Investigation of a Modified Wells Turbine for Wave Energy Extraction

Abstract

The Oscillating Water Column (OWC) is the most promising self-rectifying device for power generation from ocean waves; over the past decade, its importance has been rekindled. The bidirectional airflow inside the OWC drives the Wells turbine connected to a generator to harness energy. This study evaluated the aerodynamic performance of two hybrid airfoil (NACA0015 and NACA0025) blade designs with variable chord distribution along the span of a Wells turbine. The present work examines the aerodynamic impact of the variable chord turbine and compares it with one with a constant chord. Ideally, Wells rotor blades with variable chords perform better since they have an even axial velocity distribution on their leading edge. The variable chord rotor blade configurations differ from hub to tip with taper ratios (Chord at Tip/Chord at Hub) of 1.58 and 0.63. The computation is performed in ANSYS™ CFX 2023 R2 by solving three-dimensional, steady-state, incompressible Reynolds Averaged Navier–Stokes (RANS) equations coupled with a k-ω Shear Stress Transport (SST) turbulence model in a non-inertial reference frame rotating with the turbine. The accuracy of the numerical results was achieved by performing a grid independence study. A refined mesh showed good agreement with the available experimental and numerical data in terms of efficiency, torque, and pressure drop at different flow coefficients. A variable chord Wells turbine with a taper ratio of 1.58 had a peak efficiency of 59.6%, as opposed to the one with a taper ratio of 0.63, which had a peak efficiency of 58.2%; the constant chord Wells turbine only had a peak efficiency of 58.5%. Furthermore, the variable chord rotor with the higher taper ratio had a larger operating range than others. There are significant improvements in the aerodynamic performance of the modified Wells turbine, compared to the conventional Wells turbine, which makes it suitable for wave energy harvesting. The flow field investigation around the turbine blades was conducted and analyzed.

1. Introduction

Renewable energy sources are being explored more thoroughly than before as the world moves towards an era whose energy needs can no longer be met by fossil fuels. Ocean wave energy is one of the most viable renewable sources since it offers a clean and eco-friendly alternative to fossil fuels and can provide power up to 90% of the time at appropriate locations [1]. It has caught the high-level attention of researchers for the past two decades because of its high energy density, low health risks, and perennial availability [2]. According to recently published studies, ocean waves can generate approximately 30,000 TWh/year, sufficient to meet the global electricity demand if harnessed effectively [3].

The wave energy converter (WEC) and related technologies have been the subject of many studies in recent decades [4,5]. Due to its simplicity, the OWC WEC is the most practical wave energy device by far. The OWC device is chosen on many prototypes already deployed in the sea [6]. Among the key components of the OWC are the air chamber, the turbine, and the generator. The OWC uses an air turbine, such as a Wells turbine [7], to convert pneumatic energy into mechanical energy.

The Wells turbine is the simplest self-rectifying air turbine, and it has symmetrical airfoil blades around the hub, staggered radially at 90-degree angles, with the chord plane normal to the axis of rotation (Figure 1). The symmetrical blade profiles allow it to maintain a constant direction of the tangential force, as shown in Figure 2. Thus, it rotates and produces unidirectional torque regardless of exposure to bidirectional airflow.

However, the Wells turbine has several inherent limitations compared to conventional turbines. For instance, it has a narrow operating range, lower aerodynamic efficiency, high axial force but low tangential force, increased noise levels, and poorer starting characteristics. In response, many authors have attempted to improve the Wells turbine design over time.

In turbine efficiency analysis and optimization, numerical methods play a crucial role. Among these, Computational Fluid Dynamics (CFD) is commonly employed due to its ability to provide detailed insights into the fluid flow and performance characteristics of turbine systems. An example of the application of CFD in related hydrodynamic studies can be found in the work by Karpenko et al. [8], who investigated the hydrodynamic processes in a pipeline-fittings system. Their study demonstrates the utility of CFD methods in understanding and optimizing fluid dynamics in engineering systems.

Building on these insights, the researchers devised a variety of design modifications to improve aerodynamic performance by addressing the losses due to blade profile and secondary flow and carried out experimental and numerical evaluations. For example, optimizing the blade geometry or thickness along the blade length reduces the blade profile loss. As a result of this optimization, stalls are delayed and the operational range is extended.

A study by Takao et al. [9] has demonstrated that the shape of the blade’s leading edge significantly affects the stall characteristics of a large-scale Wells turbine. Therefore, they proposed two optimal blades with maximum thickness near the leading edge, which caused a delay in the onset of separation and stalling of the turbine. The thicker National Advisory Committee for Aeronautics (NACA) airfoils increase the starting torque considerably [10], but as the blade thickness increases, hysteresis becomes more prominent, as demonstrated by Setoguchi et al. [11] experimentally.

With a constant chord Wells turbine, the main flow drifts radially downstream of the rotor, and consequently, when the flow rate increases, the turbine blades tend to stall at the tip rather than the turbine’s hub [12]. An attempt to counteract this situation was made by Marco et al. [12], who numerically implemented a variable chord rotor in a Wells turbine and demonstrated its influence on global performance.

