Physical–Mathematical Modeling and Simulations for a Feasible Oscillating Water Column Plant

Abstract

The focus of this paper is placed on Oscillating Water Column (OWC) systems. The primary aim is to analyze, through both mathematical modeling and numerical simulations, a single module (chamber) of an OWC plant which, in addition to energy production, offers the dual advantage of large-scale integration into port infrastructures or coastal defense structures such as breakwaters, etc. The core challenge lies in optimizing the geometry of the OWC chamber and its associated ducts. A trapezoidal cross-section is adopted, with various front wall inclinations ranging from $90°$ to $45°$. This geometric parameter significantly affects both the internal compression ratio and the hydrodynamic behavior of incoming and outgoing waves. Certain inclinations revealed increased turbulence and notable interference with waves reflected from the chamber bottom which determined an unexpected drop in efficiency. The optimal performance occurred at an inclination of approximately $55°$, yielding an efficiency of around 12.8%, because it represents the most advantageous and balanced compromise between counter-trend phenomena. A detailed analysis is carried out on several key parameters for the different configurations (e.g., internal and external wave elevations, crest phase shifts, pressures, hydraulic loads, efficiency, etc.) to reach the most in-depth analysis possible of the complex phenomena that come into play. Lastly, the study also discusses the additional structural and functional benefits of inclined walls over traditional parallelepiped-shaped chambers, both from a structural and construction point of view, and for the possible use for coastal defense.

1. Introduction

Global energy consumption rose by approximately 33.8% between 2004 and 2024, rising from about $\mathrm{131,760}\mathrm{TWh}$ in 2004 to about $\mathrm{176,300}\mathrm{TWh}$ in 2024. This corresponds to roughly $20.1256\mathrm{TWyr}$, over 81% of which is derived from fossil fuels. Climate change, rising atmospheric CO₂ levels, and the energy crisis that emerged in 2022 have made the urgent need to invest in research on new renewable energy sources increasingly evident. Among the least tested yet most promising sources are those harnessing energy from seas and oceans. These can broadly be classified into the following categories: wave energy, sea current and tidal energy, oceanic thermal energy, energy from salinity gradients, and energy from biomass cultivation in sea water. Being relatively new, all the techniques and technologies related to the above categories are still in the study or initial experimentation phase. Among these, wave energy is arguably the most promising and compelling category.

The earliest patents for wave energy exploitation date back to the late 18th century [1], but we have to wait for the oil crisis of the 70s of the last century for a strong stimulus of research in this direction [2].

It is estimated that the global wave potential is greater than 2 TW [3,4]. Numerous types of Wave Energy Converters (WECs) have been developed, but whichever type of device you consider, it is either under study or in research and development. Only a small range of devices have been tested in full scale and deployed in the oceans.

Utilizing wave energy as a renewable source presents several advantages over conventional energy generation methods. For example:

• Sea waves possess the highest energy density among all renewable energy sources [5]. The waves are generated indirectly by solar energy through the winds: the typical intensity of solar energy is 0.1–0.3 kW per m² of horizontal surface, and it is converted into a concentrated and relevant wave energy flux. For example, considering a significant wave height (i.e., the average height of the highest third of the waves) $H_s$ of 1 m and a wave energy period $T_e$ of 4.5 s, we obtain a wave energy flux of 2.25 kW per meter of wave crest, which rises to 13 kW/m if $H_s=2$ m and $T_e=6.5$ s [6].
• Minimal environmental impact during operation. In [4] the potential impact and an estimate of the life cycle emissions of a typical nearshore device is analyzed. In general, offshore devices have the least potential impact.
• The natural seasonal variability of wave energy aligns well with electricity demand in temperate climates in a very efficient way [5].
• Waves can propagate over long distances with minimal energy loss.
• Studies have shown that wave energy converters can capture up to $90\%$ of available energy, compared to 20–30% for wind and solar devices [7].

A significant distinction exists between nearshore and offshore WECs, as environment, energy contents, and related issues are very different. In nearshore locations the wave directions can largely be determined in advance due to the natural phenomenon of refraction and reflection. But the waves lose much of their power as the seabed rises. In offshore or deep-water environments (as is well known, there is no universally accepted definition of deep water; some mean a water depth greater than 1/3 of the wavelength, others greater than 40 m, etc.),the wave energy is significantly more intense, but their direction is extremely variable, so devices must align themselves accordingly, or be symmetrical, in order to efficiently capture wave energy.

Another challenge is that most devices are optimized for specific wave types, and the performance drops significantly with waves that do not conform to it. But the wave motion of the seas is extremely irregular. An example is provided by the western coasts of Europe, where, quite commonly, offshore waves have an energy flux of about 30–70 kW/m; however, conditions occur that generate extreme waves with power levels even higher than 2000 kW/m to which the device must resist. There are also considerable design challenges in mitigating the action of a highly corrosive environment to which devices operating under or on the surface of the water are subjected.

A wide range of technologies exists for converting wave energy. More than a thousand wave energy conversion techniques have been patented in Japan, North America, and Europe. Despite their design variability, WECs are generally classified by installation location, structural type, and operational mode.

As regards the position, we add that, despite the higher construction costs, the floating devices in deep waters offer greater structural economy in relation to the energy produced because they exploit more powerful waves [8]. Notably, up to 95% of a wave’s energy is concentrated between the water surface and a depth of one-quarter of its wavelength [9].

Despite the great variation in design, WECs can be classified into three predominant types: attenuator, point absorber, and terminator. Within these categories, they are then distinguished by mode of operation: for example, submerged with pressure differential, oscillating wave surge converters, overlapping devices, oscillating water column (OWC) devices, and others [10,11,12,13,14].

Beyond wave-based WECs, numerous systems have been developed to harness ocean currents and tidal flows. Ocean current and tidal systems are particularly appealing due to seawater’s high density—approximately 800 times that of air—and even the flow of ocean currents can be predicted for decades with 98% accuracy.

Other noteworthy technologies include salinity gradient energy systems, called “Salinity Gradient Power” (SGP), which includes pressure-retarded osmosis (PRO), reverse electrodialysis (RED), plus other techniques in studies such as capacitive mixing (CAPMIX) and capacitive reverse electrodialysis (CRED) [15,16].

Finally, we highlight Ocean Thermal Energy Conversion (OTEC) systems. The first references date back to the second half of the 1800s. In fact, from a historical point of view, it is very interesting to note that even before James Clerk Maxwell published his theory on the unitary nature of light and electromagnetic fields in 1873, Jules Verne, in 1870, made one of the earliest documented references to harnessing oceanic thermal gradients to produce electricity in “Twenty Thousand Leagues Under the Seas”. Eleven years later, in 1881, D’Arsonval proposed using the relatively warm surface waters (between 24 °C and 30 °C) of the tropical oceans to vaporize pressurized ammonia through a heat exchanger (evaporator) and use the resulting vapor to drive a turbo generator. Then the cold ocean water brought to the surface from 800–1000 m depth, with temperatures between 4 °C and 8 °C, would have condensed the ammonia vapor through another heat exchanger (condenser).

