Geometric Evaluation of the Hydro-Pneumatic Chamber of an Oscillating Water Column Wave Energy Converter Employing an Axisymmetric Computational Model Submitted to a Realistic Sea State Data

Abstract

In this research, considering the air methodology, an axisymmetric model was developed, validated, and calibrated for the numerical simulation of an Oscillating Water Column (OWC) converter subjected to a realistic sea state, representative of the Cassino beach, in the south of Brazil. To do so, the Finite Volume Method (FVM) was used, through the Fluent software (Version 18.1), for the airflow inside the hydro-pneumatic chamber and turbine duct of the OWC. Furthermore, the influence of geometric parameters on the available power of the OWC converter was evaluated through Constructal Design combined with Exhaustive Search. For this, a search space with 100 geometric configurations for the hydro-pneumatic chamber was defined by means of the variation in two degrees of freedom: the ratio between the height and diameter of the hydro-pneumatic chamber (H₁/L₁) and the ratio between the height and diameter of the smallest base of the connection, whose surface of revolution has a trapezoidal shape, between the hydro-pneumatic chamber and the turbine duct (H₂/L₂). The ratio between the height and diameter of the turbine duct (H₃/L₃) was kept constant. The results indicated that the highest available power of the converter was achieved by the lowest values of H₁/L₁ and highest values of H₂/L₂, with the optimal case being obtained by H₁/L₁ = 0.1 and H₂/L₂ = 0.81, achieving a power 839 times greater than the worst case. The values found are impractical in real devices, making it necessary to limit the power of the converters to 500 kW to make this assessment closer to reality; thus, the highest power obtained was 15.5 times greater than that found in the worst case, these values being consistent with other studies developed. As a theoretical recommendation for practical purposes, one can infer that the ratio H₁/L₁ has a greater influence over the OWC’s available power than the ratio H₂/L₂.

1. Introduction

The transition from the traditional energy generation system, based on centralized energy generation units, to renewable energy generation systems is a current global trend, allowing diversification, decentralization, scale reduction, and spatial dispersion in energy production [1]. Several recent studies have been addressed to this topic, discussing different aspects [2,3,4,5,6,7,8,9,10,11].

Among the renewable energy sources, sea wave energy stands out for its abundance, with an estimated potential of 32,000 TWh per year offshore and 2 TWh per year in coastal regions—the latter equating to the world’s annual energy consumption [12,13].

In this context, Brazil is first in a group composed of the 20 countries with the highest estimated available power to be harnessed from the ocean wave energy. Due to its large shoreline, Brazil reaches an available wave power of 372.1 TWh/month, being, for example, about 30%, 335%, 365%, 645%, and 830% higher than the values achieved in New Zealand, South Africa, Australia, the United States, and Spain, respectively [14]. From the entire Brazilian coast, which is approximately 9 × 10³ km long, the southern region has around 1.4 × 10³ km of shoreline, with the greatest available wave energy power [15,16].

There are several proposed technologies to transform the sea wave energy into electrical energy, among which the Oscillating Water Column (OWC) Wave Energy Converter (WEC) stands out. The OWC WEC is fundamentally composed of a hydro-pneumatic chamber connected to an airflow duct. The chamber is open below the sea-water-free surface, while the air duct is open to the atmosphere. Because of the incidence of waves on the chamber, the water column inside it presents a piston-type movement that compresses and decompresses the air, causing an alternate flow through the duct. This airflow drives a turbine installed at the duct which, in turn, drives an electricity generator [17].

The OWC operating principle can be numerically studied considering the incidence of waves over the device, resulting in the water–air flow that occurs within it. To do so, the OWC device, composed here by the hydro-pneumatic chamber and turbine duct, is inserted in a numerical wave flume. For the wave generation, one of the most used methodologies is through the Volume of Fluid (VOF) method, both for regular waves [18,19,20,21,22,23,24,25] and irregular waves [26,27,28,29,30,31].

On the other hand, it is also possible to analyze the OWC operating principle considering only its internal fluid-dynamic behavior, i.e., without the wave incidence over the device. This methodology is based on the piston-type movement of the water column, which is reproduced only considering the airflow inside the hydro-pneumatic chamber of the OWC (so-called air methodology), as in the work of El Marjani et al. [32], Conde and Gato [33], El Marjani et al. [34], Bouhrim and El Marjani [35], Bouhrim and El Marjani [36], El Barakaz and El Marjani [37], and Santos et al. [38]. One can highlight that in these works, a regular alternate airflow was always adopted, reproducing a hypothetical situation of an OWC submitted to regular waves.

Considering this last approach, the present work proposes the development, validation, and calibration of an axisymmetric computational model that considers only the airflow inside the OWC WEC. After that, a geometric optimization study is carried out employing the developed numerical model associated with the Constructal Design method and the Exhaustive Search technique under realistic sea state conditions, an approach that has not been explored in the literature. To do so, realistic sea state data of the coastal region of Rio Grande City (located in Rio Grande do Sul State, in southern Brazil) are employed to generate the boundary condition of prescribed velocity, allowing reproducing the irregular alternate airflow inside the device. The goal here is to maximize the OWC’s available power.

In summary, the present study brings as original scientific contribution the improvement of the computational modeling that was already adopted in previous works [32,33,34,35,36,37,38], using an axisymmetric computational model associated with a random airflow based on realistic sea state data (imposed as inlet prescribed velocity to mimic the piston-type movement of the water column that occurs inside the hydro-pneumatic chamber of an OWC), as well as the employment of this computational model to perform a geometric evaluation of 100 geometric configurations for the hydro-pneumatic chamber of the OWC defined by means the Constructal Design method.

In order to meet these goals, the present article is outlined as follows: After the Introduction Section, the mathematical model, application of the Constructal Design method, and definition of the study region are presented. Subsequently, the numerical model is addressed, including its validation, calibration, and the tests concerning the temporal and spatial discretization for the case study. Thereafter, the results obtained for the 100 numerical simulations are presented and discussed. Finally, the conclusions, acknowledgments, and references are presented.

2. Mathematical Modeling

The airflow inside the axisymmetric OWC domain is considered incompressible, unsteady, isothermal, and turbulent. Figure 1a presents a 3D image of the OWC, Figure 1b depicts the axisymmetric domain and its dimensions (with L₁ and H₁ being the diameter and height of the hydro-pneumatic chamber, respectively, L₂ and H₂ the diameter and height, respectively, of the transition region between the hydro-pneumatic chamber and the turbine duct, and L₃ and H₃ the diameter and height, respectively, of the air duct in which the turbine is installed), and Figure 1c indicates the boundary conditions used in the numerical simulations performed.

For this 2D axisymmetric problem, the governing equations are the mass, axial momentum, and radial momentum conservation equations, respectively, given by [39]:

$${\displaystyle \frac{\partial \rho }{\partial t}}+{\displaystyle \frac{\partial \left(\rho \overline{u_x}\right)}{\partial x}}+{\displaystyle \frac{\partial \left(\rho \overline{u_r}\right)}{\partial r}}+{\displaystyle \frac{\rho \overline{u_r}}{r}}=0$$

(1)

$$\begin{array}{l}{\displaystyle \frac{\partial \left(\rho \overline{u_x}\right)}{\partial t}}+{\displaystyle \frac{1}{r}}{\displaystyle \frac{\partial \left(r\rho \overline{u_x}\overline{u_x}\right)}{\partial x}}+{\displaystyle \frac{1}{r}}{\displaystyle \frac{\partial \left(r\rho \overline{u_r}\overline{u_x}\right)}{\partial r}}=-{\displaystyle \frac{\partial \overline{p}}{\partial x}}+{\displaystyle \frac{1}{r}}{\displaystyle \frac{\partial }{\partial x}}\left[r\left(\mu +{\mu }_t\right)\left(2{\displaystyle \frac{\partial \overline{u_x}}{\partial x}}-{\displaystyle \frac{2}{3}}\left(\nabla .\overrightarrow{u}\right)\right)\right]+\\ \hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}{\displaystyle \frac{1}{r}}{\displaystyle \frac{\partial }{\partial r}}\left[r\left(\mu +{\mu }_t\right)\left({\displaystyle \frac{\partial \overline{u_x}}{\partial r}}+{\displaystyle \frac{\partial \overline{u_r}}{\partial x}}\right)\right]+F_x\end{array}$$

(2)

$$\begin{array}{l}{\displaystyle \frac{\partial \left(\rho \overline{u_r}\right)}{\partial t}}+{\displaystyle \frac{1}{r}}{\displaystyle \frac{\partial \left(r\rho \overline{u_x}\overline{u_r}\right)}{\partial x}}+{\displaystyle \frac{1}{r}}{\displaystyle \frac{\partial \left(r\rho \overline{u_r}\overline{u_r}\right)}{\partial r}}=-{\displaystyle \frac{\partial \overline{p}}{\partial r}}+{\displaystyle \frac{1}{r}}{\displaystyle \frac{\partial }{\partial r}}\left[r\left(\mu +{\mu }_t\right)\left(2{\displaystyle \frac{\partial \overline{u_r}}{\partial r}}-{\displaystyle \frac{2}{3}}\left(\nabla .\overrightarrow{u}\right)\right)\right]+\\ \hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}{\displaystyle \frac{1}{r}}{\displaystyle \frac{\partial }{\partial x}}\left[r\left(\mu +{\mu }_t\right)\left({\displaystyle \frac{\partial \overline{u_r}}{\partial x}}+{\displaystyle \frac{\partial \overline{u_x}}{\partial r}}\right)\right]-2\left(\mu +{\mu }_t\right){\displaystyle \frac{\overline{u_r}}{r^2}}+{\displaystyle \frac{2}{3}}{\displaystyle \frac{\left(\mu +{\mu }_t\right)}{r}}\left(\nabla .\overrightarrow{u}\right)+F_r\end{array}$$