Gato et al. [13] conducted experiments to evaluate several design approaches to improve turbine performance. Two sets of rotor blades were presented: (1) constant thickness blades comprising a NACA0015 profile along the span and (2) variable thickness blades with a linear change in the thickness from a NACA0021 profile at the hub to a NACA0009 profile at the tip. According to their findings, the variable thickness blades stall at lower flow coefficients, and, as a result, the efficiency drops more rapidly.

One of the most important factors influencing a turbine’s efficiency is the relative velocity distribution at the leading edges of the blades from hub to tip. In their study [14], Soltanmohamadi et al. proposed a rotor with variable chord using hybrid airfoils (NACA0012 and NACA0022) maintaining a constant taper ratio of 1.58, with no blade skew or sweep angle. The objective was to obtain a constant linear increase in the distance between the hub and the tip of two consecutive blades, along with a favorable relative velocity distribution.

Suzuki and Arakawa [15] investigated the performance of fan-shaped blades of hybrid airfoils (NACA0012 and NACA0021) with different sweep angles. They found improved efficiency for an Angle of Attack (AOA) < 7°. Takao et al. [9] examined four different blade profiles and determined that the NACA0015 was the most optimal profile for Wells turbine rotor blades. Takao et al. [16] compared three-dimensional blades (NACA0015 at the hub, NACA0020 at the mean radius, and NACA0025 at the tip) with the two-dimensional (original) blades and found that the 3D blades improved efficiency and stall characteristics. Researchers have also implemented Neural Network algorithms [17,18] to control the airflow and improve Wells turbine stalling performance.

This study investigated two variable chord rotor configurations comprising hybrid airfoils (NACA0015 and NACA0025), and their aerodynamic performance characteristics were compared with the baseline. Since the NACA0015 airfoil produces higher lift but stalls earlier than the NACA0025 airfoil for a given Reynolds number [19], a combination of both airfoils was selected. ANSYS™ CFX was used to solve the incompressible, three-dimensional, steady-state RANS equations. The non-dimensional turbine’s performance parameters were measured in terms of torque, efficiency, and pressure drop. The fidelity of our numerical results was checked through a grid independence study [20,21]. Moreover, we validated the numerical model by comparing the computed results with experimental data [22] and other CFD [23] investigations available in the literature that covered a wide range of flow rates and conditions. An extensive flow field study was also carried out to uncover the reasonings behind the aerodynamic performance improvements.

Figure 1. Schematic diagram of a Wells turbine [14].

Figure 1. Schematic diagram of a Wells turbine [14].

Figure 2. Force analysis on a Wells turbine blade [14]. Here, V_A, V_T, W_R, α, F_D, F_L, F_A and F_T represent the inlet velocity, the circumferential velocity at mean radius, the relative airflow velocity, angle of incidence, drag force, lift force, axial force and tangential force, respectively.

Figure 2. Force analysis on a Wells turbine blade [14]. Here, V_A, V_T, W_R, α, F_D, F_L, F_A and F_T represent the inlet velocity, the circumferential velocity at mean radius, the relative airflow velocity, angle of incidence, drag force, lift force, axial force and tangential force, respectively.

2. Computational Settings and Parameters

Our study uses a reference Wells turbine geometry consisting of eight blades positioned perpendicularly to the hub’s axis of rotation. A 90° stagger angle is used to construct turbine blades with symmetrical NACA0015 profiles. Tip leakage is one of the most important secondary fluid flows in turbomachinery, so tip clearance has been used in models to predict turbine performance with high accuracy. In

Table 1. Main features of the turbine models.

Parameters	Reference Turbine Blade	Hybrid Tapered Upward Blade	Hybrid Tapered Downward Blade
No. of planes	1	1	1
No. of blades	8	8	8
Chord length from hub to tip	125 mm	125–79 mm	79–125 mm
Solidity at mean radius	0.67	0.67	0.67
Tip diameter	600 mm	600 mm	600 mm
Hub diameter	400 mm	400 mm	400 mm
Tip clearance	1.25 mm	1.25 mm	1.25 mm
Blade profile from hub to tip	NACA0015	NACA0025 to NACA0015	NACA0025 to NACA0015

, the specifications of the reference turbine and hybrid turbines with varying thicknesses are listed.

Considering that WECs operate at a very low airflow frequency (f < 1 Hz), a steady-state solver is appropriate because of the negligible dynamic effects. ANSYS™ CFX was used to solve the steady, 3-dimensional RANS equations. We chose the k-ω SST turbulence closure model based on eddy viscosity because it can better capture separation and adverse pressure gradients. The equations are discretized using a fully implicit discretization method. Diffusion terms are discretized using a shape function-based approach, while advection terms employ a second-order discretization scheme.