Many techniques have been proposed since then, but the only ones that currently seem to have a solid theoretical and experimental basis are the so-called closed cycle schemes (CC-OTEC) and open cycle (OC-OTEC) [17,18].

To emphasize the importance of research on systems that draw energy from the seas, we conclude this general part of the introduction by informing the reader of current estimates according to which, on an annual basis, the amount of solar energy absorbed by the oceans is equivalent to at least 4000 times the amount currently consumed by humans. Assuming an average OTEC efficiency of merely 3% in converting ocean heat energy to electricity, we would need less than 1% of this renewable energy to satisfy all of our energy desires. Furthermore, according to some researchers, the removal of this relatively small amount of oceanic solar energy would not lead to a negative environmental impact. On the contrary, it could help mitigate the ongoing and alarming rise in ocean temperatures.

Structure and Goals of the Paper

Following this contextual overview of marine energy systems, the remainder of the paper focuses on the previously introduced OWC systems and the associated power take-off (PTO) devices that convert wave motion into electrical energy.

The main purpose of this paper is to investigate, from a physical–mathematical and modeling perspective, a specific wave energy converter (WEC) based on Oscillating Water Column (OWC) technology. This type of device appears particularly well suited to the characteristics of wave motion in the Mediterranean Sea, offering an effective compromise between simplicity, robustness, performance, and economy of contruction. As will be detailed later, the study focuses on a single module, designed as a basic unit within a larger array that could include several tens or even hundreds of such elements. In simple terms, each module consists of a reinforced concrete chamber—potentially serving secondary functions such as forming part of a breakwater or coastal protection system—within which a water column oscillates under wave action. The air occupying the upper portion of the chamber is subjected to compression and decompression cycles, which drive a Wells-type turbine via a dedicated air duct. This low-pressure turbine is designed to rotate in the same direction regardless of the reversal in airflow direction. The turbine ultimately converts mechanical power into electricity via an electric generator.

To develop a theoretical model of wave motion, the study applies both linear wave theory and, more importantly, second-order Stokes nonlinear wave theory. Based on these theoretical foundations, we identified a reference geometry for the OWC chamber and its auxiliary structures, and subsequently explored several geometric variants in which the most critical design element—the front wall of the chamber facing the incoming waves—is systematically varied. The baseline configuration features a vertical front wall ($90°$), and three alternative geometries, with inclinations of $75°$, $60°$, and $45°$, were added. All of them were examined through both theoretical analysis and numerical simulation. Numerical simulations were conducted using a “wave tank” model with a custom computational grid implemented in Fluent-Fluid Simulation Software 2021 R1, an industry-standard computational fluid dynamics (CFD) software distributed by Ansys Inc., Canonsburg, PA, USA.

From an energy efficiency perspective, the configurations with front wall inclinations of $60°$ and $45°$ yielded the best performance. These configurations also offer structural and functional advantages, such as improved integration with breakwater systems. To refine the analysis within the 45°–60° range, two intermediate configurations, $50°$ and $55°$, were added. Among all simulated configurations, the $55°$ front wall yielded the highest efficiency—approximately 12.8%, as reported in the final part of Section 4. More importantly, a comprehensive analysis was conducted on key parameters—including internal and external wave elevations, hydraulic loads, and phase shifts—highlighting how these respond to variations in wall inclination and, consequently, changes in chamber volume and compression ratio (see figures in Section 4). Notably, certain configurations—particularly those approaching a $45°$ inclination—exhibited significant interference between the incident and bottom-reflected waves, resulting in unexpected performance drops.

The structure of the paper is as follows: Section 2 presents a concise overview of OWC systems, covering current research, scale trials, and operational plants worldwide. Section 3 describes the mathematical model and leads up to the plant specifications of the proposed OWC system (see Section 3.3). The first part of Section 4 focuses on the mathematical modeling of the coupled PTO system, while Section 4 provides an in-depth analysis of the simulation results. Finally, Section 5 presents the conclusions and outlines directions for future research.

2. OWC Systems for Converting Ocean Energy

This section takes a more in-depth look at OWC systems, and intends to inform the reader about the state-of-the-art of research on them.

The Oscillating Water Column (OWC) technology is among the most promising and best suited for the typical wave conditions of the Mediterranean Sea—and beyond. One of its key strengths lies in the robustness of its primary structural element: the reinforced concrete caisson, which closely resembles those already employed in port infrastructures such as breakwaters and protective barriers. As a result, OWC systems can serve a dual purpose, simultaneously supporting structural functions and enabling electricity generation. In the context of port construction or renovation, this implies that much of the investment in OWC systems could be absorbed within the existing budget.

Current research mainly focuses on two aspects: the geometry of the caisson and its integration with the turbine—or more generally, with the mechanical systems known as power take-off (PTO) units—that drive the electric generator.

The main reference for this section is the book [19], published in 2021, which is entirely dedicated to OWC systems. For mathematical modeling, see [19,20]; for numerical simulations, see [21,22,23] and the references therein; general overviews are available in [24,25].

2.1. OWC: Operation, Scale Trials, and Plants Around the World

OWC systems are a type of Wave Energy Converter (WEC) designed to generate electricity from marine wave motion. They typically consist of a partially submerged concrete or steel structure that is open at the bottom and contains an air chamber above the free water surface. The oscillation of the water column, induced by incoming waves, compresses and decompresses the air, thereby generating a bidirectional airflow that drives a turbine connected to an electric generator.

OWC systems offer several advantages, including low environmental impact. They can be integrated into common coastal infrastructure such as breakwaters and do not significantly interfere with the marine ecosystem or human activities like navigation and fishing.

While some OWC devices are deployed offshore or on the seabed, shoreline installations offer clear advantages, including easier installation and maintenance, as well as the elimination of long submarine cables for power transmission.

Although coastal waves contain less energy than offshore waves, this can be partially compensated by strategic placement that exploits wave concentration through refraction and diffraction. Since the early 1980s, both theoretical studies and experimental evidence have shown that wave energy absorption can be enhanced by extending the chamber with protruding walls—either natural or artificial—in the direction of wave propagation (see also Section 5). This creates an inlet or manifold, a structural feature present in most existing prototypes.

2.1.1. Considerations on OWC Operation and Physical Resonance

The oscillatory motion of sea waves causes the water level inside the OWC chamber to rise and fall, thereby compressing and decompressing the air volume above, which in turn drives the turbine.

Theoretical studies on the energy efficiency of OWC systems indicate that optimal performance is achieved when the system’s natural frequency matches the dominant frequency of the incoming waves. Under this resonance condition, the wave-induced forcing exerts the maximum effect on the oscillating system. A useful analogy is that of a seesaw: if a push is applied just before the seat reaches its highest point (maximum potential energy), the resulting oscillation increases efficiently; if applied at the wrong moment, however, the effect is reduced or even counterproductive.

Like a mechanical oscillator, an OWC system has its own natural frequency, which depends primarily on:

• the geometry and dimensions of the chamber;
• the volume of air it contains;
• the damping effect introduced by the turbine.

Achieving resonance is challenging in practice, as the natural frequency of the system is often significantly higher than that of typical waves. Moreover, real sea states are composed of multiple frequencies and are subject to continuous variation due to the stochastic nature of wind forcing.