(3)

where ρ is the air density, (¯) is the time-averaged operator, u_x is the time-averaged axial velocity, u_r is the time-averaged radial velocity, t is the time, x is the axial coordinate, r is the radial coordinate, p is the time-averaged static pressure, μ is the air dynamic viscosity, μ_t is the turbulent viscosity, F_x is the axial source term, F_r is the radial source term, and $\nabla \cdot \overrightarrow{u}$ is defined as:

$$\nabla \cdot \overrightarrow{u}={\displaystyle \frac{\partial \overline{u_x}}{\partial x}}+{\displaystyle \frac{\partial \overline{u_r}}{\partial r}}+{\displaystyle \frac{\overline{u_r}}{r}}$$

(4)

Regarding the turbulence, it is worth mentioning that due to the random fluctuations that occur in several flow properties, it is necessary to employ a statistical approach. For this reason, the evolved quantities are defined as the sum of the mean and fluctuating components [40]. In this work, three Reynolds-Averaged Navier-Stokes (RANS) turbulence models were considered: standard κ-ε, standard κ-ω, and κ-ω SST, with κ being the turbulence kinetic energy, ε the turbulent dissipation rate, ω the specific turbulent dissipation rate, and SST the acronym for Shear-Stress Transport [41,42].

2.1. Standard κ-ε Model

The standard κ-ε model was proposed by Launder and Spalding [43], being one of the most used due to its robustness, economy, reasonable accuracy, and applicability for a wide range of turbulent flows. This turbulence model is based on the Boussinesq hypothesis, including one additional transport equation for the turbulent kinetic energy (κ) and another one for its dissipation rate (ε) [41], defined, respectively, as [43]:

$${\displaystyle \frac{\partial \left(\rho \kappa \right)}{\partial t}}+{\displaystyle \frac{\partial \left(\rho \kappa u_j\right)}{\partial x_j}}=\rho {\tau }_{ij}{\displaystyle \frac{\partial u_i}{\partial x_j}}+{\displaystyle \frac{\partial }{\partial x_j}}\left[\left(\mu +{\displaystyle \frac{{\mu }_t}{{\sigma }_{\kappa }}}\right){\displaystyle \frac{\partial \kappa }{\partial x_j}}\right]-\rho \epsilon$$

(5)

$${\displaystyle \frac{\partial \left(\rho \epsilon \right)}{\partial t}}+{\displaystyle \frac{\partial \left(\rho \epsilon u_j\right)}{\partial x_j}}={\displaystyle \frac{\partial }{\partial x_j}}\left[\left(\mu +{\displaystyle \frac{{\mu }_t}{{\sigma }_{\epsilon }}}\right){\displaystyle \frac{\partial \epsilon }{\partial x_j}}\right]+C_{1\epsilon }{\displaystyle \frac{\epsilon }{\kappa }}\rho {\tau }_{ij}{\displaystyle \frac{\partial u_i}{\partial x_j}}-C_{2\epsilon }\rho {\displaystyle \frac{{\epsilon }^2}{\kappa }}$$

(6)

where C_1ε = 1.44 and C_2ε = 1.92 are constants, σ_κ = 1.00 and σ_ε = 1.30 are the turbulent Prandtl numbers, respectively, for κ and ε, and τ_ij is the Reynolds stress tensor. It should be noted that the standard κ-ε model is valid only for fully turbulent flows, since its derivation considers the flow as fully turbulent and neglects the molecular viscosity effects [43]. In addition, from the association of κ and ε, the turbulent viscosity (μ_t) can be obtained by:

$${\mu }_t=\rho C_{\mu }{\displaystyle \frac{{\kappa }^2}{\epsilon }}$$

(7)

in which C_μ = 0.09 is a constant.

2.2. Standard κ-ω Model

The standard κ-ω model is an empirical turbulence model that also considered the Boussinesq hypothesis [41], applying two additional transport equations: one for the turbulence kinetic energy (κ) and one for the specific dissipation rate (ω), obtained, respectively, by [40]:

$${\displaystyle \frac{\partial \left(\rho \kappa \right)}{\partial t}}+{\displaystyle \frac{\partial \left(\rho \kappa u_j\right)}{\partial x_j}}=\rho {\tau }_{ij}{\displaystyle \frac{\partial u_i}{\partial x_j}}-{\beta }^{*}\rho \omega \kappa +{\displaystyle \frac{\partial }{\partial x_j}}\left(\mu +{\sigma }_{\kappa }{\mu }_t\right){\displaystyle \frac{\partial \kappa }{\partial x_j}}$$

(8)

$${\displaystyle \frac{\partial \left(\rho \omega \right)}{\partial t}}+{\displaystyle \frac{\partial \left(\rho \omega u_j\right)}{\partial x_j}}={\displaystyle \frac{\alpha \omega }{\kappa }}{\tau }_{ij}{\displaystyle \frac{\partial u_i}{\partial x_j}}-\beta \rho {\omega }^2+{\displaystyle \frac{\partial }{\partial x_j}}\left[\left(\mu +{\sigma }_{\omega }{\mu }_t\right){\displaystyle \frac{\partial \omega }{\partial x_j}}\right]+{\displaystyle \frac{\rho {\sigma }_d}{\omega }}{\displaystyle \frac{\partial \kappa }{\partial x_j}}{\displaystyle \frac{\partial \omega }{\partial x_j}}$$

(9)

with:

$$\beta ={\beta }_o{f}_{\beta }$$

(10)

$$f_{\beta }={\displaystyle \frac{1+85x_{\omega }}{1+100x_{\omega }}}$$

(11)

$$x_{\omega }=\left|{\displaystyle \frac{{\Omega }_{ij}{\Omega }_{jk}{\Omega }_{ki}}{{\left({\beta }^{*}\omega \right)}^3}}\right|$$

(12)

$${\Omega }_{ij}={\displaystyle \frac{1}{2}}\left({\displaystyle \frac{\partial u_i}{\partial x_j}}-{\displaystyle \frac{\partial u_j}{\partial x_i}}\right)$$

(13)

$${\sigma }_d=\left\{\begin{array}{c}0,{\displaystyle \frac{\partial \kappa }{\partial x_j}}{\displaystyle \frac{\partial \omega }{\partial x_j}}\le 0\\ {\sigma }_{do},{\displaystyle \frac{\partial \kappa }{\partial x_j}}{\displaystyle \frac{\partial \omega }{\partial x_j}}>0\end{array}\right\}$$

(14)

where σ_κ = 3/5 and σ_ω = 1/2 are the turbulent Prandtl numbers, respectively, for κ and ω, α = 13/25, β_o = 0.0708, β* = 9/100, and σ_do = 1/8. In turn, the turbulent viscosity is defined as:

$${\mu }_t={\displaystyle \frac{\rho \kappa }{\tilde{\omega }}}$$

(15)

in which:

$$\tilde{\omega }=\max \left\{\omega ,\hspace{0.17em}\hspace{0.17em}{\displaystyle \frac{7}{8}}\sqrt{{\displaystyle \frac{2S_{ij}{S}_{ij}}{{\beta }^{*}}}}\right\}$$

(16)

$$S_{ij}={\displaystyle \frac{1}{2}}\left({\displaystyle \frac{\partial u_i}{\partial x_j}}+{\displaystyle \frac{\partial u_j}{\partial x_i}}\right)$$

(17)

One can highlight that the standard κ-ω model is especially applicable for near-wall flows, while the earlier mentioned standard κ-ε model is indicated to flows far enough away from walls [40].

2.3. κ-ω SST Model

Proposed by Menter [44], the κ-ω Shear-Stress Transport (SST) turbulence model can employ the standard κ-ε model for the regions far away from walls and the standard κ-ω model for the near-wall regions [41,42]. The transport equations of the κ-ω SST model are similar to those of the standard κ-ω model, being defined as [44]:

$${\displaystyle \frac{\partial \left(\rho \kappa \right)}{\partial t}}+{\displaystyle \frac{\partial \left(\rho \kappa u_j\right)}{\partial x_j}}=\rho {\tau }_{ij}{\displaystyle \frac{\partial u_i}{\partial x_j}}-{\beta }^{*}\rho \omega \kappa +{\displaystyle \frac{\partial }{\partial x_j}}\left(\mu +{\sigma }_{\kappa }{\mu }_t\right){\displaystyle \frac{\partial \kappa }{\partial x_j}}$$

(18)

$${\displaystyle \frac{\partial \left(\rho \omega \right)}{\partial t}}+{\displaystyle \frac{\partial \left(\rho \omega u_j\right)}{\partial x_j}}={\displaystyle \frac{\alpha \omega }{\kappa }}{\tau }_{ij}{\displaystyle \frac{\partial u_i}{\partial x_j}}-\beta \rho {\omega }^2+{\displaystyle \frac{\partial }{\partial x_j}}\left[\left(\mu +{\sigma }_{\omega }{\mu }_t\right){\displaystyle \frac{\partial \omega }{\partial x_j}}\right]+2\left(1-F_1\right){\displaystyle \frac{\rho {\sigma }_{\omega 2}}{\omega }}{\displaystyle \frac{\partial \kappa }{\partial x_j}}{\displaystyle \frac{\partial \omega }{\partial x_j}}$$