The velocity and pressure field associated with turbulent flows can be divided into the mean component and the fluctuating component. A RANS equation governing mean flow is formulated through the average of Navier–Stokes equations over time. Despite their inherent nonlinearity, Navier–Stokes equations are still subjected to fluctuations in the velocity field caused by convective acceleration (Equation (1)). A nonlinear term resulting from the closure problem is known as Reynolds stress, which Boussinesq considers as a function of the mean flow components (Equation (2)).

$${\displaystyle \frac{\partial \left(\mathsf{\rho }\mathrm{U}\right)}{\partial \mathrm{t}}}+\nabla .\left(\mathsf{\rho }\mathrm{U}\mathrm{U}\right)=-\nabla \mathrm{p}+\nabla .\left[\mathsf{\mu }\left(\nabla \mathrm{U}+{\left(\nabla \mathrm{U}\right)}^{\mathrm{T}}\right)\right]+\mathsf{\rho }\mathrm{g}-\nabla \left({\displaystyle \frac{2}{3}}\mathsf{\mu }\left(\nabla .\mathrm{U}\right)\right)-\underset{\mathrm{R}\mathrm{e}\mathrm{y}\mathrm{n}\mathrm{o}\mathrm{l}\mathrm{d}\mathrm{s} \mathrm{S}\mathrm{t}\mathrm{r}\mathrm{e}\mathrm{s}\mathrm{s}}{\underbrace{\nabla .\left(\overline{\mathsf{\rho }{\mathrm{U}}^{\prime }{\mathrm{U}}^{\prime }}\right)}},$$

(1)

$$-\overline{\mathsf{\rho }{\mathrm{U}}^{\prime }{\mathrm{U}}^{\prime }}={\mathsf{\nu }}_{\mathrm{t}}\underset{\mathrm{M}\mathrm{e}\mathrm{a}\mathrm{n} \mathrm{V}\mathrm{e}\mathrm{l}\mathrm{o}\mathrm{c}\mathrm{i}\mathrm{t}\mathrm{y} \mathrm{G}\mathrm{r}\mathrm{a}\mathrm{d}\mathrm{i}\mathrm{e}\mathrm{n}\mathrm{t}\mathrm{s}}{\underbrace{\left(\nabla \mathrm{U}+{\left(\nabla \mathrm{U}\right)}^{\mathrm{T}}\right)}}-{\displaystyle \frac{2}{3}}\mathsf{\rho }\mathrm{k}{\mathsf{\delta }}_{\mathrm{i},\mathrm{j}},$$

(2)

where ${\mathsf{\delta }}_{\mathrm{i},\mathrm{j}}$ is the Kronecker delta.

The Wells turbine has a symmetrical geometry, with the blades arranged circumferentially equally apart. Therefore, we limited our analysis to one blade passage with periodic boundary conditions, as illustrated in Figure 3. The computational domain consisted of a straight duct and a 45° segment of the turbine section.

In the axial direction, the domains extended upstream and downstream were chosen at 4C and 6C, respectively. A uniform velocity profile was specified for the inlet boundary condition, while static pressure was imposed at the outlet. Additionally, no-slip wall boundary conditions were applied to the blade surface, hub, and tip. In the computational domain, lateral faces were specified with periodic boundary conditions. As observed from the inlet, the turbine rotates anticlockwise at 2000 rpm. A Moving Reference Frame (MRF) was employed to realize the rotation of the rotor blades. In this case, the air was used as the working fluid, with the buoyancy effect not being considered. The boundary conditions and details of the meshing are presented in

Table 2. Meshing and appropriate boundary conditions.

Parameter	Single Turbine
Mesh/Nature	Unstructured
No. of inflation layers	20
$y^{+}$	<1
Fluid	Air at 25 °C
Turbulence model	k-$\omega$ SST
Inlet	Uniform inlet velocity
Outlet	Pressure Outlet
Hub, tip, and blade	No-slip wall
Number of revolutions	2000 rpm
Convergence criterion	1 × 10−5
Mass imbalance	0.001

.

A CFD analysis requires discretization of the computational domain to solve the governing fluid flow equations [24]. In this situation, CFD results may be subject to discretization errors.

Computational grid quality is a significant source of errors in numerical analysis. The ANSYS™ meshing software generated an unstructured mesh for discretizing the flow domain in this study (Figure 4). For an accurate prediction of the dynamic stall point of a turbine, boundary layer flows surrounding the blades needed to be sufficiently resolved. Due to its sensitivity to the mesh, the k-ω SST turbulence model required a high degree of resolution (y⁺ < 1) to simulate the viscous sublayer region near the wall accurately. The prism layers around the blade surface captured boundary layer flows.

The dimensionless wall distance, ${\mathrm{y}}^{+}$, is defined as follows:

$${\mathrm{y}}^{+}={\displaystyle \frac{\mathrm{y}{\mathrm{u}}_{\mathrm{T}}}{\mathsf{\nu }}},$$

(3)

Here ${\mathrm{u}}_{\mathrm{T}}, \mathrm{y}, \mathrm{a}\mathrm{n}\mathrm{d} \mathsf{\nu }$ represent the friction velocity, the first layer distance (absolute) from the wall, and the kinematic viscosity of the fluid, respectively. According to Equation (3), the first layer height equals $1.1\times {10}^{-5}$ for y⁺ < 1. There were 20 layers of prism elements, with a layer growth ratio of 1.2. A wide range of non-dimensional flow coefficients, $\mathsf{\varphi }$, is considered for obtaining aerodynamic performance where

$$\mathsf{\varphi }={\displaystyle \frac{\mathrm{v}}{{\mathrm{U}}_{\mathrm{t}\mathrm{i}\mathrm{p}}}},$$

(4)

We changed the inlet axial velocity v to vary the flow coefficient while maintaining a constant circumferential velocity, ${\mathrm{U}}_{\mathrm{t}\mathrm{i}\mathrm{p}}$, at the tip radius.