For this reason, the resonant frequency of the OWC chamber is a key design parameter, along with the fluid dynamic behavior of the resulting airflow. Overall system efficiency depends on the proper coupling between wave frequency, chamber geometry, and turbine characteristics. Several innovative solutions have been proposed to address the difficulty of frequency matching, although most are still in early stages of development. These include multi-resonant cavities, active control systems, and U-OWCs. The first approach, developed by Orecon Ltd., Exeter Devon, UK, employs multiple chambers of equal cross-section but differing volume to achieve a broader spectral response. Active control systems adapt the device’s characteristics dynamically to the incident wave frequency. The U-OWC system, designed at the Mediterranea University of Reggio Calabria, uses wave pressure as an external forcing mechanism without direct wave entry, thus improving efficiency [24,25,26,27].

Despite their apparent conceptual and structural simplicity, OWC systems pose significant challenges in both theoretical modeling and experimental implementation. These difficulties are evidenced by numerous practical trials worldwide and by the many unresolved issues identified above.

2.1.2. Working OWC Plants, Past and Ongoing Trials

As already mentioned, even from an experimental point of view, OWC technology is still taking its first steps. In Europe, the countries most committed to this are the United Kingdom, Spain, Portugal, Italy, Norway, and Ireland. Outside Europe, these include in particular Japan, India, China, and Australia.

An important example of an OWC device is LIMPET (Land Installed Marine Pneumatic Energy Transfer), developed from research conducted by Queen’s University Belfast in 1985. LIMPET500 is a 500 kW plant built near Portnahaven, on the coast of the Isle of Islay, Scotland. Since becoming operational in 2000, it has supplied a significant portion of the island’s electricity needs.

In the Basque Country, an OWC system is integrated into the breakwater of Mutriku Bay, in the Bay of Biscay. Operational since 2011, the plant consists of 16 chambers and 16 Wells turbine units, each coupled with an 18.5 kW generator, for a total capacity of 296 kW.

The Mediterranea University of Reggio Calabria has conducted research on OWC devices for many years. One of the objectives is also to attenuate wave energy impacting the coast, thereby reducing erosion. Several devices have been patented: REWEC1, fully submerged, with a pressurized air chamber and a hydraulic turbine; REWEC2, partially emerged from the water, equipped with an air turbine; and REWEC3, based on the U-OWC concept (see [26] and the references therein). A small-scale REWEC3 prototype is currently being tested at the university’s Marine Laboratory. A full-scale prototype is under construction at the new dock in Civitavecchia (Lazio), with a 650-m frontage and an estimated power output of approximately 650 kW. Several other projects have been developed to insert these devices both in Italy and on the Pacific Ocean coast.

2.2. Power Take-Off Devices Applicable to OWC Systems

The energy contained in the airflow generated by OWC systems must be converted into mechanical energy in order to produce electricity. This airflow is characterized by alternation—i.e., a reversal in direction every half-period—and by significant fluctuations in intensity due to the inherent variability of wave phenomena over time. Therefore, it is necessary to couple OWC systems with PTO machines having specific characteristics—capable not only of maintaining the direction of rotation despite changes in airflow direction, but also of ensuring satisfactory performance under varying operating conditions.

The devices used to extract energy from the airflow generally fall into the following categories:

-

conventional turbines with suitable flow rectification systems using non-return valves and auxiliary channels;

-

self-rectifying turbines, mainly including the Wells turbine in its various versions;

-

the action turbines with two distributors;

-

some special turbines including a Savonius rotor and Denniss–Auld turbine.

Further research is needed to effectively combine a PTO system with an OWC device, beginning with theoretical studies and physical-mathematical modeling. For example, it is often unclear which type of turbine should be selected for a given OWC configuration—nor which variant within a given category—when considering various factors such as efficiency, start-up torque, performance at different speeds, size, noise levels, and more. Even less is known when changes to the geometry of the OWC chamber are introduced simultaneously. What is certain is that the chamber geometry has a significant impact on the system’s final performance (see Section 2.1.1). Conversely, the choice of turbine strongly affects the resonance frequency and overall physical behavior of the OWC system.

3. From Mathematical Modeling to the Definition of Plant Specifications

Taking Italy as an example—a country for which the authors have detailed data—it becomes evident how significantly the widespread use of OWC systems, such as those described in the previous section, could affect the national energy balance.

Italy has between 7500 and 10,000 km of coastline according to the most cited sources. From a strictly mathematical point of view, as is well known, we cannot speak of length of coastlines because they are all infinite as they are fractal figures [28,29]. A numerical data of 9226 Km for the Italian coasts is provided, for example, by the World Resources Institute (WRI), but it is not clear which measurement step was used to obtain it (for more details on fractals, infinite lengths and infinite computing, the interested readers can see [30,31,32,33]). Italy has also 534 commercial ports (approximately one every 14–15 km), around 300 private or emergency mooring facilities, 374,693 m of port approaches, over 160,000 boat berths, and approximately 8 million m² of dock area.

These conditions necessarily imply the presence of extensive lengths of breakwaters and other structures potentially well suited to hosting OWC systems and generating green electricity.

3.1. The Mathematical Modeling of Wave Motion

Referring to Figure 1 below, let L denote the wavelength, H the wave height (i.e., the z-difference between a crest and a trough), and A the wave amplitude, defined as the z-difference between a crest and the still water level (where $z=0$). Note that, in general, $A\ne H/2$. Let h be the water depth and $\eta (x,t)$ the surface elevation at position x and time t.

As is well known, if

$$\overrightarrow{v}(x,z,t)=\left(u(x,z,t),w(x,z,t)\right)$$

is the decomposition of the velocity vector into the two components u and w along the x and z axes, respectively, the assumption of irrotational flow allows the introduction of a scalar function $\Phi (x,z,t)$, called the velocity potential, such that

$$\nabla \Phi (x,z,t)=\left({\displaystyle \frac{\partial \Phi }{\partial x}},{\displaystyle \frac{\partial \Phi }{\partial z}}\right)=\left(u(x,z,t),w(x,z,t)\right).$$

(1)

Two governing Equations, (2) and (3), describe the motion. Under the assumptions of incompressibility and irrotationality, the continuity equation (mass conservation) reduces to Laplace’s equation $\Delta \Phi ={\nabla }^2\Phi =0$, i.e.,

$${\displaystyle \frac{{\partial }^2\Phi }{\partial x^2}}+{\displaystyle \frac{{\partial }^2\Phi }{\partial z^2}}=0.$$

(2)

For the equation of motion we use the generalized Bernoulli equation

$$gz+{\displaystyle \frac{p}{\rho }}+{\displaystyle \frac{1}{2}}\left[{\left({\displaystyle \frac{\partial \Phi }{\partial x}}\right)}^2+{\left({\displaystyle \frac{\partial \Phi }{\partial z}}\right)}^2\right]+{\displaystyle \frac{\partial \Phi }{\partial t}}=0,$$

(3)

where p is the total pressure, $\rho$ the fluid density, and g the gravitational acceleration.