(19)

where:

$$F_1=\tanh \left\{{\left\{\min \left[\max \left({\displaystyle \frac{\sqrt{\kappa }}{0.09\omega y}},{\displaystyle \frac{500\nu }{y^2\omega }}\right),{\displaystyle \frac{4\rho {\sigma }_{\omega 2}\kappa }{C{D}_{\kappa \omega }{y}^2}}\right]\right\}}^4\right\}$$

(20)

$$C{D}_{\kappa \omega }=\max \left(2\rho {\sigma }_{\omega 2}{\displaystyle \frac{1}{\omega }}{\displaystyle \frac{\partial \kappa }{\partial x_j}}{\displaystyle \frac{\partial \omega }{\partial x_j}},{10}^{-20}\right)$$

(21)

where F₁ is a blending function that is equal to 1 near the wall (activating the κ-ω model) and equal to 0 in the free stream (employing the κ-ε model), y is the distance to the next surface, and CD_κω is the positive portion of the cross-diffusion term. Moreover, the turbulent viscosity for this model is given by:

$${\mu }_t={\displaystyle \frac{\rho \kappa {\alpha }_1}{\max \left({\alpha }_1\omega ,\Omega F_2\right)}}$$

(22)

in which:

$$F_2=\tanh \left\{{\left[\max \left({\displaystyle \frac{2\sqrt{\kappa }}{0.09\omega y}},{\displaystyle \frac{500\nu }{y^2\omega }}\right)\right]}^2\right\}$$

(23)

where Ω is the absolute value of the vorticity and F₂ is a blending function. In addition, the following constants are also considered: σ_κ1 = 0.85, σ_ω1 = 0.5, σ_κ2 = 1.0, σ_ω2 = 0.856, α₁ = 0.31, β₁ = 0.075, β₂ = 0.0828, and β* = 0.09.

3. Geometric Evaluation Approach and Study Region

The developed axisymmetric computational model was used together with the Constructal Design method and the Exhaustive Search technique to evaluate and optimize the geometric configuration of an OWC WEC submitted to the realistic sea state characteristics from the coastal region of Rio Grande City, in southern Brazil.

The Constructal Design method is based on the Constructal Law, which states: “For a finite-size flow system to persist in time (to live/to survive), its configuration must evolve in such a way that it provides an easier access to the imposed (global) currents that flow through it” [45,46]. The Constructal Design has been widely used in heat transfer and/or fluid mechanics systems, including renewable energy applications, as presented by Dos Santos et al. [47]. More specifically, regarding the geometric evaluation of WECs, one can highlight some studies addressed to the Submerged Horizontal Plate [48] and Overtopping Device [49,50], as well as to the OWC [18,30].

In agreement with Dos Santos et al. [47], the Constructal Design application requires the definition of constraints (global and/or local), degrees of freedom (varying freely, but respecting the constraints), and at least one performance indicator (which must be maximized or minimized, seeking superior performance). To do so, an OWC WEC was considered, with dimensions based on those adopted in the European Wave Energy Pilot Plant on the Island of Pico, Azores, Portugal [51]: H₁ = 7.5 m, H₂ = 2.5 m, H₃ = 5.0 m, L₁ = L₂ = 12.0 m, and L₃ = 2.3 m (Figure 2a). From this geometric configuration and varying the degrees of freedom, several other OWC geometries were defined (for instance, Figure 2b,c), forming the search space of the analysis.

From Figure 2, three constraints were imposed:

$$A_1=H_1{L}_1=90.0 {\mathrm{m}}^2$$

(24)

$$A_2=H_2{\displaystyle \frac{\left(L_1+L_2\right)}{2}}=30.0 {\mathrm{m}}^2$$

(25)

$$A_3=H_3{L}_3=11.5 {\mathrm{m}}^2$$

(26)

where A₁ is the area of the inferior portion of the hydro-pneumatic chamber, A₂ is the area of the transition region between the hydro-pneumatic chamber and the turbine duct, and A₃ is the area of the turbine duct. Therefore, the total longitudinal section area of the OWC is defined as:

$$A_T=A_1+A_2+A_3=\left[\left(H_1{L}_1\right)\right]+\left[H_2{\displaystyle \frac{\left(L_1+L_2\right)}{2}}\right]+\left[\left(H_3{L}_3\right)\right]$$

(27)

Moreover, three degrees of freedom were adopted: H₁/L₁ and H₂/L₂ (variables) and H₃/L₃ (constant). Thus, Equation (27) can be rewritten in terms of the degrees of freedom as:

$${\left(L_2\right)}^2+\left(L_2{L}_1\right)+{\left[{\displaystyle \frac{H_2}{L_2}}\right]}^{-1}\left\{2\left[{\displaystyle \frac{H_1}{L_1}}\right]{\left(L_1\right)}^2+2\left[{\displaystyle \frac{H_3}{L_3}}\right]{\left(L_3\right)}^2-2A_T\right\}=0$$

(28)

From Equation (28), it is possible to obtain the values of L₂ through the values of degrees of freedom, while the values of H₂ can be defined by:

$$H_2=\left[{\displaystyle \frac{H_2}{L_2}}\right]L_2$$

(29)

In a similar way, and considering Equation (24), the values of L₁ and H₁ can be obtained, respectively, as:

$$L_1=\sqrt{A_1.{\left[{\displaystyle \frac{H_1}{L_1}}\right]}^{-1}}$$

(30)

$$H_1=\left[{\displaystyle \frac{H_1}{L_1}}\right]L_1$$

(31)

In summary, the geometric evaluation consisted of considering ten values for the degree of freedom H₁/L₁, from 0.1 to 1.0 with an increment of 0.1. So, for each value of H₁/L₁, ten values for the degree of freedom H₂/L₂ were assumed, varying from L₂ = L₁ (see Figure 2a) to L₂ = L₃ (see Figure 2c). In turn, the degree of freedom H₃/L₃ was kept constant and equal to 2.17. Therefore, 100 different geometric configurations for the hydro-pneumatic chamber of the OWC WEC were determined through the Constructal Design application, defining the search space, as illustrated in Figure 3. In sequence, the fluid-dynamic behavior of each case was numerically simulated and compared among each other by means of the Exhaustive Search technique, seeking to maximize the performance indicator.

As the performance indicator, the OWC’s available power was adopted. Based on El Barakaz and El Marjani [37], one can assume that the water column has a piston-type movement inside the hydro-pneumatic chamber. Hence, the theoretical power absorbed by the OWC can be calculated as:

$$P=F\cdot {\displaystyle \frac{S_c}{t}}$$

(32)

where P is the OWC’s theoretical power, F is the force imposed by the water surface, S_c is the water column displacement, and t is the interval time of the displacement. So, if F is considered as the product of total pressure of the water surface by the cross-sectional area of the hydro-pneumatic chamber, and if the ratio between the water column displacement and time is assumed as the water column velocity, it is possible to rewrite Equation (32) as:

$$P=p_{total}\cdot A\cdot u=p_{total}·{\displaystyle \frac{\dot{m}}{\rho }}$$

(33)

where p_total is the total pressure inside the chamber, A is the cross-sectional area of the chamber, u is the airflow velocity inside the chamber, $\dot{m}$ is the air mass flow rate, and ρ is the air density.

However, Equations (32) and (33) provide instantaneous values for the OWC’s available power. If a global value is desired, it is necessary to calculate its Root Mean Square (RMS) value, as [18]:

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

(34)

where P_RMS is the RMS value of the OWC’s available power and N is the number of monitored instantaneous values.

It is noteworthy that the adoption of the Exhaustive Search technique together with the Constructal Design method is a widely used approach because, in this way, it is possible not only to determine the optimal geometry, but also to evaluate and understand how the degrees of freedom influence the performance indicator.

Regarding the study region, of all the Brazilian coastline, the southern region stands out for having the highest energy potential when compared to other regions of the country [15,16]. While in the south, power values along the coast reach 11 kW/m of wave front, in northern Brazil, these values range between 5 and 8 kW/m of wave front [16]. Furthermore, in agreement with the Brazilian Institute of Geography and Statistics (IBGE), one of the regions in the state of Rio Grande do Sul (RS) with a higher population density and, consequently, higher electricity consumption, is formed by the cities of Rio Grande and Pelotas. Based on that, as Rio Grande is a coastal city, the study region was defined as Cassino beach, as illustrated in Figure 4.

To impose on the OWC WEC a realistic prescribed velocity as the inlet boundary condition (air methodology), a database containing information about the sea state of the coast of RS for the year 2014 was employed. This database was developed by Oleinik [52], including data from 14 points located along the coast of RS. Among the available points, the one located at Cassino beach in the city of Rio Grande was selected (see Figure 4), with coordinates 52°17′47.25″ W and 32°22′30.95″ S, at a depth of 10 m and approximately 2 km from the coast.