To minimize rounding errors, all simulations were performed with double precision. In this study, root–mean–square residuals (RMS) for the governing equations were set at $1.0\times {10}^{-5}$ in order to measure convergence. In addition, as part of the convergence verification process, the turbine’s torque output and pressure drop were monitored. In several cases, it was necessary to decrease the set values assigned to each residual’s convergence criterion to achieve constant monitor quantities.

Seven different steady flow configurations were tested to validate the numerical modeling. The turbine operation range was considered as $0.075\le \mathsf{\varphi }\le 0.275$, or equivalently $4°\le \mathsf{\alpha }\le 15°$. A variety of non-dimensional coefficients have been used in previous literature to evaluate the performance of turbines and are given by the following:

The torque coefficient, C_T

$${\mathrm{C}}_{\mathrm{T}}={\displaystyle \frac{\mathrm{T}}{\mathsf{\rho }{\mathsf{\omega }}^2{\mathrm{R}}^5}},$$

(5)

The pressure drop coefficient, $\mathsf{\Delta }{\mathrm{P}}_0^{\mathrm{*}}$

$$\mathsf{\Delta }{\mathrm{P}}_0^{\mathrm{*}}={\displaystyle \frac{\mathsf{\Delta }{\mathrm{p}}_0}{\mathsf{\rho }{\mathsf{\omega }}^2{\mathrm{R}}^2}},$$

(6)

The efficiency, $\mathsf{\eta }$

$$\mathsf{\eta }={\displaystyle \frac{\mathrm{T}\mathsf{\omega }}{\mathsf{\Delta }{\mathrm{p}}_0\mathrm{Q}}},$$

(7)

The AOA α is related to the flow coefficient ϕ as follows:

$$\mathsf{\alpha }={\mathrm{t}\mathrm{a}\mathrm{n}}^{-1}\left(\mathsf{\varphi }\right),$$

(8)

where $\mathrm{T}, \mathrm{Q}, \mathsf{\Delta }{\mathrm{p}}_0, {\mathrm{U}}_{\mathrm{R}}, \mathrm{b}, \mathrm{c}, \mathrm{z}, \mathrm{v}, \mathsf{\omega }, \mathrm{a}\mathrm{n}\mathrm{d} \mathsf{\rho }$ represent the blade torque, volumetric flow rate, static pressure drops across the turbine, the circumferential velocity at blade mean radius, blade height, blade chord length, number of blades, axial inlet velocity, the angular velocity of the turbine, and air density, respectively. The Reynolds number is calculated using the formula as follows:

$$\mathrm{R}\mathrm{e}={\displaystyle \frac{\mathsf{\rho }\mathrm{c}\sqrt{\left({\mathrm{v}}^2+{\mathrm{U}}_{\mathrm{t}\mathrm{i}\mathrm{p}}^2\right)}}{\mathsf{\mu }}},$$

(9)

Grid Independence Study

Grid independence is essential to ensure accurate results in numerical analysis. Therefore, meshes with different cell sizes were used as the starting point for studying grid dependency. The numerical and experimental torque coefficient for all meshes was compared at $\mathsf{\varphi }=0.225$. Notably, the torque coefficient for the coarse grid (3.6 million cells) was significantly lower than that for the medium grid (7.6 million cells).

Nevertheless, the number of cells for the fine mesh nearly doubled after the cell size was reduced by half, and the variation of the resulting torque coefficient was considerably reduced. As a last resort, the extra-fine grid was selected, which resulted in corresponding negligible changes. The disparity between the medium, fine, and extra-fine grids was ~24%, ~4%, and ~0.5%, respectively. A reasonable degree of accuracy could be achieved since there was no significant variation in the results when using the extra-fine grid. Therefore, the extra-fine grid was used for subsequent simulations. Statistical information about cell sizes and the grid dependence study can be found in

Table 3. Grid sizes.

Mesh	No. of Cells (Millions)	Cell Size (mm)	Turbulence Model
Coarse	3.6	10.0	k-$\omega$ SST
Medium	7.6	5.0
Fine	18.1	2.5
Extra Fine	32	1.25

and Figure 5, respectively.

3. Results

A constant rotational speed of 2000 rpm was used in this study; however, different axial inlet speeds were used to assess the effectiveness of the turbine. In this case, flow coefficients, $\mathsf{\varphi },$ were used to capture the variation in α of the flow, and $0.075\le \mathsf{\varphi }\le 0.275$. Accordingly,

Table 4. Inlet axial velocity and flow coefficient ranges.

$\mathbf{Flow}\mathbf{Coefficient},\mathsf{\varphi }$	Inlet Axial Velocity, v (m/s)	Reynolds Number, Re
0.075	4.71	$5.25\times {10}^5$
0.125	7.85	$5.27\times {10}^5$
0.175	10.99	$5.31\times {10}^5$
0.20	12.94	$5.34\times {10}^5$
0.225	14.14	$5.36\times {10}^5$
0.25	15.52	$5.39\times {10}^5$
0.275	17.28	$5.43\times {10}^5$

shows the change in inlet velocity and Reynolds number.

