Performance Enhancement for a High-Blockage Vertical Axis Autorotation Turbine VAACT by Blade Design Improvement

Highlights

What are the main findings?

• A CFD-based optimization combining parametric analysis, response surface modeling, and a genetic algorithm identified an improved VAACT blade with 40 maximum camber, 65% camber position, and an optimal tip-speed ratio of 1.06.
• Under a 21 blockage ratio, the optimized blade achieved a power coefficient of 40.68%, representing an approximately 21% improvement over the conventional S-shaped VAACT profile.

What are the implications of the main findings?

• The performance improvement is primarily associated with reduced resisting torque from the returning blade, together with weaker and delayed vortex shedding and more favorable pressure distributions.
• The results demonstrate that controlling the returning-blade flow dynamics through blade-shape optimization can provide an effective strategy for improving the intracycle performance of hydrokinetic turbines.

Abstract

This study presents a comprehensive parametric optimization of the blade geometry for the so-called Vertical Axis Autorotation Current Turbine (VAACT) using Computational Fluid Dynamics (CFD) simulations at high-blockage conditions. The optimization process aimed to minimize the resisting torque produced by the returning blade, the major contributor to performance loss in drag-type vertical axis turbines. The investigation was first conducted to assess the effects of maximum camber and its position on turbine efficiency. A response surface relating the power coefficient ($C_P$) to maximum camber (M), camber position (P), and tip-speed ratio ($TSR$) was then generated and used as input for the optimization routine. The optimal configuration was M = 40%, P = 65%, and $TSR$ = 1.06, achieving an efficiency of 40.68% at 21.4% blockage ratio. The enhanced blade demonstrated a 21% improvement in performance compared to the conventional S-shaped profile, primarily due to the reduction in resisting torque during the returning phase. The intracycle analysis showed that the optimized turbine allowed a better distribution of hydrodynamic effects along the cycle, improving efficiency. Flow field analysis confirmed that the optimized geometry delays vortex shedding and reduces pressure gradients along the returning blade, leading to improved hydrodynamic behavior. The results demonstrate this procedure offers an effective way for the design of vertical axis hydrokinetic turbines.

1. Introduction

The investigation of vertical axis turbines (VATs) is extensively employed for energy harvesting in sites where conventional horizontal turbines are not feasible, such as roads and urban areas [1,2,3,4], and low-flow-rate rivers [5,6,7,8]. Although they have lower efficiency than horizontal axis turbines [9,10], their omnidirectional behavior [11,12,13], good self-starting capability associated with drag devices [14,15,16], and low levelized cost of energy (LCOE) [17,18] justify their study.

In the literature, these vertical turbines are classified as cross-flow turbines [19]. That means they capture the kinetic energy of a moving flow with spinning blades oriented perpendicular to the direction of flow [20]. Furthermore, the VATs can be subdivided into drag-driven [21] and lift-driven devices [22]. The drag-driven devices are simple machines that usually operate at low velocities and high torques. The maximum tip-speed ratio ($TSR$) is 1.0, and they are less efficient than the lift-driven devices, which operate at higher rotational speeds and achieve higher efficiency values [23]. Nevertheless, drag devices have economic and operational benefits that justify their study and application over lift devices (low cost, simpler manufacturing, and better self-starting properties) [23]. Equations (1) and (2) depict the tip-speed ratio and power coefficient expressions,

$$TSR={\displaystyle \frac{\omega ·D}{2·U_{\infty }}}$$

(1)

$$C_P={\displaystyle \frac{P_{PTO}}{\frac{1}{2}\rho A{U}_{\infty }^3}}$$

(2)

where $\omega$ is the rotation speed; D is the turbine diameter; $U_{\infty }$ is the flow velocity measured 0.75 m ($2.5·D$) upstream of the turbine model; P is the power output; $\rho$ is the fluid density; and A is the frontal area (or the projected area).

There are also the so-called hybrid turbines [24]. These devices are manufactured by coupling lift- and drag-driven devices on the same axis to improve their overall properties. Recent investigations indicate these hybrid concepts can increase the design tip-speed ratio and the optimum rotor efficiency against the conventional Savonius and Darrieus rotors [25,26].