Three boundary conditions must be added to (2) and (3): two are kinematic ((4) and (6)) and one is dynamic (i.e., (7)).

The kinematic condition at the seabed assumes impermeability: $w(x,z,t)=0$ for $z=-h$ and all $x,t$. Recalling (1), this can be rewritten as follows:

$${\left[{\displaystyle \frac{\partial \Phi (x,z,t)}{\partial z}}\right]}_{z=-h}=0.$$

(4)

A second kinematic condition is imposed at the free water surface to ensure that no mass crosses it. This is achieved by requiring that the component of the velocity vector orthogonal to the water surface be equal to the same component of the velocity of the surface itself. In other words, noting that $z-\eta (x,t)=0$ defines the set of points describing the free surface of the water, we must impose that ${\displaystyle \frac{\partial }{\partial t}}(z-\eta (x,t))=0$ on this surface:

$$0={\displaystyle \frac{\partial \left(z-\eta (x,t)\right)}{\partial t}}=w(x,z,t)-\left({\displaystyle \frac{\partial \eta }{\partial x}}·u(x,z,t)+{\displaystyle \frac{\partial \eta }{\partial t}}\right),$$

(5)

where we used the chain rule in the right hand side. Recalling (1) we can rewrite (5) as

$${\left[{\displaystyle \frac{\partial \Phi (x,z,t)}{\partial z}}-\left({\displaystyle \frac{\partial \eta (x,t)}{\partial x}}·{\displaystyle \frac{\partial \Phi (x,z,t)}{\partial x}}+{\displaystyle \frac{\partial \eta (x,t)}{\partial t}}\right)\right]}_{z=\eta (x,t)}=0.$$

(6)

The third, dynamic condition is derived from (3) by assuming the pressure p is constant across the free surface and equal to the atmospheric pressure $p_{\mathrm{atm}}$. Setting the last one equal to zero, we obtain

$${\left[{\displaystyle \frac{\partial \Phi (x,z,t)}{\partial t}}+{\displaystyle \frac{1}{2}}\left[{\left({\displaystyle \frac{\partial \Phi (x,z,t)}{\partial x}}\right)}^2+{\left({\displaystyle \frac{\partial \Phi (x,z,t)}{\partial z}}\right)}^2\right]+gz\right]}_{z=\eta (x,t)}=0.$$

(7)

In addition to the boundary conditions, regularity conditions must be imposed on the flow field to reflect the periodicity in x and t:

$$\left\{\begin{array}{c}\Phi (x,z,t)=\Phi (x+L,z,t)\hfill \\ \Phi (x,z,t)=\Phi (x,z,t+T)\hfill \end{array}\right.,$$

(8)

where T is the period of the wave.

We note that the system of equations underlying the water wave problem is nonlinear in the unknowns $\Phi$ and $\eta$. Moreover, except for the bottom boundary condition (Equation (4)), those on the free surface are given in implicit form (note that the wave profile is in fact expressed by an unknown function $\eta (x,t)$). All these complications inherent in the differential problem make the analytical solution of the system particularly challenging. A first simplification is obtained by using a Maclaurin series expansion of the functions appearing in the boundary-value equations and discarding terms deemed negligible. However, despite this mathematical artifice that succeeds in rendering the boundary conditions explicitly, an analytical solution of the differential system remains highly challenging. We therefore resort to the so-called perturbation theory [34], assuming that the unknowns of the problem can be expressed as power series in a small parameter $\epsilon$, called the perturbation parameter. We can then consider approximate solutions by neglecting powers of $\epsilon$ above a certain degree. This yields first-, second-, third-order solutions, and so on. In Stokes theory the perturbation parameter is represented by $\epsilon =A/L$. By applying a long series of analytical manipulations to our system of differential equations (for which we refer the interested reader to one of the numerous specialized texts, e.g., [35,36]), we obtain the following second-order solutions for Stokes waves expressing the potential and the free surface profile:

$$\Phi (x,z,t)={\displaystyle \frac{Ag}{\omega }}·{\displaystyle \frac{cosh\left(k(h+z)\right)}{cosh\left(kh\right)}}·sin(kx-\omega t)+{\displaystyle \frac{3k{A}^{2}g}{8\omega }}·{\displaystyle \frac{cosh\left(2k(h+z)\right)}{{sinh}^3\left(2kh\right)cosh\left(kh\right)}}·sin\left(2(kx-\omega t)\right),$$

(9)

$$\eta (x,t)=Acos(kx-\omega t)+{\displaystyle \frac{k{A}^2}{4\omega }}·{\displaystyle \frac{[2+cosh(2kh\left)\right]cosh\left(kh\right)}{{sinh}^3\left(kh\right)}}·cos\left(2(kx-\omega t)\right),$$

(10)

where $k=2\pi /L$ is the wave number and $\omega =2\pi /T$ is the angular frequency.

Now that we have established some of the basic equations, we can proceed to the next subsection, which describes a numerical simulation apparatus suitable for our purposes.

3.2. Numerical Simulation of an Induced Wave Motion

For the numerical simulations, we adopt the so-called “wave tank” scheme, in which a fluid mass is set in motion by a wave generator, as illustrated in Figure 2.

Three boundary conditions are imposed: the first sets the pressure at the top of the tank equal to atmospheric pressure; the second sets the fluid velocity to zero on both the bottom and the right wall; and the third, on the left wall, depends on the method chosen to generate the waves. In our case, we selected a rigid wall generator floating at a fixed angle, whose motion resembles that of a piston. To control its movement according to the desired wave characteristics, we use the following relation (transfer function):

$${\displaystyle \frac{H}{S}}={\displaystyle \frac{2\left[\cosh \right(2kh)-1]}{sinh\left(2kh\right)+2kh}},$$

where h is the depth of still water in the tank and S is the displacement of the piston [35]. We write the equation of motion of the floating wall as

$$x\left(t\right)={\displaystyle \frac{S_0}{2}}\left(1-e^{-{\displaystyle \frac{5t}{2T}}}\right)sin\left(\omega t\right),$$

(11)

where $S_0$ is the maximum piston displacement [21,22]. Differentiating (11) yields the piston velocity equation:

$$v\left(t\right)={\displaystyle \frac{dx\left(t\right)}{dt}}={\displaystyle \frac{S_0}{2}}\left[\omega \left(1-e^{-{\displaystyle \frac{5t}{2T}}}\right)cos\left(\omega t\right)+{\displaystyle \frac{5}{2T}}{e}^{-{\displaystyle \frac{5t}{2T}}}sin\left(\omega t\right)\right].$$

The numerical simulations were carried out using a software package provided by Ansys Inc., Canonsburg, PA, USA, specifically the Fluent 2021 R1 module, which belongs to the category of CFD (Computational Fluid Dynamics) tools. The following subsections describe the steps in detail.

3.2.1. Definition of the Domain and of the Calculation Grid

Having standard computing powers normally available on the market (Intel Core i9 processors), we opted for two-dimensional simulations of the problem. Indeed, it is well established in the literature that three-dimensional simulations would yield negligible gains in accuracy, while dramatically increasing computation times (already very long, on the order of several days for a single run).