The database created by Oleinik [52] was obtained through the Open Telemac-Mascaret modeling system, using the TOMAWAC spectral model developed by Benoit et al. [53]. For the generation of the database, sea state parameters used in TOMAWAC were obtained from historical data of the Wave Watch III wave model from NOAA (National Oceanic and Atmospheric Administration). The surface wind boundary condition in TOMAWAC was derived from the NOAA Reanalysis 1 project, and bathymetric data representing the seafloor were acquired from nautical charts from the Directorate of Hydrography and Navigation of the Brazilian Navy and the General Bathymetric Chart of the Oceans [52]. The sea state simulation was run for a one-year time interval, starting on 1 January 2014 and ending on 31 December 2014, with computational model validation presented by Oleinik et al. [54].

The transformation of the sea state spectrum obtained in TOMAWAC into a time series of free surface elevation was achieved through the inverse Fourier transform. Further details about this procedure can be found in Machado et al. [29] and Oleinik et al. [55]. Therefore, from the realistic sea state free surface elevation (η) of the Cassino beach, it was possible to obtain the vertical velocity (w) necessary to employ the air methodology in the OWC analysis. To do so, considering that velocity is the derivative of position with respect to time, obtaining the vertical velocity of the free surface involves calculating the derivative of the time series of elevation with respect to time [56], i.e.,

$$w={\displaystyle \frac{\partial \eta \left(t\right)}{\partial t}}$$

(35)

To apply Equation (35), it was necessary to define an interval time during the year of 2014 based on the most frequent significant wave height (Hs) and mean wave period (Ts). For that, Figure 5 presents a bivariate histogram showing the frequency of occurrence of pairs of different values of Hs and Ts, making it possible to identify that one of the most frequent was the one with Hs = 0.54 m and Ts = 6.94 s.

Once the characteristics deemed representative of the sea state in the Cassino beach were established, it was also necessary to define the duration of the sea state to which the OWC WEC would be subjected during the geometric evaluation study. Since the flow is irregular, a small time step may be required for solving the conservation equations; moreover, it was proposed to evaluate 100 different geometric configurations of its hydro-pneumatic chamber. Therefore, taking these aspects into consideration, a velocity profile with a duration of 15 min (900 s) was defined.

Knowing Hs, Tm, and the time interval, it was necessary to search within the database for a time series of free surface elevation that represented these parameters as faithfully as possible. Thus, a series of 900 s free surface elevations with Hs = 0.54 m and Tm = 7.27 s was adopted, in which Equation (35) was applied, generating the transient irregular velocity variation (shown in Figure 6) that was imposed as a boundary condition of prescribed velocity in the OWC WEC.

4. Numerical Modeling

One of the first steps in the validation process is the definition of the computational model that will be used. In the present work, the choice of the model was carried out during the validation process itself, where the main equations available for solving this type of problem, in the Ansys Fluent software (version 18.1), were combined, and the results obtained were compared with experimental data in order to assess the accuracy of each model tested. To this end, not only the accuracy of each model in relation to the experimental data was considered, but also the processing time required to obtain the results. Since during the geometric evaluation the aim was to study 100 different domains, it means that processing time was an important factor in completing this research.

The simplified approach adopted here for the numerical simulation of an OWC-type converter (air methodology) allowed the problem to be reduced to a 2D domain with a change in area, through which an internal, transient, and turbulent airflow occurs. Considering the operating principle of the OWC converter, it was possible to use a prescribed velocity as an entry condition in the domain to represent the movement of waves arriving to the interior of the device. At the exit, the imposition of a prescribed pressure is a good choice, as most OWC devices have one or more openings to the atmosphere, through the turbine duct, where the pressure is 1 atm. The other contours of the domain can be defined as a wall or as an axis of symmetry, depending on how the domain is defined. Such characteristics as the shape of the domain, flow, and boundary conditions were found in three different experiments already carried out, which were used in the validation process and choice/definition of the numerical model used in this research.

The first experiment reproduced deals with a stationary and turbulent airflow through a 2D internal domain with a change in area. The second experiment reproduced also deals with a stationary and turbulent airflow, however, through the internal region of a 3D domain. Finally, the third experiment deals with a transient and turbulent flow of air through the internal region of a 3D domain. Different turbulence models (standard κ-ε and κ-ω SST), different wall functions (Scalable Wall Function and Enhanced Wall Treatment), and various coupling algorithms (SIMPLE, SIMPLEC, PISO, and Coupled) were tested, as indicated in

Table 1. Numerical models tested.

Model	Turbulence Model	Wall Function	Pressure–Velocity Coupling
1	Standard κ-ε	Standard Wall Function	SIMPLE
2	Standard κ-ε	Standard Wall Function	SIMPLEC
3	Standard κ-ε	Standard Wall Function	PISO
4	Standard κ-ε	Standard Wall Function	Coupled
5	Standard κ-ε	Scalable Wall Function	SIMPLE
6	Standard κ-ε	Scalable Wall Function	SIMPLEC
7	Standard κ-ε	Scalable Wall Function	PISO
8	Standard κ-ε	Scalable Wall Function	Coupled
9	Standard κ-ε	Enhanced Wall Treatment	SIMPLE
10	Standard κ-ε	Enhanced Wall Treatment	SIMPLEC
11	Standard κ-ε	Enhanced Wall Treatment	PISO
12	Standard κ-ε	Enhanced Wall Treatment	Coupled
13	κ-ω SST	-	SIMPLE
14	κ-ω SST	-	SIMPLEC
15	κ-ω SST	-	PISO
16	κ-ω SST	-	Coupled

. Furthermore, in all models tested, the Second-Order Upwind interpolation scheme was used in the spatial discretization of pressure and in the treatment of other advective terms, while the First-Order Upwind interpolation scheme was used in the transient formulation of the models, in which flow depends on time. The residue of solving the mass and momentum conservation equations, as well as the turbulence model, was set to 10⁻⁶.

4.1. Validation 1

The first validation was carried out with an experiment in which an internal, turbulent, incompressible, and stationary airflow occurred over a descending step in a 2D domain. The domain geometry, presented in Figure 7, as well as the flow conditions, are those defined in case C30 of the Ercoftac Classic Collection Database [57].

The experimental tests, developed by Driver and Seegmiller [58], were carried out in a wind tunnel with a cross-section of 15.1 × 10.16 cm² (D₁ = 10.16 cm; D₂ = 11.37 cm). The descending step, with height H = 1.27 cm, was located 1 m after the tunnel entrance section. In this section, a 12.5 cm-long strip of abrasive sheet was placed across the entire width of the tunnel, so that it was possible to obtain a completely turbulent boundary layer upstream of the step.

The experiment was carried out at a constant undisturbed flow velocity of U_R = 44.2 m/s, and at ambient pressure and temperature, these conditions correspond to a Mach number Ma = 0.128. The thickness of the boundary layer of the wall was δ = 1.9 cm and the Reynolds number was Re = 5000 in the section located at a distance 4H upstream of the step. This high Reynolds number was chosen to ensure that the boundary layer was fully turbulent before passing through the step.

Driver and Seegmiller [58] carried out measurements of the components of the average velocity and Reynolds stresses by Laser Doppler Anemometry (LDA) in several cross-sections. They also measured the evolution of pressure on the floor and ceiling and the surface shear tension using a Laser Interferometry technique on an oil film on the lower surface. During the experiments, the authors evaluated the effect of the slope of the wind tunnel roof, defined by α, on the flow velocity, pressure, and shear stress fields.

In the numerical simulations carried out to validate the model, the case in which the ceiling of the domain, under which the flow occurs, was horizontal (α = 0°) and the coordinates were non-dimensionalized by the height of the step, H, and the velocity components of the input velocity, U_R, was considered. The origin of the reference was placed in the lower corner of the step; thus, it was considered that, horizontally, the domain was comprised between −4 < x/H < 40, and vertically between 0 < y/H < 9, parameters also adopted by Eça and Hoekstra [59]. The boundary conditions and domain dimensions can be seen in Figure 7b.

As an input condition, a prescribed velocity profile was imposed, as illustrated in Figure 7a, associated with profiles of the components of the turbulence models, namely, the turbulence kinetic energy and its dissipation rate. These profiles were obtained through a non-linear regression of data made available by the Ercoftac Classic Database, which were subsequently spatially discretized for the simulated domain mesh. The equations obtained from the non-linear regressions to the velocity components in x and y directions are, respectively, given by:

$$u=-2.30\times {10}^{-3}{x}^4+4.59\times {10}^{-2}{x}^3-3.30\times {10}^{-1}{x}^2+1.01x-9.20\times {10}^{-2}$$

(36)

$$\begin{array}{l}v=1.99\times {10}^{-5}{x}^6-5.98\times {10}^{-4}{x}^5+6.98\times {10}^{-3}{x}^4-4.00\times {10}^{-2}{x}^3+\\ \hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}1.16\times {10}^{-1}{x}^2-1.57\times {10}^{-1}x+7.71\times {10}^{-2}\end{array}$$

(37)

In a similar way, profiles were obtained for the turbulence kinetic energy, κ, and its dissipation rate, ε and ω, referring, respectively, to the κ-ε and κ-ω turbulence models, as follows [40]:

$$\kappa ={\displaystyle \frac{1}{2}}\left(\overline{{u^{\prime }}^2}+\overline{{v^{\prime }}^2}\right)$$

(38)

with $u^{\prime }$ and $v^{\prime }$ also being obtained through non-linear regressions from Ercoftac Classic Collection Database [57] and defined as:

$$u^{\prime }=5.16\times {10}^{-4}{x}^6-1.55\times {10}^{-2}{x}^5+\hspace{0.17em}2.11\times {10}^{-1}{x}^4-1.64x^3+7.38x^2-1.77\times {10}^{1}x+1.77\times {10}^1$$

(39)

$$v^{\prime }=-1.31\times {10}^{-3}{y}^6+3.94\times {10}^{-2}{y}^5-4.52\times {10}^{-1}{y}^4+2.49y^3-6.525y^2+\hspace{0.17em}6.13y+6.06\times {10}^{-1}$$

(40)

and

$$\epsilon =C_{\mu }^{3/4}{\displaystyle \frac{{\kappa }^{3/2}}{l}}$$

(41)

$$\omega ={\displaystyle \frac{{\kappa }^{1/2}}{C_{\mu }^{1/4}l}}$$

(42)

where C_μ is a constant (which for the standard model was 0.09) and l is the turbulence length scale.