A comparison was made between the current CFD results and experimental and numerical data to validate numerical modeling. According to Figure 6, the computed results and experimental [22] and other CFD results [23] are in good agreement until $\mathsf{\varphi }<0.225$, in terms of pressure drop coefficient ($\mathsf{\Delta }{\mathrm{P}}_0^{\mathrm{*}}$), torque coefficient (${\mathrm{C}}_{\mathrm{T}}$), and efficiency ($\mathsf{\eta }$) at different flow coefficients. For $\mathsf{\varphi }\ge 0.225$, it appears that the incident flow has a relatively high α (12.68°). By looking at the flow separation at the blade’s leading edge, it is evident that a deep stalling in the flow has occurred, resulting in a rapid reduction in blade torque as well as turbine efficiency. In the case of adverse pressure gradient flow, the RANS solver tends to overestimate the torque of separated flows and is vulnerable to errors. The numerical results are, therefore, increasingly difficult to reconcile with the experimental results [22]. Nevertheless, given the same geometrical features, the present study has comparable results to those found in other CFD studies [21].

Once the numerical model had been validated, we simulated our two sets of turbine designs using variable chord rotors for $0.075\le \mathsf{\varphi }\le 0.3$, or equivalently 4.3° ≤ $\mathsf{\alpha }$ ≤ 16.7°. Wells turbine blades with two different designs, (1) tapered upward and (2) tapered downward, consisting of hybrid airfoils (NACA0015 and NACA0025) with a variable chord distribution over the blade span are explored and compared to one with a constant chord (baseline). Compared to the baseline design, the tapered upward design had the worst torque coefficient, followed by the tapered downward design. However, the hybrid tapered downward design delayed stall by 1° α, since the torque coefficient started to drop off at $\mathsf{\varphi }=0.275$, rather than at $\mathsf{\varphi }=0.25$ for the other designs, as shown in Figure 7a.

According to Figure 7b, the tapered downward design has the highest peak efficiency, 59.6%, and is the most efficient for the flow coefficient, $\mathsf{\varphi }>0.125$. The pressure drop coefficient, $\mathsf{\Delta }{\mathrm{P}}_0^{\mathrm{*}}$, across the turbine rotor maintains a linear relationship with the flow coefficient $\mathsf{\varphi }$ and is lowest for the hybrid tapered upward design and highest for the baseline, as shown in Figure 7c.

Figure 8 illustrates the axial velocity profiles along the normalized span locations, 10 mm downstream of the turbine rotor for $\mathsf{\varphi }=0.225$, for all the turbine designs. As axial velocity decreases in the vicinity of the blade tip, the relative velocity and α at the blade’s leading edge are lower in the variable chord rotor designs than in the constant chord turbines. The blades’ leading edges thus experienced more uniform axial velocity distribution. In addition, the hybrid tapered downward design is superior to the hybrid tapered upward design, as the axial velocity of the latter was higher throughout the span, which resulted in the early separation and stall.

Turbulence Kinetic Energy (TKE) can be used to estimate the location of the stall and the turbulence flow increase. Figure 9 shows the TKE profiles along the normalized span at 10 mm downstream of the turbine rotor for $\mathsf{\varphi }=0.225$ for various designs. Hybrid tapered downward designs exhibit a TKE curve that coincides with the baseline up to a normalized span of 0.20.

The baseline TKE was the lowest until the normalized span was equal to 0.65. However, the hybrid upward design maintained a constant TKE profile for 0.1 < normalized span $\le 0.7$. Additionally, there was an increase in turbulence at the near tip of all designs except for the hybrid tapered upward design, i.e., 0.65 < normalized span $\le 1$; TKE was lower for the hybrid tapered upward than the hybrid tapered downward or the baseline design.

Figure 10 illustrates three designs for different flow coefficients with streamlines at the blade’s suction surface. For example, in the case of the baseline, when the flow coefficient is high ($\mathsf{\varphi }=0.225$), the flow begins to detach from the hub near the leading edge of the blade’s suction side (SS). Hybrid tapered upward designs, however, exhibit flow separation near the mid-chord (x/c = 0.5) location, resulting in vortices at around 40% span locations indicating blade stall. As a result of the vortices, the turbine has the worst aerodynamic performance when compared to the other turbines. On the contrary, the hybrid tapered downward turbine shows no evidence of recirculation, although a flow separation line is evident on its suction side.

An increased flow coefficient of $\mathsf{\varphi }=0.250$ (corresponding to a large incident α (14°)) produces a strong vortex formation near the leading edge of the baseline turbine blade; the center of the vortex lies approximately mid-span. Consequently, the baseline stalled, resulting in a significant reduction in torque and efficiency. In addition, the hybrid tapered upward turbine is in the post-stall phase. However, the flow is still attached to the blade surface of the hybrid tapered downward turbine, although it starts to recirculate near the leading edge close to the hub.

At the flow coefficient $\mathsf{\varphi }=0.275$, the reference turbine and the hybrid tapered upward turbine are in the deep stall conditions, as evidenced by the vortices shift towards the leading edge. As indicated by the vortices shifting toward the leading edge at the flow coefficient $\mathsf{\varphi }=0.275$, both baseline and hybrid tapered upward turbines are in deep stall conditions. The hybrid tapered downward turbine blades also exhibit strong vortices at the mid-chord and near-hub (30%) locations, confirming their stall.