A schematic of the computational domain is shown in Figure 3: the tank is 115 m long, 10 m high, with a still water level of 4 m.

Selecting an appropriate computational grid is essential for accurately simulating the phenomenon. We opted for a non-uniform, heterogeneous grid, subdivided into three distinct zones, as shown in Figure 4. The region most affected by motion is the L-shaped zone (rotated by −90°), which includes a band straddling the water surface and another on the left, where the floating wall moves. In this region, a very fine mesh was employed, with cells measuring 2.5 cm × 5 cm. In the upper zone (air) and the lower zone (water), labeled zones 2 and 3 in Figure 4, a coarser discretization was applied, using square cells with 20 cm sides.

3.2.2. Numerical Simulation Settings

Once the simulation type has been defined in Fluent (pressure-based, transient, with gravitational acceleration in the y-direction), the first step is to define the water–air interaction. To this end, we use the Volume of Fluid (VOF) model in its implicit formulation [21,22,36,37]. A fixed control volume is considered, containing, in general, n non-interpenetrating fluids (or phases). Denoted by ${\alpha }_q\in [0,1]$ the volume fraction of the q-th phase in the given cell ($q=1,2,\dots ,n$), the continuity equation is written for it as

$${\displaystyle \frac{1}{{\rho }_q}}\left[{\displaystyle \frac{\partial }{\partial t}}\left({\alpha }_q{\rho }_q\right)+\nabla ·\left({\alpha }_q{\rho }_q{\overrightarrow{v}}_q\right)\right]={\displaystyle \frac{1}{{\rho }_q}}\left[S_{{\alpha }_q}\sum _{p=1}^n({\dot{m}}_{pq}-{\dot{m}}_{qp})\right],$$

(12)

where ${\rho }_q$ is the density of the q-th phase, ${\overrightarrow{v}}_q$ its velocity vector, ${\dot{m}}_{pq}$ the mass transfer from phase p to phase q, ${\dot{m}}_{qp}$ the reverse, and the different phases are subject to the following constraint

$$\sum _{q=1}^n{\alpha }_q=1.$$

Since there is no mass transfer in our case, Equation (12) simplifies to the following:

$${\displaystyle \frac{1}{{\rho }_q}}\left[{\displaystyle \frac{\partial }{\partial t}}\left({\alpha }_q{\rho }_q\right)+\nabla ·\left({\alpha }_q{\rho }_q{\overrightarrow{v}}_q\right)\right]=0.$$

In each cell, the volume fractions and fluid interface are computed using a piecewise linear interpolation method. To do so, it must solve the following governing equations:

• the conservation of mass (or continuity) equation

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

  which holds for both compressible and incompressible flows;
• the momentum balance equation

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

  where p is the pressure, $\overrightarrow{F}$ is a source term (for example, related to a porous medium) and $\overrightarrow{\overrightarrow{\tau }}$ is the stress tensor equal to

  $$\overrightarrow{\overrightarrow{\tau }}=\mu \left[\left(\nabla \overrightarrow{v}+\nabla {\overrightarrow{v}}^t\right)-{\displaystyle \frac{2}{3}}\left(\nabla ·\overrightarrow{v}\right)\overrightarrow{\overrightarrow{I}}\right],$$

  where $\mu$ is the molecular viscosity;
• the energy conservation equation

  $$\begin{array}{c}{\displaystyle \frac{\partial }{\partial t}}\left[\rho \left(e+{\displaystyle \frac{1}{2}}{\left|\overrightarrow{v}\right|}^2\right)\right]+\nabla ·\left[\rho \left(e+{\displaystyle \frac{1}{2}}{\left|\overrightarrow{v}\right|}^2\right)\overrightarrow{v}\right]\hfill \\ =\nabla ·(k_t\nabla T)+\nabla ·\left(-\rho \overrightarrow{v}+\overrightarrow{\overrightarrow{\tau }}·\overrightarrow{v}\right)+\overrightarrow{v}·\overrightarrow{F}+Q_h,\hfill \end{array}$$

  where e is the specific energy, $k_t$ the thermal conductivity, T the temperature, and $Q_h$ a possible heat source.

Using the RANS (Reynolds Averaged Navier–Stokes) approach for solving the Navier–Stokes equations, each generic variable $\Psi$ is decomposed into a mean component $\overline{\Psi }$ (over a fairly small time interval) and a fluctuating one ${\Psi }^{\prime }$ (also called Reynolds decomposition). As a result, each nonlinear term introduces additional variables that must be modeled, such as the Reynolds stress tensor$-\overline{{\overrightarrow{v}}^{\prime }{\overrightarrow{v}}^{\prime }}$ arising from the momentum balance equation. Being symmetric, this tensor requires modeling of six components, and the Boussinesq hypothesis is used for the closure problem (to relate Reynolds stresses to mean velocity gradients) together with two transport equations for the turbulent kinetic energy k and the specific turbulent dissipation rate $\omega$, according to the standard k-$\omega$ turbulence model [37,38].

Boundary conditions were then applied as described in Section 3.1, along with the User-Defined Function (UDF) describing the motion of the movable wall. This function is implemented using the Dynamic Mesh technique, which continuously adapts the computational grid to follow the wall’s motion.

The PISO (Pressure-Implicit with Splitting Operators) algorithm was used for pressure-velocity coupling. It belongs to the SIMPLE family of algorithms. A known limitation of the SIMPLE and SIMPLEC (Semi-Implicit Method for Pressure Linked Equations-Consistent, a variant of SIMPLE) algorithms is that, after solving the pressure correction equation, the updated velocities and flows do not immediately satisfy momentum conservation. To address this, the calculation must be repeated iteratively until equilibrium is achieved. To improve computational efficiency, the PISO algorithm performs two additional corrections involving neighboring cells and asymmetry adjustments.

The Green–Gauss cell-based method was employed to evaluate gradients and discretize convection terms in the governing equations. The “PRESTO!” (PREssure STaggering Option) scheme was selected for pressure interpolation. The remaining convection-diffusion equations (e.g., momentum or energy equations) were discretized using second-order interpolation schemes.

Finally, to maintain a Courant number below 1 and enhance solution convergence, a time step of $\Delta t=0.01$ s was adopted. Additionally, the maximum number of iterations allowed per time step to achieve system convergence was set to 60.

The graph in Figure 5 shows the results of a simulation invilving a wave with height $H=0.5$ m, length $L=18.886$ m, period $T=3.73$ s and depth $h=$ 4 m. It compares the surface elevation $\eta (x,t)$ obtained from our numerical simulation with the theoretical profile predicted by the Navier–Stokes equations. After an initial startup phase, during which the system stabilizes, the two profiles become nearly identical. The small discrepancy observed is due to the still water condition assumed at the beginning of the simulation.