Additionally, as an exit condition, a prescribed pressure of 101,325 Pa was imposed, a value representative of atmospheric pressure, and in the remaining contours, the wall condition was used.

After the simulations were carried out, an error was calculated to estimate the difference between the numerical solution and the experimental data. To calculate the error, the magnitude of the vertical and horizontal velocities was monitored at 458 points spread across the domain; subsequently, the absolute difference between the experimental and numerical results was calculated for each point. This difference, in turn, was non-dimensionalized by the reference velocity U_R = 44.2 m/s.

This methodology, of non-dimensionalization of absolute point differences by a reference velocity, was used because dividing the point difference by the experimental value obtained at each point ends up resulting in extremely large errors when used at low-magnitude values. The vertical velocity field has, for the most part, magnitudes smaller than one unit and, therefore, when used in the error calculation, they end up resulting in an error so large that it could completely influence, in an undesirable way, the choice of a model.

The errors of all monitored points in the domain were then averaged for each of the models, allowing them to be compared appropriately. Through Figure 8, it is possible to compare both the error and the processing time required for each of the 16 models (see Table 1). The processing time, in turn, was also non-dimensionalized; thus, the time required to solve the equations in each model was divided by the longest time obtained in the simulations (10 h).

From Figure 8, one can note that models 3, 4, 8, 11, and 15 reached 100% of the processing time, that is, they took 10 h to complete the simulation. This occurred because these cases simply did not reach convergence for the stipulated residues, and the processing time was limited to precisely 10 h to prevent these simulations from taking too long. It is also possible to note that all models achieved a similar error, below 3%, even those that did not reach convergence for the chosen residuals. The differences among these models were due to the processing time, where models 12 and 16 stood out for requiring less than 10% of the proposed maximum time, and both have in common the solution of pressure and velocity in a coupled way.

4.2. Validation 2

The second validation was carried out with an experiment in which an internal, turbulent, incompressible, and stationary airflow occurs through axisymmetric expansions in a 3D domain. The domain geometry, shown in Figure 9, as well as the flow conditions, are those defined in case C75 of the Ercoftac Classic Collection Database [57].

The experimental tests developed by Stieglmeier et al. [60] were conducted in a vertically mounted test facility. The flow was directed from a main tank to a tube at the diffuser inlet, with diameter D₁ = 50 mm. A trip wire was placed immediately downstream of the inlet to ensure fully developed turbulent flow in the test section. The test section consisted of interchangeable diffusers with half angles of α = 14°, 18°, or 90° (see Figure 9a). The downstream diameter of the angular section had the dimension D₂ = 80 mm, resulting in a diameter ratio of D₂/D₁ = 1.6. The maximum possible Reynolds number, based on the average flow velocity, was Re = 15,600. The velocity on the center line upstream of the diffuser, U_R, used in the non-dimensionalization of the results, was equal to 2.51 m/s for all tested geometries.

The test section was enclosed in an external, square containment vessel extending 100 mm upstream and 600 mm downstream of the diffuser inlet. The volume between the test section and the external containment was filled with the working fluid, which was selected to have a refractive index equal to that of the containment glass. Thus, the test section tube was mechanically present, but optically transparent, facilitating the use of a Laser Doppler Anemometer for velocity measurements.

The Laser Doppler Anemometer system was mounted on a support table equipped with gauges that guaranteed high relative positioning accuracy. The absolute positioning of the measurement volume was determined by traversing the wall boundaries and detecting the changing frequency. The absolute positioning accuracy was estimated to be ± 100 µm in the z direction, ±100 µm in the radial direction perpendicular to the optical axis, and ±1 mm in the radial direction parallel to the optical axis. The latter was less accurate due to the small length of 2 mm of the control volume.

In the numerical simulations carried out in the present work to validate the computational model, a case in which the axisymmetric expansion occurs suddenly (α = 90°, as represented in Figure 9b) was considered. In addition, the data used in the comparison were from the radial direction perpendicular to the optical axis, as they had greater precision. The coordinates were non-dimensionalized by the step height obtained from the difference between the diffuser inlet and outlet diameters, H = 15 mm, and the velocity components, by the center line velocity, U_R = 2.51 m/s. The simulated domain was 2D and axisymmetric, whose reference origin was placed on the axisymmetric axis, below the step; thus, it was considered horizontal, with the domain comprised between −2 < x/H < 40, and vertically between 0 < r/H < 8/3. The boundary conditions and dimensions of the simulated domain can be seen in Figure 9b.

In a similar way to validation 1, a prescribed velocity profile was imposed as an inlet boundary condition (see Figure 9a) together with the turbulence model’s components. To do so, non-linear regressions of experimental data from the Ercoftac Classic Collection Database [57] were obtained. The velocity components in x and y directions are defined, respectively, as:

$$\begin{array}{l}u=-6.29\times {10}^{-8}{x}^6+8.00\times {10}^{-7}{x}^5+5.82\times {10}^{-5}{x}^4-1.50\times {10}^{-3}{x}^3+\\ \hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}1.07\times {10}^{-2}{x}^2-3.69\times {10}^{-2}x+2.49\end{array}$$

(43)

$$\begin{array}{l}v=-2.51\times {10}^{-8}{y}^6+1.82\times {10}^{-6}{y}^5-5.02\times {10}^{-5}{y}^4+6.49\times {10}^{-4}{y}^3-\\ \hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}3.79\times {10}^{-3}{y}^2+5.97\times {10}^{-3}y+3.21\times {10}^{-2}\end{array}$$

(44)

In turn, the profiles for the turbulence kinetic energy, κ, and its dissipation rate, ε and ω, referring, respectively, to the κ-ε and κ-ω turbulence models, were obtained employing Equations (38), (41), and (42), in addition to:

$$\begin{array}{l}{u}^{\prime }=-8.82\times {10}^{-8}{x}^6+6.49\times {10}^{-6}{x}^5-1.80\times {10}^{-4}{x}^4+2.30\times {10}^{-3}{x}^3-\\ \hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}1.34\times {10}^{-2}{x}^2+2.81\times {10}^{-2}+7.63\times {10}^{-3}\end{array}$$

(45)

$$\begin{array}{l}{v}^{\prime }=\begin{array}{r}\hfill -3.38\end{array}\times {10}^{-8}{y}^6+2.49\times {10}^{-6}{y}^5-6.87\times {10}^{-5}{y}^4+8.76\times {10}^{-4}{y}^3-\\ \hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}5.042\times {10}^{-3}{y}^2+\hspace{0.17em}\begin{array}{r}\hfill 9.85\times {10}^{-3}\end{array}y+6.82\times {10}^{-3}\end{array}$$

(46)

Moreover, a prescribed pressure of 101,325 Pa was imposed as an outlet condition, representing the atmospheric pressure, while in the upper contour, the wall boundary condition was used, and in the lower contour, the axisymmetric boundary condition was adopted.

After carrying out simulations for all proposed models, the error was calculated using the same methodology mentioned previously. This time, 142 points spread across the domain were monitored. The absolute difference found at each point was non-dimensionalized by the reference velocity U_R = 2.51 m/s. Through Figure 10, it is possible to compare both the error and the processing time required for each of the 16 models (see Table 1). The processing time, as before, was dimensionless by a maximum time, adopted here with the value of 11 h.

It is noted in Figure 10 that models 3, 4, 7, 11, and 15 reached 100% processing time and 10% error. This occurred because these five models diverged and, in fact, there was no way to estimate the error between the numerical solution and the experimental result, nor the time required to reach the solution because the solution could not be found. Therefore, just to illustrate the divergence that occurred between these models, we chose to display them on the graph with maximum processing time and maximum error (here represented by 10%). It is also possible to note that there was a greater error variation among the models that reached convergence, ranging from 1.26% to 4.11%. In this sense, models 1 and 2 were more prominent and reached the lower error value of 1.26%, having the standard κ-ε turbulence model as a common characteristic. However, if processing time was considered, models 8, 12, and 16 achieved better values, having in common, again, the solution of pressure and velocity in a coupled way.

The aim for this research was to find the best relationship between processing time and error obtained; in this regard, the models that were most adequate in the second validation were numbers 8 and 16.

4.3. Validation 3

The third and final validation was carried out with an experiment in which an internal, turbulent, incompressible, and transient airflow occurs through a 3D divergent conical duct, which was the closest to the object of study of this research (an OWC WEC, as represented in Figure 1a, with an irregular and alternating airflow). The domain geometry, presented in Figure 11a, as well as the flow conditions, were those obtained from Sumida [61].