In-depth analyses of the turbine flow field at different conditions (i.e., flow coefficient) help to provide a fundamental understanding of aerodynamic performance between the baseline and hybrid turbines. For example, Figure 11 shows the contours of the static pressure coefficient at various span locations (40% and 80% of the span distance from the hub) at three different flow coefficients. The static pressure coefficient, ${\mathrm{C}}_{\mathrm{p}}$, is defined as follows:

$${\mathrm{C}}_{\mathrm{p}}={\displaystyle \frac{\mathrm{P}-{\mathrm{P}}_{\mathrm{a}}}{\frac{1}{2}\mathsf{\rho }{\mathrm{v}}^2}},$$

(10)

A stagnation point occurs when the flow impinges on the blade’s leading edge. The static pressure and velocity are at their highest and zero, respectively. A high-pressure zone is located on the lower surface of the airfoil and is typically called the pressure side (PS). This high-pressure area enlarges with an increasing flow coefficient and corresponds to rising inlet axial velocities. A low-pressure zone is also formed on the suction surface (SS), the airfoil’s top surface. Air flows from the PS to the SS while attempting to remain attached to the blade surface.

At a flow coefficient of $0.225$, the baseline has a larger low-pressure zone than the hybrid turbines on the SS of the blade. The baseline turbine has an additional low-pressure zone that the others do not, as seen at the 80% span location. No appreciable differences are observed on the PS of the blade between the baseline and the hybrid downward turbine for the assigned span locations. The hybrid upward turbine has smaller high-pressure zones and produces the lowest pressure drop across the rotor compared to the other turbines.

Moreover, the hybrid upward turbine has stalled while the other two have not. We observe that the low-pressure region covers 80% of the chord at the 40% span location and 100% at the 80% span. Furthermore, the high-pressure zone begins to bleed onto the SS, causing an adverse pressure gradient, leading to the flow separation and blade stall.

At a flow coefficient of $0.250$, the baseline turbine has an extended low-pressure zone (approximately 90% and 100% from the blade LE to TE) for the 40% and 80% span locations, respectively. The high-pressure gradient causes the baseline configuration to stall and results in decreased aerodynamic performance. For the hybrid upward turbine, the low-pressure region covers 90% of the chord at a 40% span location and 100% at 80% span. As a result, flow bleeding to the SS has increased, and a strong vortex is observed in the wake of the 80% span location. However, qualitatively, the static pressure contour on the suction surface of the hybrid downward turbine blade is similar to that of $\mathsf{\varphi }=0.225$. In addition, the low-pressure region is concentrated close to the blade LE, which can be interpreted as flow separation and reattachment, which has delayed blade stall (Figure 10).

For a flow coefficient of $0.275$, the baseline and hybrid upward turbines have reached the deep stall. Here, their aerodynamic performance has drastically decreased compared to the unstalled conditions. Further, the hybrid downward turbine’s low-pressure region encompasses most of the chords on the blade suction surface, indicating recirculation and flow separation. This causes the formation of a strong clockwise vortex near the LE of the blade. Therefore, the turbine blade stalls, resulting in a considerable drop in torque and efficiency.

An illustration of the flow phenomenon along the blade chord is shown in Figure 12. Here, we describe the flow field at different flow coefficients using velocity-colored streamlines at planes located at 40% and 80% of the blade’s span. At $\mathsf{\varphi }=0.225$, the baseline has the most prominent high-velocity region corresponding to the largest low-pressure zones over the blade’s suction surface. Additionally, for the baseline and the hybrid downward turbine, the flow remains mostly attached along the blade chord, and there are no appreciable changes in the streamline patterns observed at the 80% span location. However, at the 40% span location, a small vortex is observed near the TE of the hybrid downward turbine blade. On the other hand, the hybrid upward turbine shows evidence of vortex shedding at 80% span. This configuration shows a clockwise vortex near the blade LE and a counterclockwise vortex near the blade TE. However, the streamlines are similar to the other turbines at a 40% span. Additionally, it has the smallest high-velocity regions because of the small low-pressure zones on its suction side. This reiterates that the turbine has stalled and results in a reduction of torque and efficiency.

The baseline turbine shows near-hub flow separation at $\mathsf{\varphi }=0.250$. The separation area expands as it approaches the tip of the rotor blade. A clockwise vortex created at the blade TE is observed at the 40% span location; it shifts towards the LE with greater strength (the center of the vortex is approximately at the mid-chord) at 80% span. Consequently, a drastic drop in turbine torque is calculated, indicating blade stall. At this flow coefficient, the hybrid upward turbine reaches its post-stall condition, and a small clockwise vortex is seen at the blade TE at the 40% span location. Moreover, multiple vortices are shed with increased strength at the 80% span location. For the baseline and the hybrid downward turbine, a strong vortex forms close to the blade TE. This indicates the beginning of the flow separation at the 40% span location of the hybrid downward turbine blade, while at 80% span, the flow has reattached to the blade surface along the chord.