Hybrid Darrieus–Savonius turbines have emerged as a promising alternative for overcoming the self-starting limitations of conventional Darrieus rotors. In this configuration, the drag-driven Savonius rotor provides the starting torque required for operation at low flow velocities, while the lift-driven Darrieus rotor ensures higher energy conversion efficiency at larger tip-speed ratios. Numerical investigations carried out by Inácio et al. [26] suggest hybrid configurations can increase the power coefficient by up to 70% at low-TSR conditions and still achieve performance gains of approximately 6.5% at the optimum operating TSR, indicating a favorable balance between the additional power generated by the Savonius rotor and the aerodynamic losses caused by flow interaction effects.

The present approach focuses on the so-called Vertical Axis Autorotation Current Turbine (VAACT). This is an S-shaped configuration that resembles the classical Savonius geometry. The research has already investigated the performance of a flat plate [27], a flapped plate [28,29], an S-shaped profile [30,31], and currently an optimized S-shape shown in this study.

This investigation also focuses on the methodology and main findings of the intracycle performance improvement of vertical axis turbines, as analyzed by examining the VAACT. This methodology includes an extensive Computational Fluid Dynamics (CFD) campaign to determine the device’s power coefficient as a function of the tip-speed ratio and the geometry, creating a database that allowed for applying the single-objective optimization method on a response surface achieved with this database.

The results obtained with the enhanced profile were compared with the classical VAACT performance by using polar diagrams of hydrodynamic coefficients versus azimuthal angle and post-processing CFD images, which made it possible to understand how the geometry and the resisting torque affect the intracycle performance of this device.

Blade-shape optimization procedures have demonstrated the potential to substantially improve the performance of drag-driven vertical axis turbines. For the Vertical Axis Autorotation Current Turbine (VAACT), this novelty CFD-based optimization study shows the blade camber and camber position can significantly influence the hydrodynamic loads acting on the rotor. By combining response surface methodology and optimization algorithms, an optimized blade profile was identified with an efficiency improvement of about 21%.

Moreover, this study also details the physical mechanism behind the efficiency improvement, such as the reduction in the resisting torque generated by the returning blade, the vortex-shedding pattern, and the pressure distributions around the rotor, reducing hydrodynamic losses throughout the rotational cycle. These findings indicate that controlling the flow dynamics on the returning side of the turbine is a key mechanism for improving the efficiency of drag-based hydrokinetic devices and provides valuable guidelines for future blade design developments.

2. Literature Review

Most recently, vertical axis turbine studies have focused on performance optimization either by blade optimization or operational performance enhancement. Blade optimization is employed since the turbine profile can be improved to boost turbine characteristics and minimize efficiency drop in the wake. This is usually performed by applying optimization techniques or parametric analysis.

On the other hand, operational enhancement assumes the turbine geometry is already optimal, or close to an optimal solution, and external aspects are changed to achieve better performance metrics. This approach may be accomplished by adding an obstacle close to the blade to create more favorable gradients of pressure and velocity near the blades. Moreover, hybrid solutions combining lift- and drag-driven devices are also adopted to improve the intracycle performance with the drag device and operate at larger tip-speed ratio values because of the lift device.

Obstacles have been widely explored in the literature to improve the operational performance of vertical axis turbines. Typical applications intend to add deflectors or a flow lens to improve the flow through the device [32,33,34,35]. Wind lenses were installed by Hesami & Nikseresht [35] to study the effect of flanged diffusers on the performance improvement of a Savonius turbine. Six diffuser layouts were analyzed to achieve the best configuration by CFD simulations. The optimum configuration occurs when the diffuser and the flange lengths are four and six times the turbine diameters, respectively. It is worth noting that these wind lens layouts create constrained fluid flow, which allows for performance greater than the Betz Limit, defined for unbounded flow [36].