Sumida [61] developed an experimental investigation of a pulsating turbulent flow in a diverging conical duct with an angle of 12°. The experiments were performed under conditions of Womersley numbers between 10 and 40, mean Reynolds number Re_ta = 20,000, and oscillatory Reynolds number Re_os = 10,000. The static pressure on the wall and the axial velocity were measured in several sections. The objective was to evaluate the increase between the pressures at the inlet and outlet of the divergent duct due to the increase in flow rate, as well as comparing the phase-averaged velocity profiles and the intensity of turbulence in the cross-section, with an equivalent constant flow.

During the experiments, a system composed of a pulsating flow generator, a conical-shaped test section, and measuring devices was used. The pulsating flow, generated by the system, is composed of a constant velocity profile and an oscillating velocity profile. Constant flow was provided, through a surge tank, by a suction blower, which ensured that the flow was independent of the superimposed oscillation and the length of the test section. On the other hand, the oscillating flow was introduced by a piston, with a diameter of 300 mm and an adjustable stroke length from 0 to 1000 mm, controlled by a computer.

The duct used in the tests has a total divergence angle of 2α = 12° and an area ratio of A₂/A₁ = 6.25, with the domain entry radius r = 40 mm, and the exit radius 2.5r = 100 mm. The divergent duct was constructed from transparent acrylic material, precision machined and connected via a slip ring. Additionally, the ring had static pressure holes, 0.8 mm in diameter and spaced at 90°, and a small hole for inserting a hot-air probe. Straight transparent glass ducts with lengths of 3700 mm and 4200 mm were connected to the inlet and outlet of the divergent duct, respectively.

Furthermore, the static pressure on the wall was measured using a diffuse-type semiconductor pressure transducer, in 11 sections between z = −44.2r in the upstream straight duct and z = 43.8r in the downstream straight duct, where z is the length measured along the axisymmetric axis that originates at the inlet of the divergent duct. Velocity measurements were performed with a hot-wire anemometer, obtained in 8 different sections, in the range of −4.0 < z/r < 19.2.

To facilitate subsequent analyses, the coordinates were dimensionless by the entrance radius, r = 40 mm, and the velocity components, by the constant flow velocity imposed at the entrance of the test section, U_R = 3.675 m/s. The simulated domain was 2D and axisymmetric, whose reference origin was placed on the axisymmetric axis, at the entrance to the conical region; thus, it was considered that, horizontally, the domain was comprised between −4.00 < z/r < 39.20, and vertically between 0 < y/r < 2.5. The boundary conditions as well as the dimensions of the simulated domain can be seen in Figure 11b.

As an inlet condition, a velocity profile with a hydraulic diameter of 80 mm and a turbulence intensity of 5% were imposed. This velocity profile was obtained from a mass flow equation that discretizes the flow as a function of time. As a complement, a spatial discretization was developed based on non-linear regressions of experimental data provided by Sumida [61]. Furthermore, as a boundary condition at the exit, a prescribed pressure of 101,325 Pa was imposed, representative of the atmospheric pressure, with a hydraulic diameter of 200 mm and turbulence intensity of 5%. In the upper wall, the wall boundary condition was used, and in the lower contour, the axisymmetric boundary condition was used.

The boundary condition imposed on the domain inlet, z/r = −4, is defined as:

$$u_r\left(t,r\right)=\left[3.675+\left(1.875\sin \left(3.675t\right)\right)\right]\Delta u$$

(47)

where Δu represents the spatial discretization of the velocity, given by:

$$\Delta u=\left\{\left[1-1.03\times {10}^{-2}\exp \left(-5.37\times {10}^7(r^{-7\times {10}^3})\right)\right]-8.39\times {10}^{-2}\tan \left({3.7110}^{-2}r\right)\right\}$$

(48)

where t is the time and r is the local radius of the simulated domain (in mm).

After carrying out the simulations for all 16 proposed models, the error was calculated using the same methodology used in the first 2 validations; this time, 96 different points spread across the domain were monitored. The absolute difference found at each point was dimensionalized by the reference velocity U_R = 3.675 m/s.

Through Figure 12, it is possible to compare both the error and the processing time required for each of the proposed models (see Table 1). The processing time, as before, was dimensionalized by a maximum time, adopted here with the value of 79 h.

One can observe in Figure 12 that model 8 reached 100% of the processing time. This time, there was no divergence or lack of convergence: this model took 79 h to complete the simulation and, this time, was used as a non-dimensionalization parameter.

Moreover, if compared with the validation cases 1 and 2, it is possible to note that there was a considerable increase in the error found in all models, ranging from 5.14% to 7.00%. This increase can be explained by the way in which the experimental data were obtained. Unlike previous cases, in which the speed profiles were obtained from a database, in this case, the data were extracted manually from the graphs presented by Sumida [61]. Furthermore, another aggravating factor was the input boundary condition in the simulated domain, which, in addition to requiring an equation that discretizes the velocity profile over time, also requires a second equation to perform the spatial discretization, without which the velocity generated by the numerical solution would not be as close to the experimental data, due to the shear suffered by the flow in the domain wall.

As for the error obtained, there was no difference between the pressure–velocity coupling methods, with the turbulence models being responsible for the resulting differences. The difference between the largest error, obtained in models 13 to 16, and the smallest, observed in models 1 to 4, was 1.86%.

As in the first validation, the biggest discrepancy among the models was related to the processing time. In this regard, model 9, with an error of 5.76% and a processing time of 5.6 h, and model 13, with an error of 6.99% and a processing time of 5.3 h, were the most promising.

4.4. Numerical Model Adopted

From the validation 1, in which an internal and stationary flow was reproduced in a 2D domain with changes in area, there was no significant difference among the errors obtained by each of the 16 models proposed in Table 1. Regarding the processing time, those that achieved the best results were those in which the standard κ-ε turbulence model was used in association with the Enhanced Wall Treatment wall function, as well as those that used the κ-ω SST turbulence model, with greater emphasis on those that solved the solutions in a coupled way.

In the validation 2, where an internal and stationary flow was reproduced in a 3D domain with change in area, again, the greatest emphasis was on the models that solved the solutions in a coupled way, namely, 8, 12, and 16 (see Table 1), with the latter having the smallest error obtained. In general, considering the error obtained in relation to the experimental results and the time required to process the numerical solution, model 16 is the most recommended when the evaluated flow is independent of time. However, for transient flows, the scenario changes a little.

The validation 3, with an internal and transient flow reproduced in a 3D domain with changes in area, did not show such a large difference in the error obtained by each model, with 1.86% being the difference between the best and worst cases. Regarding the processing time, as in the validation 1, the models that achieved the best results were those in which the standard κ-ε turbulence model was used in association with the Enhanced Wall Treatment wall function, and the models that used the κ-ω SST turbulence, with emphasis on those that solved the pressure–velocity coupling with the SIMPLE method.

Taking all these results into consideration, the models that presented the best conditions to be used in simulations of transient flows during the geometric evaluation were models 9 and 13 (see Table 1). As a final test, these two models were used in simulations in which an irregular flow, with a realistic sea state velocity profile, was imposed as an entry condition. During testing, model 9 showed convergence only for very small time steps, diverging for steps greater than 0.1 s. Model 13 showed convergence for a wide range of time step sizes, even those as large as 1 s. Due to this stability, model 13, which uses the κ-ω SST turbulence model and solves the pressure–velocity coupling through the SIMPLE method, was then chosen to simulate the transient, irregular, incompressible, and turbulent flow that will be applied in the evaluation geometry of the OWC converter.

4.5. Time Step, Residue, and Mesh Independence Tests

As the irregular flow is transient, it is essential to use an appropriate time step (Δt) during numerical simulations. It is well known that usually, the smaller the time step, the more accurate the results obtained numerically; however, this accuracy is paid for with the time required to obtain the solution. Therefore, it is essential to choose a time step that allows a good balance between numerical accuracy and processing time. Furthermore, it is common to establish a residue as a convergence criterion to obtain numerical solutions and, just like the time step, the smaller the residue, the more accurate the solution will be and, consequently, the longer the processing time will be. Therefore, the choice of the size of this residue must also be conditioned on the accuracy you want to achieve and the time you have available to carry out the research. Taking this into account, a time step and residue independence test was proposed, where seven different time steps associated with three residue sizes were compared, as shown in

Table 2. Time step and residue independence test.

Δt (s)	Residue
0.001	1 × 10−3	1 × 10−4	1 × 10−5
0.005	1 × 10−3	1 × 10−4	1 × 10−5
0.010	1 × 10−3	1 × 10−4	1 × 10−5
0.050	1 × 10−3	1 × 10−4	1 × 10−5
0.100	1 × 10−3	1 × 10−4	1 × 10−5
0.500	1 × 10−3	1 × 10−4	1 × 10−5
1.000	1 × 10−3	1 × 10−4	1 × 10−5

. The numerical model used in the tests was number 13 (see Table 1), as chosen in the previous analysis.

During the simulations, a 2D and axisymmetric OWC WEC domain was used, whose equivalent 3D geometry (see Figure 1), obtained by revolution, had the inlet diameter of L₁ = L₂ = 10 m, outlet diameter of L₃ = 2.3 m, hydro-pneumatic chamber height of H₁ + H₂ = 12 m, and turbine duct height of H₃ = 5 m. As an input condition, a transient and irregular alternating velocity profile lasting 1 min was imposed. For the process of comparing the tested parameters, a monitor was created in the center of the turbine duct, where the dynamic, static, and total pressures, the mass flow rate, and the axial and radial components of the velocity were monitored. The processing time required to obtain the solution for each case is shown in Figure 13.