At a flow coefficient of $\mathsf{\varphi }=0.275$, multiple clockwise vortices form along the blade chord of the baseline turbine; the flow has fully separated and reached deep stall conditions. Closer to the tip, there is a massive area of flow separation, and strong vortices are shown. Hence, these strong vortices cover the whole streamwise area at the 80% span location and most of the area at 40% span. The hybrid upward turbine is now in the deep stall, and vortex shedding is shown near the TE at 80% span. For the hybrid downward turbine, at the 40% span location, a huge clockwise vortex forms close to the blade LE and a small clockwise vortex near the TE. Furthermore, the flow is completely separated, and the clockwise vortex becomes stronger at the 80% span location. Reversed flow is visible at the blade TE, and the turbine blade experiences stall; this results in reduced turbine torque, power output, and efficiency.

Using Cp contours and streamlines at planes passing through 25%, 50%, and 75% of the blade’s chord, an illustration of the flow field is shown in Figure 13. The low-pressure zones are expanded with increasing flow coefficients. The baseline turbine has the lowest suction pressure across the span for all of the designated chord locations. At a flow coefficient of $0.225$, the streamlines on the SS are largely similar for all turbines at 25% chord. However, a small clockwise vortex forms close to the tip of the baseline turbine blade, unlike the other turbines. That vortex becomes stronger at 50% chord, and another counterrotating vortex is visible close to the hub. This vortex serves as a precursor to flow separation. However, the flow begins to separate near the tip at 75% chord. Similar to the baseline configuration, a small clockwise vortex is observed near the blade tip at 50% chord and intensifies at the 75% chord location for the hybrid upward turbine. This vortex covers the mid-span to the tip, confirming that the blade is stalled. For the hybrid downward turbine, a strong vortex is generated at the tip at 50% chord, but the flow reattaches along the span at 50% and 75% chord locations.

At a flow coefficient of $0.250$, the low-pressure zones continue to increase, causing an adverse pressure gradient. This leads to flow separation and blade stall for the baseline turbine. As a result, multiple vortices are found and cover the entire blade span at the 50% and 75% chord locations. Similarly, the hybrid upward turbine is in post-stall. We observe flow recirculation and vortex formation at 75% chord. The hybrid downward turbine shows smooth streamlines at 25% chord, but the flow starts to detach from the surface, since multiple vortices are observed at 50% and 75%. However, the flow shows evidence of reattachment across the span.

For the baseline turbine, at a flow coefficient of $\mathsf{\varphi }=0.275$, the suction side pressure has decreased further, and an additional small low-pressure region appears at a 25% chord near the tip downstream of the turbine blade. The baseline turbine and the hybrid upward turbine have reached deep stall conditions. Similarly, in the streamline patterns for the flow coefficient $\mathsf{\varphi }=0.250$, the baseline has multiple clockwise vortices covering the entire span. For the hybrid upward turbine, vortex shedding is shown at 50% chord, covering the whole span with separated flow regions. For the hybrid downward turbine, the most prominent low-pressure areas have covered the blade from the hub to 50% and 25% span. At the 50% chord location, a strong clockwise vortex at the tip expands throughout the blade span, and a counterclockwise vortex near the hub causes vortex shedding, which tends to cause higher noise production, flow separation, and blade stall. Furthermore, a clockwise vortex is observed at 75% chord covering almost 70% span from the tip.

A comparison of blade loading, at three different flow coefficients ($\mathsf{\varphi }=0.225, 0.250, \mathrm{a}\mathrm{n}\mathrm{d} 0.275$) for the three turbines at 40% and 80% span is shown in Figure 14. For all of the flow coefficients, the pressure distribution on the PS is similar for all of the turbines. The baseline has the lowest suction pressure, followed by the hybrid downward and then the hybrid upward turbine. This is the reason for higher torque production by the baseline compared to the hybrid downward and upward turbines. However, this comes with a considerably high pressure drop across the baseline rotor, and thus a reduction in efficiency. The hybrid downward turbine operates at the highest efficiency for most flow coefficients.

Using plots of wall shear, we can determine the location of separation and reattachment, as evident in Figure 15. Mostly the flow starts to separate from the blade surface at a flow coefficient of $\mathsf{\varphi }=0.225$. For the baseline case, at the 10% span, flow separation occurs at ${\displaystyle \frac{\mathrm{x}}{\mathrm{c}}}=0.61$, 0.78, and 0.80 for flow coefficients $\mathsf{\varphi }=0.225, 0.250, \mathrm{a}\mathrm{n}\mathrm{d} 0.275$, respectively, and reattaches close to the TE. However, for the hybrid upward turbine, flow separation is evident even at earlier chord locations, i.e., at ${\displaystyle \frac{\mathrm{x}}{\mathrm{c}}}=0.50, 0.54, \mathrm{a}\mathrm{n}\mathrm{d} 0.62$ for $\mathsf{\varphi }=0.225,0.250, \mathrm{a}\mathrm{n}\mathrm{d} 0.275$, respectively, consistent with turbine blade stall. On the other hand, for the hybrid downward turbine, wall shear equals zero, representing flow separation at ${\displaystyle \frac{\mathrm{x}}{\mathrm{c}}}=0.38$ for $\mathsf{\varphi }=0.225 \mathrm{a}\mathrm{n}\mathrm{d} 0.250$ but ${\displaystyle \frac{\mathrm{x}}{\mathrm{c}}}=0.74$ for $\mathsf{\varphi }=0.275$.