3.3. OWC Plant Specifications

In addition to the theoretical and simulation framework described in the previous subsections, we also conducted field investigations involving real wave motion and actual seabeds. These investigations were carried out using the equipment available at the Marine Experimental Station of Capo Tirone in Belvedere Marittimo (CS, Italy). Moreover, less than 500 m from the station lies the tourist port of Belvedere Marittimo. Resolution no. 97/2017 of the competent municipal administration approved preliminary studies aimed at a large-scale renovation and reinforcement of the port infrastructure. The plan includes expanding the port basin and constructing new, larger protective structures, docks, and breakwaters—making it an ideal site to study the potential integration of OWC chambers into these works, offering the added benefit of wave energy production, as discussed in Section 2.

Figure 6b and

Table 1. The measurements related to Figure 6b. We recall that $h=4$ m and $L=18.886$ m.

Symbol	Reference	Formula	Value
a	Front wall sinking	$a=h/5$	0.8 m
b	Chamber width	$b=L/10$	1.9 m
c	Visible external height	-	4 m
d	Front wall thickness	$0.15b<d<0.55b$	0.5 m
f	Bottom step height	$f=2h/5$	1.6 m
I	Bottom step length	$I=L/2$	9 m
s	Bottom step inclination	-	90°

show the dimensions of an OWC chamber suitable both for the virtual tank described earlier and for the real setting of the port of Belvedere. Figure 6a, on the other hand, presents a schematic cross-section of the chamber as integrated into a breakwater structure of appropriate size for the Belvedere port environment. The authors are confident that the results presented in this study—based on the numerical simulation settings described above—can, within a few years, be compared with real-world outcomes from nearshore construction.

To select the appropriate wave characteristics for simulation in our virtual tank, we conducted an extensive marine meteorological study at the Capo Tirone Experimental Station. After compiling data on the frequency, type, and physical characteristics of waves at the site, we found that the predominant wave direction ranges between 255° and 285°, and that the most frequent wave type (57%) has a height $H=0.5$ m, length $L=18.886$ m and period $T=3.73$ s. Based on the literature surveys (e.g., [39]), we determined the optimal characteristics of the OWC chamber for our data. These are summarized in Table 1, which corresponds to Figure 6b, where $h=4$ m denotes the seabed depth.

The final step is to define how to simulate the vent duct for the PTO system responsible for generating electrical energy. One simulation approach involves replacing the PTO system with an equivalent porous medium. This medium interacts with the airflow, inducing a pressure variation within the OWC chamber. This pressure variation allows for the evaluation of the energy conversion efficiency, $\eta$, defined as follows:

$$\eta ={\displaystyle \frac{P_{\mathrm{out}}}{P_{\mathrm{in}}}}={\displaystyle \frac{{\displaystyle \frac{1}{T}{\int }_0^T\Delta P\left(t\right)q\left(t\right)dt}}{{\displaystyle \frac{\rho gL}{16T}{H}^2\ell \left[1+\frac{2kh}{sin\left(2kh\right)}\right]\left[1+\frac{9H^2}{64k^4{h}^6}\right]}}},$$

where $\Delta P\left(t\right)$ is the pressure variation inside the chamber over time, $q\left(t\right)$ is the airflow rate through the PTO system, and ℓ is the wavefront width considered.

4. Mathematical Modeling of the PTO and Simulation Results

Before beginning the simulations, it was necessary to study the PTO system in order to define an equivalent porous medium, as mentioned in Section 3.3. For this purpose, the total pressure variation $\Delta P$ was decomposed into three contributions:

$$\Delta P=\Delta P_t+\Delta P_d+\Delta P_{\mathrm{i}+\mathrm{o}}$$

(13)

where $\Delta P_t$, due to the turbine, and $\Delta P_d$, due to pressure losses in the air duct, are the dominant terms. The third contribution, $\Delta P_{\mathrm{i}+\mathrm{o}}$, accounts for minor losses at the duct’s in let and out let.

Since the pressure variation $\Delta P_t$ across the Wells turbine is almost proportional to the axial flow velocity $V_a$, we can assume a linear relationship of the type $\Delta P^{\ast }={\displaystyle \frac{1}{2}}{B}_t{U}^{\ast }$ between the turbine parameters $\Delta P^{\ast }$ and $U^{\ast }$ (dimensionless pressure variation and dimensionless flow coefficient, respectively), where $B_t$ is the damping factor of the turbine. Therefore

$$\Delta P_t=\Delta P^{\ast }·{\rho }_a{U}_{\mathrm{tip}}^2={\displaystyle \frac{1}{2}}{B}_t{U}^{\ast }·{\rho }_a{U}_{\mathrm{tip}}^2={\displaystyle \frac{1}{2}}{B}_t{\rho }_a{U}_{\mathrm{tip}}{V}_a,$$

(14)

where $U_{\mathrm{tip}}$ is the turbine blade tip speed and in the last equality we used the relation $V_a=U^{\ast }{U}_{\mathrm{tip}}$.

For the second term in (13), since the airflow in the duct is fully turbulent, we apply the Darcy–Weisbach equation

$$\Delta P_d={\displaystyle \frac{{\rho }_a{F}_d{V}_d^2}{2D}},$$

(15)

where D is the duct diameter, $V_d$ is the airflow velocity and $F_d$ is the friction coefficient calculated using the Tsal correlation [40]:

$$F_d=0.11{\left({\displaystyle \frac{\epsilon }{D}}+{\displaystyle \frac{68}{{\mathbf{R}}_{\mathbf{e}}}}\right)}^{0.25},$$

where $\epsilon$ is the average surface roughness and ${\mathbf{R}}_{\mathbf{e}}$ is the Reynolds number.

The third term in (13), representing minor pressure losses at the duct inlet and outlet, is expressed as follows:

$$\Delta P_{\mathrm{i}+\mathrm{o}}={\displaystyle \frac{1}{2}}{\rho }_a{F}_{\mathrm{i}+\mathrm{o}}{V}_d^2,$$

(16)

where $F_{\mathrm{i}+\mathrm{o}}=F_{\mathrm{in}}+F_{\mathrm{out}}$ is the sum of the loss coefficients at the inlet and outlet of the duct.

By replacing the PTO with a porous medium, the total pressure drop becomes the following:

$$\Delta P=\Delta P_v+\Delta P_i,$$

(17)

where $\Delta P_v$ and $\Delta P_i$ represent the viscous and inertial pressure drops, respectively, across the porous medium. We write the first term in (17) as

$$\Delta P_v={\mu }_a{R}_v{V}_p\Delta l,$$

(18)

where ${\mu }_a$ is the dynamic viscosity of air, $\Delta l$ is the porous medium length, $V_p$ is the airflow speed through the medium, and $R_v$ is the viscous resistance coefficient. The second term is written as follows:

$$\Delta P_i={\displaystyle \frac{1}{2}}{\rho }_a{R}_i{V}_p^2\Delta l,$$

(19)

where $R_i$ is the inertial resistance coefficient of the porous medium.