Hajong et al. [33] and Khoury et al. [34] utilized a shielding strategy to induce favorable pressure gradients and flow conditions to minimize the effect of resisting torque in a hybrid Savonius–Darrieus and in a single Savonius rotor, respectively. The CFD study conducted by Ref. [33] led to power coefficient enhancements larger than 6.0% for hybrid turbines operating with deflectors. On the other hand, Khoury et al. [34] accomplished experimental and numerical analyses with a shielding technique for redirecting the flow from the returning blade to the advancing blade. This study clarifies that the ${30}^{\circ }$ and ${60}^{\circ }$ curtain shield configuration shows a good improvement in the power coefficient ($C_P$) based on the unshielded profile.

Parametric studies have also been taken into consideration to enhance the Savonius turbine’s performance by shape modification [37,38,39,40,41,42]. Chullai et al. [37] conducted their research evaluating the effect of the aspect ratio and the wind speed on the performance of a conventional semicircular Savonius turbine. Cabalo & Marcelo [38] performed their analysis changing the number of blades (2–4) and the overlap ratio (0.0–0.2) for a Savonius turbine for operation in an irrigation canal in the Philippines. Similarly, Francisco et al. [39] tested two-, three-, and six-bladed Savonius turbines on roofs at 2 m/s of wind speed. Their results indicate that the three-bladed configuration achieved the highest performance [37,39].

Tham et al. [40] analyzed a two-stage Savonius turbine by varying the blade configurations at four flow velocities. The optimum layout found in the investigation occurs when the turbines are perpendicular to each other, which minimizes the intracycle fluctuations. Jamal et al. [41] accomplished a broader parametric analysis, analyzing the effect of geometry (aspect ratio and blade shape) and operational (tip-speed ratio, end plates, guide vanes, and deflecting blades) issues to enhance and boost a novel Savonius turbine performance. The novel approach proposes flexible blades that achieve a 17% improvement over a conventional Savonius rotor.

Moreover, the so-called Taguchi method is widely employed to determine the experimental factors that most affect the performance of vertical axis turbines [43,44,45,46,47,48]. This is a statistical approach to improve the product and process quality against external disturbances represented by the turbine’s geometric and operational factors. The most recent research in the literature shows this methodology has been applied to enhance Savonius and hybrid Darrieus–Savonius turbine performance.

Kaya & Acir [43] investigated the effect of blade type, overlap ratio, blade separation, separation distance, and blade thickness on the power coefficient of a two-bladed Savonius turbine. They found that the blade separation and the overlap ratio further influence the turbine’s performance. Moreover, the best turbine model has elliptical blades with 2 mm thickness, an overlap ratio of 0.15, and a blade separation of 7.5 mm.

Jiang et al. [44] analyzed a Bach turbine to enhance the global performance of a dual-rotor configuration. A three-turbine layout was evaluated by changing turbine spacing, configuration angle, and rotation angle. The analysis shows that the configuration angle has a large effect on the power coefficient, followed by the spacing and the direction of rotation speed. The optimal layout leads to a 64% improvement in the global efficiency. Note that a Bach turbine has also been tested by Zadeh et al. [45], getting similar performance improvement at $TSR$ = 2.5.

Hybrid Savonius–Darrieus solutions were tested by Abdel-Razak et al. [46] for vertical axis turbines incorporated at water storage tank buildings. They conducted numerical simulations to apply the Taguchi method, considering a methodology similar to that developed by Ref. [44]. The study also agreed that the rotation direction and the turbine distance most affect the global performance.

Meanwhile, Mirzaeian et al. [47] and Nath et al. [48] analyzed modifications in Darrieus and Savonius turbines, respectively. Mirzaeian et al. [47] present a novel dual-row vertical axis wind turbine combining inner J-shaped and outer conventional blades to exploit both drag and lift forces, thereby improving performance especially at low-$TSR$ values. They explore five key geometric parameters and identify that the tip-speed ratio has the greatest influence, with other important factors including airfoil shape, radial ratio, solidity ratio, and angular spacing. The optimized configuration achieved a maximum power coefficient of approximately 0.52, significantly outperforming comparable hybrid or single-row designs.