Regarding the processing time (Figure 13), it is possible to see that, as expected, the time increased the smaller the time step and residue chosen, reaching approximately 62 h in the numerical simulation in which a time step of 0.001 s was used in conjunction with a residue of 1 × 10⁻⁵. The 1 × 10⁻³ residue was more prominent in relation to the processing time; however, due to the unreliable results achieved for the monitored variables, it was decided to discard the possibility of its use. Furthermore, there was not a big difference between the solutions achieved with the residues of 1 × 10⁻⁴ and 1 × 10⁻⁵, compared to a significant gain in time for the first residue, and it was based on this fact that the choice of the residue of 1 × 10⁻⁴ was made for later solutions. In addition, the smallest acceptable time step for this residue, based on the results achieved for the monitored parameters, was 0.100 s. This aspect can be exemplified with one of the monitored parameters, the mass flow rate (ṁ), shown in Figure 14.

One can note in Figure 14 that there was no relevant difference in the magnitude of ṁ for values of Δt ≤ 0.100 s among the three residues investigated. However, for other monitored parameters (such as dynamic, static, and total pressures), this same trend did not occur, i.e., significant differences were found when the residue was 1 × 10⁻³, justifying the use of the residue of 1 × 10⁻⁴ and Δt = 0.100 s.

The last testing stage concerned the spatial discretization of the domains to be analyzed. To do so, using the time step and residue previously defined, six different mesh refinements were used, and the same flow characteristics measured in the previous test were compared. A deviation was calculated between the results obtained for each refinement and then, the deviations obtained for each variable were averaged to obtain a global value for each refinement. The size of each mesh used in the test, the processing time, and the deviation obtained can be found in

Table 3. Mesh independence test.

Mesh	Average Cell Size (mm)	Number of Volumes	Processing Time (h)	Deviation (%)
1	60	26,641	0.9	1.46
2	50	35,892	0.8	0.87
3	40	53,014	1.1	0.65
4	30	88,044	2.0	0.62
5	20	187,026	6.7	0.60
6	15	320,655	7.9	-

.

From mesh 2 onwards, the global deviation was not significant, remaining below 1%, and from mesh 4 onwards, there was a substantial increase in the number of volumes and the processing time. Although the results indicated that the refinement obtained by mesh 2 would already be sufficient, it was decided to choose the most refined possible spatial discretization before there was a substantial increase in processing time, i.e., the mesh number 4, where the elements had an average size of 30 mm. Besides that, it is worth mentioning that the spatial-averaged y+ was lower than 1.0 for the simulations with κ-ω SST.

5. Results and Discussions

As already mentioned, during the geometric evaluation, two degrees of freedom were varied, namely, the relationship between the height and width of the OWC’s hydro-pneumatic chamber (H₁/L₁) and the relationship between the height and width of the smallest base of its trapezoidal region (H₂/L₂), inserted between the lower part of the chamber and the turbine duct (see Figure 2b). The aim was to evaluate the influence of these geometric parameters on the available power of the OWC WEC.

The estimate of the theoretical available power was achieved with the product of the total pressure and the volume flow rate, measured in the region where the turbine was intended to be installed. The volume flow rate, in turn, could be obtained by the ratio between the mass flow rate and the density of the fluid—in this case, air. For this reason, the mass flow rate and total pressure were measured using a monitor inserted in the middle of the turbine duct in each of the 100 cases numerically simulated. Figure 15 and Figure 16 present, respectively, the variation in the mass flow rate and total pressure due to the variation in the H₂/L₂ ratio for each H₁/L₁ value.

Based on the results in Figure 15, it is possible to affirm that the air density variation inside the OWC can be neglected in the computational modeling, allowing to consider the airflow as incompressible because its Mach number was lower than 0.3 [62]. The only exception occurred for the 10 geometric configurations with H₁/L₁ = 0.1, in which the Mach number reached 0.318 (considering the maximum velocity locally reached in the domain, due to the highest value of $\dot{m}$ = 557.89 m/s) in the turbine duct. However, these cases were kept in the manuscript for comparison purposes, since they in the Mach number ~0.3 range, where the compressibility effects are still barely felt.

From Figure 15 and Figure 16, it was not possible to notice a significant influence of the degree of freedom H₂/L₂ over the monitored parameters. This fact occurred because the effect of H₁/L₁ over these parameters was so relevant that the variations caused by the changes in the H₂/L₂ ratio became almost imperceptible. This considerable influence of the H₁/L₁ can be explained mainly by the alteration imposed on the OWC inlet cross-sectional area (A_in), allowing the entry of a greater amount of air and augmenting the airflow velocity in the turbine duct, since the ratio between inlet and outlet (A_out) cross-sectional areas (A_in/A_out) also increased. The gain in flow rate and pressure was directly proportional to the reduction in the degree of freedom H₁/L₁ and, consequently, proportional to the increase in the A_in/A_out ratio, as can be seen in Figure 17.

After the results presented in Figure 15 and Figure 16, the available power variation caused by the H₂/L₂ changes for each H₁/L₁ value is shown in Figure 18. As expected, a similar behavior that occurred in Figure 15 and Figure 16 could be observed for the available power. However, in Figure 18, the difference among the results is even more discrepant: the best case (defined by H₁/L₁ = 0.1 and H₂/L₂ = 0.81) reached an available power of 16,954.8 kW (or 16.9548 MW), which is nearly 839 times higher than the worst case (defined by H₁/L₁ = 1.0 and H₂/L₂ = 0.33), which achieved 20.2 kW (or 0.0202 MW).

However, it is worth mentioning that such high-pressure values are impractical in real cases. OWC devices normally have a relief valve that limits the pressure inside the device; therefore, the turbines used in this type of WEC work under less severe operational conditions [17].

Additionally, it is important to mention that the results of Figure 18 might suggest that the degree of freedom H₂/L₂ has no effect on the available power. Nonetheless, separately analyzing a specific H₁/L₁ value, it is possible to observe that higher values of available power were obtained with higher values of the degree of freedom H₂/L₂, as illustrated in Figure 19. Although it is difficult to notice this behavior in Figure 18, this same effect identified in Figure 19 was also observed for all H₁/L₁ values.

To further evaluate the influence of the degree of freedom H₂/L₂ on the monitored variables and respective available powers achieved, the magnitude percentage difference between the geometrically extreme cases of this parameter was calculated, i.e., when L₂ = L₁ (see Figure 2a) and L₂ = L₃ (see Figure 2c), for each H₁/L₁ value. Figure 20 shows these differences obtained for the air mass flow rate, total pressure, and available power of the OWC.

Regarding the mass flow rate, Figure 20 indicates that no considerable effect of the degree of freedom H₂/L₂ on its magnitude was observed, with the highest difference observed being 0.06% for H₁/L₁ = 0.8. Therefore, the shape of the transition region (area A₂ in Figure 2) apparently did not influence the air mass flow rate in a considerable way.

Figure 20 also indicates that the greatest effect of the degree of freedom H₂/L₂ on the total pressure occurred for the degree of freedom H₁/L₁ = 0.6, achieving an increase of approximately 4.2% for the total pressure. This means that, two-dimensionally speaking, a transition region with a trapezoidal shape is more effective in relation to a region with a rectangular shape, in obtaining the total pressure when the height of the hydro-pneumatic chamber has a magnitude of 0.6 times the width (or diameter) of the OWC converter inlet (see Figure 2).

Moreover, still analyzing Figure 20, one can observe that the effect of the degree of freedom H₂/L₂ on the theoretical power led to a maximum percentage difference of 5.94% at H₁/L₁ = 0.7. It is possible to note that the smaller the magnitude of the H₁/L₁ ratio, that is, the greater the area ratio, the smaller the influence of the degree of freedom H₂/L₂ on the generated power. It is also possible to state that, for the cases evaluated here, the change in the transition region, from a rectangular to a trapezoidal shape (see Figure 2), resulted in a more evident increase in theoretical power only in those devices where the width (or diameter) of the hydro-pneumatic chamber was not greater than its height.

After these preliminary investigations, in order to bring the numerically simulated OWC WECs even closer to a real case, the power of the device was limited to 500 kW, the same value observed in the Energetech and Limpet plants [63,64]. Therefore, the available power achieved for the extreme values of H₂/L₂ (i.e., for L₂ = L₁ and L₂ = L₃) can be seen in Figure 21, in which the effect of the H₁/L₁ ratio in the power magnitude variation is evident, while the influence of the H₂/L₂ ratio was not possible to visually perceive.

The data shown in Figure 21 reveal that the maximum available power achieved for the best case of 313.8 kW (with H₁/L₁ = 0.1 and H₂/L₂ = 0.81 for L₂ = L₃) was approximately 15.5 times the magnitude reached in the worst case of 20.2 kW (with H₁/L₁ = 1.0 and H₂/L₂ = 0.33 for L₂ = L₁).

Subsequently, to better understand the influence of the degree of freedom H₂/L₂ over the corrected theoretical available power of the extreme cases presented in Figure 21, the percentage difference between these cases for each H₁/L₁ value can be seen in Figure 22.

