

Furthermore, the best geometric configuration with two ramps was the case 38 with H₁/L₁ = H₂/L₂ = 0.667, H_g = 0.10 m, L_g = 0.30 m, and ($\overline{Q}$)_mm = 0.044, having $\overline{R_E}$ = 6.09% if compared to the best case for the OTD-WEC with one ramp (case 1), i.e., case 38 was the geometry that presented the lowest resistance to water flow from the incident waves on the OTD-WEC with two ramps, providing the highest average dimensionless overtopping flow, considering all the conditions imposed for this work.

Figure 28 shows the hydrodynamic behavior of transient fluid flow for the instants t = 92–100 s for the best configuration, case 38, together with the case 59, second worst case, allowing to compare the effect of the greater vertical distance between the ramps (H_g = 0.20 m), since the best and the worst case presented H_g = 0.10 m.

The instants t = 92, 94, and 100 s show that case 38 (Figure 28a) presented the geometry with the lowest resistance to the water flow over the ramps, given the greater volume overtopped and hence inside the reservoir. Furthermore, at instant t = 98 s it was observed that the water movement inside the reservoir led to the water run-down over the ramps, with part of it returning to the flume due to the large volume of water inside the reservoirs of the case 38. On the other hand, the smaller volume of water in the reservoirs of the case 59 (Figure 28b) associated with the greater height of the bottom ramp prevented the return of water to the flume. Moreover, at instants t = 92 and 94 s in case 59 (Figure 28b) one can note that the greater distance between the ramps (H_g = 0.20 m), i.e., the bottom ramp higher concerning the MWL, provided a kind of wave breaking (t = 92 s) creating a region of air between the two ramps (t = 94 s). Meanwhile, at the instant t = 92 s of Figure 28a (case 38), a continuous water movement can be noticed, providing a greater volume of water during overtopping.

5. Conclusions

This paper presents a numerical study of an overtopping device integrated into the east breakwater of the city of São José do Norte, Rio Grande do Sul, Brazil, to determine the maximum magnitude of the average dimensionless overtopping discharge (($\overline{Q}$)_mm). The work evaluated three degrees of freedom of the ramp geometry, the ratio between the height and length of the ramp (H_i/L_i)—for one and two ramps OTD-WEC, the vertical distance between ramps (H_g), and the horizontal distance between ramps (H_g)—for two ramps OTD-WEC. The geometric evaluation was performed numerically using the Constructal Design method.

The conclusion was that using a simulation time equal to t = 100 s can adequately represent the physical phenomenon since the comparison with a time t = 400 s presented a difference of only 8.59% between the relative errors of $\overline{Q}$ and $\overline{Q}$_EurOtop.

The analyses of the geometry evaluation of an OTD-WEC with one ramp and one degree of freedom (H₁/L₁) showed for case 1 the maximum magnitude for the average dimensionless overtopping flow, having ($\overline{Q}$)_mm = 0.041 and H₁/L₁ = 0.508. Case 1 reaches a superior performance with a $\overline{R_E}$ = 1.59% if compared to case 2 (H₁/L₁ = 0.667) and with a $\overline{R_E}$ = 9.60% when confronted with case 3 (H₁/L₁ = 0.852).

In addition, for the OTD-WEC with two ramps and four degrees of freedom (H₁/L₁, H₂/L₂, H_g, and L_g), the best geometric configuration was the case 38 with H₁/L₁ = H₂/L₂ = 0.667, H_g = 0.10 m, L_g = 0.30 m and ($\overline{Q}$)_mm = 0.044, achieving a $\overline{R_E}$ = 6.09% when compared to the best case for the OTD-WEC with one ramp (case 1).

In addition, the OTD-WEC with two ramps and slopes of both equal to 33.69° (same original angle of the breakwater and close to that found in literature studies (34°) [8,9,11,12]), was the case that presented the best performance indicator ($\overline{Q}$)_mm = 0.044, among those studied. However, for the OTD-WEC with one ramp and slope of 26.95° (also close to the one found in the literature (26.56°) [13]), the value of ($\overline{Q}$)_mm = 0.041 was obtained, i.e., a difference of 0.003. Therefore, it can be concluded that the use of a one ramp OTD-WEC (or several of them, side by side with angles between 26.95 and 33.69), is the most suitable for a better use of wave energy, in addition to presenting less complexity construction when compared to an OTD-WEC with two ramps.

Finally, the Constructal Design method showed the importance of geometric evaluation for rationalizing resources, as it became clear how small changes in geometry significantly influence the performance indicator chosen for the problem.

For future works, it is suggested to employ different approaches for the irregular waves generation, for instance the WaveMIMO methodology [61] that allows to numerically reproduce a realistic sea state of the region of interest; as well as the investigation of other type of WEC integrated to the breakwater, for example the Oscillating Water Colum (OWC).