In the CFD simulations, the viscous pressure drop $\Delta P_v$ was set equal to that generated by the Wells turbine, $\Delta P_t$. Likewise, $\Delta P_i$ was taken as equal to $\Delta P_d+\Delta P_{\mathrm{i}+\mathrm{o}}$. By comparing Equations (14)–(16), (18), and (19), and using the mass flow relations

$${\rho }_a{A}_p{V}_p={\rho }_a{A}_t{V}_a,\mathrm{and}{\rho }_a{A}_p{V}_p={\rho }_a{A}_d{V}_d,$$

with $A_d=\pi {(D/2)}^2$ as the duct cross-sectional area, we derive the following expressions for the porous medium parameters $R_v$ and $R_i$:

$$\left\{\begin{array}{ccc}{R}_v\hfill & =\hfill & {\displaystyle \frac{A_p{B}_t{\rho }_a{U}_{\mathrm{tip}}}{2A_t{\mu }_a\Delta l}}\hfill \\ R_i\hfill & =\hfill & {\displaystyle \frac{16A_p^2}{{\pi }^2{D}^4}\left(\frac{F_d}{D}+\frac{F_{\mathrm{i}+\mathrm{o}}}{\Delta l}\right)}\hfill \end{array}\right..$$

(20)

With the PTO system replaced by a porous medium, the problem reduces to identifying the two parameters $R_v$ and $R_i$. These depend on the characteristics of the actual PTO system and must be evaluated experimentally. It should be noted that energy production efficiency is not directly investigated in this study; rather, it serves as a control parameter to assess the influence of the front wall inclination on the OWC system. Accordingly, we fixed a specific efficiency value and determined the corresponding parameters $R_v$ and $R_i$ required to achieve it, using the final formulas (20). This process is iterative, as it relies on input values (e.g., velocity $V_p$) that themselves depend on the parameters $R_v$ and $R_i$.

Specifically, we defined the characteristics of the PTO system and carried out initial simulations without including the porous medium. Based on the resulting airflow velocities, an initial estimate of $R_v$ and $R_i$ was made. The simulation was then repeated using the initial $(R_v,R_i)$ pair. This yielded new flow and velocity values, which in turn led to a second estimate of $(R_v,R_i)$, and the process was iterated accordingly. The iteration continued until the target efficiency of $\eta =10\%$ was achieved, corresponding to the pair $R_v=70.45·{10}^6$ m⁻² and $R_i=25$ m⁻¹.

Only now, starting from this triple of values for $\eta ,R_v$ and $R_i$, it is possible to carry out the conclusive simulations of this research by varying the inclination of the front wall of the OWC chamber to evaluate its effects on energy efficiency of the system.

The Final Simulation Results

As previously discussed, the research setup is now complete, and we can proceed with simulations to evaluate how the front wall inclination—ranging from $90°$ to $45°$—affects various parameters. Specifically, we selected four initial inclinations at $15°$ intervals: $90°$, $75°$, $60°$, and $45°$ (see Figure 7).

For the purposes of this research, we focused on the following four performance aspects:

(a)

Elevation trends of the water surface inside and outside the OWC chamber;

(b)

Trends in internal and external hydraulic loads;

(c)

Phase shift of the wave crest between the internal and external domains;

(d)

Pressure profiles and the performance of the PTO system.

Each of the four aspects (a)–(d) plays a key role in assessing the performance of an OWC plant. Starting with point (a), we generated the plots shown in Figure 8 and Figure 9 for each of the four initial configurations.

First, it is worth noting that during the 50-s simulation interval (from 50 to 100 s, allowing for stabilization), just under 13.5 full wave cycles are visible in Figure 8 and Figure 9, which aligns with the wave period $T=3.73$ (see Section 3.3). Regarding the external wave elevation (blue curves in Figure 8 and Figure 9), the upper crest appears largely unaffected by the wall inclination. In contrast, the lower trough behaves quite differently. Although the minimum points remain always just below $-0.2$ m for the first three configurations ($90°$, $75°$ and $60°$ wall), the waveform near the trough begins to change as if to undergo a certain “flattening”. In contrast, the $45°$ wall configuration (see Figure 9b) exhibits significant and unexpected changes. As the inclination decreases from $60°$ to $45°$, the negative excursion of the external wave—measured from the still water level—nearly doubles, reaching approximately $-0.4$ m. The lower portion of the waveform also undergoes a complete transformation, with the appearance of a reversed peak growth and two relative minima. This phenomenon is mainly attributed to the reflected wave exiting the chamber, which significantly disrupts the rising phase of the external wave in the $45°$ wall configuration.

Regarding the trend of the wave elevations inside the chamber (red curve in Figure 8 and Figure 9), there is an asymmetric reduction in height from approximately 0.4 m to 0.3 m ($\approx -25\%$) when transitioning from the $90°$ configuration to the $75°$ one. Notably, the reduction in the downward excursion (which slightly exceeds $-0.1$ m in Figure 8b is much more evident than the upward one. When moving from $75°$ to $60°$, the profile of the internal wave does not undergo any significant changes, and this is also an interesting fact. With the $45°$ wall, the trend reverses, and the height extension of the internal wave increases until it reaches and slightly surpasses the 0.4 m value observed with the $90°$ vertical wall. The phenomena affecting the internal wave observed thus far are partly attributed to the different compression ratios of the OWC chamber corresponding to the four different inclinations. Specifically, recall the marked reduction in wave excursion, particularly downwards, during the transition from $90°$ to $75°$. The effects of increasing the compression ratio from $75°$ to $60°$ appear to be counteracted by other opposing phenomena resulting from the change in linear dimensions. The $45°$ case is sui generis because the relevant interactions between the external and reflected waves, as seen previously, must be considered in addition to the further increase in the compression ratio.

Point (b) concerns both the internal and external hydraulic loads, shown in Figure 10 and Figure 11: violet curves indicate internal load, yellow curves the external one.

It should be noted that the external load is mainly used here as a reference for analyzing the internal hydraulic load, which is the one directly involved in energy generation. In fact, the yellow curves of Figure 10 and Figure 11 coincide with an appropriate section of the blue curves of Figure 8 and Figure 9, respectively. In the $90°$ vertical wall configuration (see Figure 10a), the peak-to-peak distance for internal and external loads is very similar—approximately 0.6 m. The internal load waveform appears smoother and more regular than the external one, which already shows slight asymmetries and irregularities. These irregularities become more pronounced as the wall inclination decreases, although the internal waveform remains quite regular across all configurations. The internal pressure peak excursion slightly decreases at $75°$ but returns to about 0.6 m at $60°$. It is also worth noting that flow beneath the chamber’s front wall consistently causes a reduction in internal hydraulic load—estimated at around 18%—in the first three configurations. The $45°$ configuration again diverges from this trend: here, the maximum internal pressure peak equals or slightly exceeds the corresponding external peak (see Figure 11b). This is accompanied by a wider internal pressure excursion, greater than in the previous three cases, reaching approximately 0.7 m.

Point (c) concerns the phase shift between the internal and external wave crests in the OWC chamber. Figure 12 and Figure 13a show a consistent phase shift of approximately 0.8 s for the $90°,75°$, and $60°$ cases. In contrast, the $45°$ configuration differs from them once again: it exhibits a larger shift of about 1.0 s (see Figure 13b).

Regarding point (d)—the performance of the PTO system—special attention must be paid to the internal pressures and volumetric flows through the duct. Figure 14 compares the internal pressures for the four configurations.