Furthermore, Nath et al. [48] studied the hydrodynamic behavior of a hybrid-bladed Savonius turbine under low-flow conditions using Computational Fluid Dynamics and a Taguchi optimization framework, aiming to improve its torque and efficiency in hydrokinetic applications. By blending curved (airfoil-shaped) blades with straight ones, the study identifies optimal geometric parameters and operating conditions that enhance starting torque, reduce flow separation, and maximize power output over a range of Reynold numbers.

These hybrid concepts were employed by Nhambiu et al. [25] and Inácio et al. [26]. The first investigation focused on a hybrid Savonius–Darrieus turbine solution for rural Mozambique fields, while the second only intended to improve the device performance. Ref. [25] determined the lift-to-drag ratio for achieving the optimum turbine setup, so that it presented $C_P$ = 45% at $TSR$ = 2.0. On the other hand, the CFD model proposed in Ref. [26] led to a significant performance improvement (≈65%) at $TSR$ = 1.5. These studies also suggest that the hybrid turbine performance becomes better than a conventional Darrieus turbine. Turbines in tandem can also be adopted to improve the operational performance of vertical axis turbines [49].

Other approaches recently considered in the literature are the application of design of experiments (DOEs) and factorial design [50,51,52], artificial neural network surrogate models [53], and torque fluctuation reduction techniques [54]. They also lead to promising findings on the shape and operational optimization of Savonius rotors.

The relevant operational parameters from the literature discussed in this section are summarized in Table 1.

Table 1. Recent investigation on vertical axis turbines. $C_P$ and $TSR$ values are provided by the respective authors. The subscripts 2b, 3b, 4b, and 6b mean 2-, 3-, 4-, and 6-bladed turbines.

Turbine	Ref.	Method	Operational Range	Design Point
Savonius	[35]	CFD	$C_P$ = 9–71% $TSR$ = 0.4–1.3	$C_P$ = 70.2% $TSR=1.0$
Hybrid Darrieus–Savonius	[33]	CFD	$C_P$ = 10–40% $TSR$ = 1.0–3.5	$C_P=40\%$ $TSR=1.5$
Savonius	[34]	CFD + Exp.	$C_P$ = 0–25% $TSR$ = 0.1–1.0	$C_P=25\%$ $TSR=0.77$
Savonius	[37]	CFD	-	$C_P=21.7\%$ $TSR=0.80$
Savonius	[38]	CFD + Exp.	-	$C_{P,2b}=10.2\%$ $C_{P,3b}=21.2\%$ $C_{P,4b}=17.5\%$ $TSR=0.90$
Savonius	[39]	Exp.	-	$C_{P,3b}=24.5\%$ $TS{R}_{3b}=0.94$ $C_{P,4b}=22.7\%$ $TS{R}_{4b}=0.87$ $C_{P,6b}=22.3\%$ $TS{R}_{6b}=0.80$
Savonius	[40]	CFD	$C_P$ = 5–25% $TSR$ = 0.4–1.0	$C_P=25\%$ $TSR=0.8$
Savonius	[41]	CFD	$C_P$ = 20–45% $TSR$ = 0.15–0.20	$C_P=45\%$ $TSR=0.16$
Savonius	[43]	CFD	-	$C_P=17.9\%$ $TSR=0.80$
Bach Savonius	[44]	CFD	-	$C_P=49.9\%$
Bach Savonius	[45]	CFD	$C_P$ = 0–15% $TSR$ = 0.05–1.2	$C_P=15\%$ $TSR=0.65$
Hybrid Darrieus–Savonius	[46]	CFD	-	$C_P=26.1\%$ $TSR=2.86$
Hybrid Bladed-Darrieus	[47]	CFD	$C_P$ = 4–50% $TSR$ = 0.5–3.5	$C_P=50\%$ $TSR=2.3$
Hybrid Bladed-Savonius	[48]	CFD	-	$C_P=27.2\%$ $TSR=0.68$
Hybrid Darrieus–Savonius	[25]	CFD	-	-
Hybrid Darrieus–Savonius	[26]	CFD	$C_P$ = 2–22% $TSR$ = 0.2–2.0	$C_P=22\%$ $TSR=1.5$
Savonius	[51]	CFD	$C_P$ = 7–34% $TSR$ = 0.2–1.2	$C_P=34\%$ $TSR=0.80$
Savonius	[50]	Exp.	$C_P$ = 0–25% $TSR$ = 0.0–1.0	$C_P=24.3\%$ $TSR=0.67$
Savonius	[53]	CFD	-	-
Savonius	[54]	CFD + Exp.	-	-
Flapped VAACT	[52]	CFD + Exp	-	$C_P=15.4\%$