From Figure 22, one can infer that the degree of freedom H₂/L₂ influenced the corrected theoretical available power, similar to that observed previously for the OWC available power without limitations (see Figure 20). Here, the transition of the region inserted between the hydro-pneumatic chamber and the turbine duct (see Figure 2), from a rectangular shape (L₂ = L₁) to a trapezoidal shape (L₂ = L₃), had a greater effect on the corrected available power as the magnitude of the H₁/L₁ ratio increased.

In addition, to better explore the augmentation in available power obtained due to smaller magnitudes of the degree of freedom H₁/L₁, when considering the action of a relief valve in the OWC WEC, limiting it to a power of 500 kW (see Figure 22), the percentage difference between the theoretical power obtained by the OWC device without limitations (see Figure 18) with the OWC device with limitations for geometries that have extreme values of the degree of freedom H₂/L₂ (L₂ = L₃) was calculated and illustrated in Figure 23.

Figure 23 evidently indicates that, despite the better results obtained due to smaller magnitudes of the degree of freedom H₁/L₁, only a small fraction of this power could actually be used due to the limitations of the turbine-generator system. As an example, among the cases specifically evaluated here, there was a reduction of almost 100% in available power, from 16,954.8 kW to 313.8 kW, in the geometry with H₁/L₁ = 0.1, when a relief valve was considered in the OWC device. One can also infer that greater use was observed in geometries with H₁/L₁ ≥ 0.7, in which less than 20% of the theoretical available power generated was wasted.

Therefore, the results obtained with the approach of power limitation in Figure 21 were much closer to reality and corresponded well with other studies previously developed. For instance, Lisboa et al. [65] carried out a geometric evaluation of an OWC WEC subject to waves with different characteristics found in the sea state of the coast of Rio Grande, RS, achieving theoretical powers of approximately 2.5 to 330 kW. In turn, the study developed by El Marjani et al. [32] used an OWC device with dimensions similar to those on Pico Island, subject to waves with characteristics approximate to the sea state found on the coast of Morocco, and these authors achieved theoretical powers with values between 20 and 170 kW, approximately.

Finally, it is worth highlighting that of the 100 geometric configurations evaluated, the one that presented the greatest power was that obtained by the degrees of freedom H₁/L₁ = 0.1 and H₂/L₂ = 0.81 (extreme condition of L₂ = L₃), achieving an available power of 16,954.8 kW for the case without a relief valve and 313.8 kW for the case with a relief valve. On the other hand, the geometric configuration that presented the lowest power was that with the degrees of freedom H₁/L₁ = 1.0 and H₂/L₂ = 0.33 (extreme condition of L₂ = L₁), reaching an available power of 20.2 kW both for the cases with and without a relief valve.

To illustrate the best and worst OWC geometries, Figure 24 and Figure 25 depict the fields of total pressure and velocity, respectively. Both geometric configurations were plotted on the same scale, and the best cases reached magnitudes considerably superior, while the worst cases presented magnitudes of nearly zero. Despite these magnitude discrepancies, Figure 24 and Figure 25 are a good way to visually compare the differences in the geometric dimensions of best and worst cases.

Figure 24 and Figure 25 depict the same instant of time, in which the piston movement of the water column inside the OWC WEC was compressing the air into the hydro-pneumatic chamber. It was possible to identify this fact due to the positive pressure in the hydro-pneumatic chamber (Figure 24) and positive velocity in the turbine duct (Figure 25), mainly for the best OWC geometric configuration. We highlight that due to using the same color scale for pressure and velocity, respectively, in Figure 24 and Figure 25, it seems that there was no variation in these parameters in Figure 24b and Figure 25b; however, in Figure 24b, a pressure variation between 360 and 2220 Pa occurred, while there was a velocity variation between 0 and 10.4 m/s in Figure 25b.

6. Conclusions

In the present work, a numerical study was carried out to investigate the influence of geometric parameters on the theoretical available power of an OWC-type WEC. The main objectives here were to develop, for an OWC converter, an axisymmetric computational model that allows the imposition of realistic sea state data, to use it in a geometric evaluation of the device through the Constructal Design method combined with the Exhaustive Search technique, aiming to maximize the OWC’s available power and obtain recommendations regarding its geometry. To do so, the time-averaged equations of conservation of mass, momentum, and treatment of Reynolds stresses were solved numerically using the finite volume method, in the ANSYS Fluent software.

Initially, the choice and calibration of the computational model were carried out in conjunction with its validation. From this, for a 2D axisymmetric OWC domain, in which a turbulent alternating airflow occurs, the results allowed us to recommend the following numerical parameters: the κ-ω SST turbulence model, pressure–velocity coupling SIMPLE method, spatial discretization of advective terms Second-Order Upwind method, transient formulation First-Order Upwind method, time step equal to 0.1 s, residue from the solution of the conservation equations equal to 1 × 10⁻⁴, and average size of the mesh elements equal to 30 mm. The results achieved during the model definition and calibration process demonstrated that it is possible to simplify the OWC analysis as an airflow through an axisymmetric 2D domain. A good agreement was reached when comparing the solution obtained numerically and the experimental results presented in the literature, validating the proposed computational model with a maximum error of approximately 7%.

Among the advantages observed when using the axisymmetric model was the possibility of evaluating 3D flows from a 2D domain. This is extremely appreciated because a 2D domain is less complex to generate and requires fewer elements in the mesh; consequently, the computational time needed to obtain the numerical solution is substantially reduced. It was also possible, as performed in this work, to insert realistic sea state data directly into the input of the evaluated converter, using the air methodology (a simplified approach to numerically simulate the OWC WEC), having greater control over the input condition than for the VOF methodology, for example.

Among the limitations of this model was the format of the evaluated geometry, which, as it was obtained by revolving the 2D domain, did not allow for very complex details, such as the coupling of a turbine to the converter; furthermore, as the name of the model suggests, it must necessarily have symmetry. Another limitation was the impossibility of considering the effects of the interaction between waves and the structure of the device, as it was not possible to simulate the sea state outside the converter.

After that, the influence of the geometric configuration of the hydro-pneumatic chamber and a transition region, with a trapezoidal shape, inserted between the chamber and the turbine duct of an OWC device, on the power made available by the converter was evaluated. This analysis took place through the variation in two degrees of freedom, namely, the relationship between the height and width of the hydro-pneumatic chamber (H₁/L₁) and the relationship between the height and width of the smallest base of the trapezoidal region (H₂/L₂).

In all cases simulated here, considering the realistic sea state from Cassino beach in southern Brazil, an incompressible, turbulent, transient, and irregular flow was considered for an axisymmetric 2D domain. The results showed that the OWC power was much more sensitive to variations in the H₁/L₁ degree of freedom than the H₂/L₂ degree of freedom. This occurred because the H₁/L₁ ratio influenced the size of the converter inlet area, allowing an increase or decrease in airflow, while the H₂/L₂ ratio influenced the shape of the transition region between the chamber and the turbine duct, which could generate only a small increase in total pressure in the turbine region. As an example, keeping L₂ = L₃, the lowest power obtained, P = 21.3 kW, occurred with H₁/L₁ = 1.0, while the highest, P = 16,954.8 kW, occurred with H₁/L₁ = 0.1; that is, the highest power was approximately 795 times greater than the lowest power. Meanwhile, keeping H₁/L₁ = 0.7, the lowest power, P = 58.1 kW, was obtained for H₂/L₂ = 1.91, and the highest power, P = 54.8 kW, for H₂/L₂ = 0.23; that is, the variation in the H₂/L₂ parameter resulted in an increase of approximately 6% in the power obtained

To bring the simulated converters even closer to a real case, the available power of the device was limited to 500 kW, the same value observed in the Energetech and Limpet plants, as if a relief valve were acting on the system. As a result, the lowest power obtained was P = 20.2 kW for H₁/L₁ = 1.0 and H₂/L₂ = 0.33 (L₂ = L₁), while the highest power, P = 313.8 kW, was obtained for H₁/L₁ = 0.1 and H₂/L₂ = 0.81 (L₂ = L₃). These results are consistent with the values found by Lisboa et al. [65], who carried out a study on a converter in the same region.

Finally, it was evident that, although the lower values of the degree of freedom H₁/L₁ led to the best results, only a small fraction of this power could actually be used due to the limitations of the turbine-generator system used by the converter. As an example, there was a reduction of almost 100% in the power obtained by the geometry generated by H₁/L₁ = 0.1 and H₂/L₂ = 0.81 (L₂ = L₃), from 16,954.8 kW to 313.8 kW. On the other hand, greater use was observed in geometries where the H₁/L₁ ratio was greater than 0.7, where less than 20% of the theoretical power was dispensed.

It is worth highlighting that the study using the Constructal Design combined with a realistic sea state carried out with an axisymmetric computational model is an unprecedented scientific contribution of this research. As a proposal for future work, it would be interesting to include the effect of a turbine to this computational model. To achieve this, it is suggested to consider a physical restriction in the turbine duct that represents the pressure drop that would be caused by the presence of the turbine. In this way, it would be possible to determine an even more real value of the OWC converter power. To do so, possibly the computational modeling needs to be improved to consider compressible and non-isothermal airflow. Furthermore, it is suggested to compare the results obtained by this simplified approach with those of an OWC converter subject to the effects of wave incidence in a realistic sea state, as this is a limitation of the axisymmetric model developed here. Finally, as a last suggestion, the variation in the degree of freedom H₃/L₃ must be studied to evaluate its influence, together with the degrees of freedom H₁/L₁ and H₂/L₂, on the available power of the OWC converter.