The pink curve ($45°$ wall) reaches the highest pressure peak, approximately 2100 Pa. The other pressure peaks are roughly equidistant and inversely ordered by wall inclination: orange ($60°$) ∼2050 Pa, cyan ($75°$) ∼2000 Pa, and green ($90°$) ∼1950 Pa. The minimum pressure values are around $-2000$ Pa for the $60°$ and $75°$ configurations, the middle ones, and about $-1900$ Pa for $45°$ and $90°$. The largest pressure excursion—about 4050 Pa—is recorded for the $60°$ wall (orange curve). The phase shift is, however, more evident for the pink curve relating to the $45°$ wall.

Finally,

Table 2. The efficiency $\eta$ for the four initial configurations.

Frontal Wall	$90°$	$75°$	$60°$	$45°$
$\eta$	≈10.4%	≈11.6%	≈12.5%	≈11.5%

reports the efficiency obtained for each of the four initial configurations.

Note that the $75°$ and $45°$ configurations yield nearly identical efficiencies (∼11.5%), while the $60°$ configuration achieves the highest efficiency (∼12.5%), about one percentage point higher. The $90°$ configuration shows the lowest efficiency, more than one percentage point below both the $75°$ and $45°$ setups. This suggests that efficiency increases (as a nearly linear trend?) with decreasing wall inclination from $90°$ to $60°$. This trend does not hold for the $45°$ case, which shows an 8% drop in efficiency compared to $60°$, due to the disturbance effects previously observed for this configuration.

It is now pertinent to focus the research on a narrower, more relevant interval. Although the efficiency at $75°$ is approximately 1% greater than at $45°$, based on the preceding considerations, we anticipate that the most promising interval for identifying absolute maximum efficiencies lies between $60°$ and $45°$. Furthermore, understanding the mechanism and onset of the disturbance and interference phenomena observed at $45°$ is of interest. To this end, we conducted further investigations, examining configurations with the front wall at $55°$ and $50°$ to cover the chosen interval with a $5°$ spacing.

Bearing in mind also for the latter configurations the list of four points (a)–(d) outlined at the beginning of this subsection, we direct the reader’s attention to Figure 15, which illustrates the trend of the external and internal wave elevation over the same 50-s time interval considered previously (cf. Figure 8 and Figure 9).

Already in the case of the $55°$ wall, a slight disturbance in the trough of the external wave begins to be noticeable (see Figure 15a). With the $50°$ angle, the trough lowers and becomes increasingly irregular, culminating in the case of the $45°$ wall already analyzed in Figure 9b. In contrast, the upper peak of the outer wave is not significantly affected by the change in inclination between $60°$, $55°$, $50°$, and $45°$. In the transition from $60°$ to $45°$, the internal wave maintains a rather regular profile, as confirmed at $55°$ and $50°$. Instead, its extension between its extremes progressively increases as the angle of inclination of the wall decreases.

Regarding point (b), Figure 16 displays the internal and external hydraulic loads.

Among the four configurations—$60°$, $55°$, $50°$, and $45°$—the deepest troughs, though only slightly, are observed at $50°$, reaching nearly $-0.31$ m. The internal pressure peaks vary noticeably and almost linearly, from approximately $0.32$ m for the $60°$ wall to nearly $0.4$ m for the $45°$ wall (cf. Figure 11 and Figure 16).

Figure 17 shows the phase shift of the wave crest for the new configurations.

Recall that, for the initial configurations of $90°$, $75°$, and $60°$, the phase shift was approximately 0.8 s, while for the $45°$ wall it reached about 1.0 s (Figure 12 and Figure 13). For the new configurations with $55°$ and $50°$ walls, the phase shift unexpectedly assumes almost identical values: the average shift is approximately 0.91 s for the $55°$ case and 0.92 s for the $50°$ case.

As for the efficiency of these two additional configurations, the corresponding values are reported in

Table 3. The central columns yield the efficiency $\eta$ for the two new system configurations.

Frontal Wall	$60°$	$55°$	$50°$	$45°$
$\eta$	≈12.5%	≈12.8%	≈12.15%	≈11.5%

.

The highest efficiency among the six configurations studied is approximately 12.8%, achieved by the one with the $55°$ wall. In Figure 18, we can observe a polynomial function that best approximates the trend of the system’s efficiency as the inclination of the wall varies.

Regarding the efficiency of the OWC system, we can conclude that it increases as the front wall inclination decreases, up to the point where wave interference effects begin to reverse the trend. Peak efficiency is observed around $55°$, where initial signs of such disturbances appear, though not yet sufficient to cause a significant reduction in performance.

The following section presents the overall conclusions of this study.

5. Conclusions

Our main contribution begins in Section 2, which provides the essential reference framework for understanding OWC systems. We also presented key state-of-the-art information on existing OWC plants worldwide, whether experimental, full-scale, or in regular operation. Importantly, the study is not only theoretical; it also has practical implications, particularly for the potential construction of an OWC plant near the “Capo Tirone” Marine Experimental Station in Belvedere Marittimo, southern Italy. This station supplied extensive data on bathymetry, sea currents, wave conditions, and related parameters. In Section 3, we developed the physical and mathematical models and derived many of the parameters, equations, and input data required for the simulations presented and analyzed in Section 4.

We now turn to the final results obtained from the simulations. First, recall that the average hydraulic head loss associated with the front wall of the OWC chamber was 18% (see Section 4, point (b)). This value is consistent with those reported in the literature for flow passing beneath a thin wall, confirming that our numerical results reflect a well-known physical phenomenon. We also found that adopting an inclined front wall improves the overall performance of the system, partly due to the increased compression ratio resulting from the trapezoidal shape of the chamber.

The importance of carefully evaluating the shape and dimensions of the OWC chamber clearly emerged, as this helps prevent the reflected wave from the chamber bottom from excessively interfering with the incoming external wave—an effect that would lead to a significant reduction in system efficiency (see, for example, the results in Section 4 for the $45°$ and $50°$ configurations).

Adopting a chamber section with an inclined front wall not only improves efficiency and energy output, but also offers important advantages in terms of structural construction and safety. In breakwater construction, trapezoidal cross-sections are often preferred, as they are both easier to build in marine environments and more resistant to external forces. An additional benefit of the inclined wall is that it can be more easily covered with porous material to dissipate high-energy incident waves during storm surges—an essential factor in protecting the port basin.

Finally, we note that the entire discussion refers to a single chamber with a typical width of 4 to 5 m. In practical applications—such as breakwater barriers for port defense at the Belvedere Marittimo site, or coastal protection in similar locations—multiple contiguous chambers would be deployed, resulting in substantial energy production.

As for future research directions, it would be highly valuable to construct a laboratory-scale experimental setup and compare the resulting data with our numerical simulations. Additional wave characteristics—such as height, period, and others typical of different locations—could also be considered, in order to evaluate the performance of the same chamber geometry across a broader range of conditions. Moreover, alternative chamber cross-sections that are not necessarily trapezoidal could be investigated. In particular, appropriately designed curved front walls could be explored, as their response to external marine forcing would differ significantly from that of flat walls—an aspect that may prove especially relevant under severe storm conditions.