At the 40% span location, the hybrid downward design shows earlier flow separation compared to the other turbines at a flow coefficient of $\mathsf{\varphi }=0.225$. Post-stall conditions for baseline and hybrid upward turbines are evident at $\mathsf{\varphi }=0.250$. For instance, for the baseline, the flow detaches near the LE and reattaches at ${\displaystyle \frac{\mathrm{x}}{\mathrm{c}}}=0.20$ until ${\displaystyle \frac{\mathrm{x}}{\mathrm{c}}}=0.84$. This is also consistent for the hybrid upward turbine where flow detaches, reattaches, and again separates near the TE. However, the wall shear curve for $\mathsf{\varphi }=0.250$ is very similar to what we observed at $\mathsf{\varphi }=0.225$. At this flow coefficient, the hybrid downward turbine shows the start of blade stall. Finally, at $\mathsf{\varphi }=0.275$, there is massive separation for all turbine configurations, reite rating blade stall.

At 80% span, similar wall shear curves are observed for various flow coefficients. However, there is a more highly separated flow near the 80% span location than the 40% span location.

4. Discussion

While the Wells turbine has a narrow operating range and lower aerodynamic efficiency, the hybrid airfoil blades increased efficiency slightly, which is a positive development. However, the most significant advantage of the hybrid design appears to be the extension of the operating range of the Wells turbine, particularly with the hybrid tapered downward design.

By delaying stall by 1° α, the hybrid tapered downward design is able to operate at higher angles of attack, which is particularly advantageous in actual sea wave conditions. This is because waves can create a range of flow conditions, and the ability to operate at higher angles of attack allows for better performance in more fluctuating flow conditions. The extension of the Wells turbine’s operating range through a hybrid design is, therefore, a substantial improvement and could result in significant benefits in terms of power output and reliability in real-world conditions.

Regarding the cost aspect, it is important to consider the trade-offs between efficiency, power output, and cost. While a hybrid tapered blade design may not provide the highest power output compared to other blade designs, it may still provide a good balance of performance across a wider range of operating conditions. In addition, the cost of a blade design depends not only on its performance but also on other factors such as the materials used, manufacturing process, and maintenance requirements. Therefore, a careful cost–benefit analysis is required to determine the most suitable blade design for a given application.

Additionally, the hybrid tapered downward blade experiences a larger pressure drop and torque coefficient than the hybrid tapered upward blade.

One possible reason could be related to the direction of the flow. When the flow hits the blade near the tip, the hybrid tapered downward blade will have a larger cross-sectional area, which could help to generate higher pressure drop and torque. On the other hand, the hybrid tapered upward blade will have a smaller cross-sectional area near the tip, which could decrease the pressure drop and torque.

Another possible reason could be related to the blade angle. The angle of the blade with respect to the flow direction can affect the lift and drag forces acting on the blade. For example, in the case of the hybrid tapered downward blade, the angle of the blade may be such that the flow is more perpendicular to the blade surface, resulting in higher drag and pressure drop. On the other hand, the hybrid tapered upward blade may have a more favorable angle that allows the flow to separate more smoothly from the blade surface, resulting in lower drag and pressure drop.

Furthermore, Wells turbines generally operate at a relatively low frequency, in the range of ${10}^{-3}$ [25,26]. Thus, any dynamic effects on the Wells turbine can be ignored. Hysteresis occurs in the Wells turbine due to the capacitive behavior of the OWC chamber, rather than turbine aerodynamics [27]. Consequently, steady and unsteady conditions should result in negligible differences in turbine performance. Therefore, the simulations for the current study were performed only in a steady-state condition.

5. Conclusions

A variable chord Wells turbine’s aerodynamic performance was evaluated numerically in this study. The steady-state three-dimensional, incompressible RANS equations coupled with the k-ω SST turbulence model were solved using ANSYS™ CFX. The design and analysis of two hybrid turbines comprising NACA0015 and NACA0025 profiles were carried out with hub-to-tip ratios of 1.58 (hybrid upward) and 0.63 (hybrid downward). Compared to the baseline, the hybrid downward configuration increased peak efficiency by 1.1%. In contrast, the hybrid upward design experienced a 0.3% decrease in peak efficiency. The baseline showed a higher torque coefficient than the hybrid configurations over a flow coefficient range of 0.075 ≤ ϕ ≤ 0.275. However, it is essential to note that the hybrid downward configuration extended the operating range by 1° α. An analysis of the flow field topology was presented. The results revealed that the improved aerodynamic performance of the hybrid downward design was primarily due to the reduction of flow separation, particularly at high flow coefficients.

A higher axial velocity along the span, resulting in an early separation and stall, contributed to the decrease in performance of the hybrid upward design. Through the hybrid downward design, the strong vortices that form close to the leading edge of the blade tip were mitigated to delay separation.

A more realistic representation of the ocean environment, including unsteady inflow boundary conditions, should be included in future numerical simulations of Wells turbines for Oscillating Water Column applications. Furthermore, to maximize turbine performance, we recommend optimizing the taper ratio and profile of the airfoil.