3. Materials and Methods

3.1. Domain and Boundary Conditions

Table 2 shows the control volume dimensions. The boundary conditions are the following: the inlet and the outlet have velocity and pressure gauge boundary conditions, whereas the turbine and the wall have no-slip boundary conditions. Figure 1 illustrates the CFD domain and the boundary conditions.

Table 2. Control volume dimensions in the turbine shape parameterization study.

Parameter	Symbol	Value
Upstream Length	$L_U$	1.50 m
Downstream Length	$L_D$	3.00 m
Control Volume Width	W	1.40 m
Rotation Region Diameter	$D_r$	0.35 m
Blockage Ratio	$ϵ=D/W$	21%

Figure 1. CFD domain and boundary conditions: (a) turbine geometry 0.30 m diameter, 3 mm thickness, maximum camber of 37% turbine radius, and camber position at 64% turbine radius; (b) domain dimensions and boundary conditions.

The CFD simulations were performed using the commercial software ANSYS Fluent 2024 R1 [55]. A two-dimensional approach was adopted in the present study. This simplification is supported by the experimental configuration, in which the turbine is positioned close to the bottom of the flume, with a small bottom gap that restricts the development of significant vertical flow motions beneath the rotor and promotes the predominance of the horizontal flow component.

This particularity avoids three-dimensional flow structures that may develop near the blade tips and at the rotor base. These structures, including tip vortices and spanwise flow, could contribute to additional momentum losses and consequently reduce the actual performance of a finite-span turbine that could make the present 2D model overpredict the performance results. Recent studies involving Darrieus- and Savonius-type vertical axis turbines have reported differences of approximately 30–35% in the maximum power coefficient between two- and three-dimensional CFD predictions [56,57,58].

This range should not be interpreted as a universal correction factor, since the magnitude of three-dimensional effects depends on the turbine geometry, aspect ratio, blade-tip configuration, operating conditions, and blockage. Despite this limitation, two-dimensional CFD models are widely employed because they are considerably less computationally demanding than three-dimensional models while providing robust information for assessing blade-section hydrodynamics, comparing different configurations, and supporting preliminary turbine design and optimization.

Finally, the relatively low free-surface deformation and limited wave generation under the investigated operating conditions further support the assumption that the flow is predominantly governed by its horizontal-plane dynamics. Therefore, the two-dimensional formulation provides an appropriate representation of the flow around the blade section for the comparative analysis and optimization performed in this study.

3.2. Grid Generation

The grid for these simulations is illustrated in Figure 2. It has about 136,000 elements, where 64,000 are triangular cells, and 72,000 are quadrilateral cells. The dimensionless distance to the wall ($Y^{+}$) at the turbine geometry is between 0.54 and 1.30, which complies with the recommendations for using the turbulence model SST k-$\omega$ to model the boundary layer ($Y^{+}\approx 1$) [59]. The other grid quality metrics, such as skewness, aspect ratio, and orthogonal quality, also present adequate values for the CFD simulations with these grid characteristics. These properties are summarized in Table 3. The time series of $Y^{+}$ is seen in Figure 3.

Figure 2. Grid around the profile: (a) control volume; (b) rotating region; (c) turbine geometry; (d) turbine blades.

Table 3. Mesh information for simulations with NACA profiles.

Parameter	Value
Number of Nodes	212,756
Number of Elements	135,788
Triangular Mesh	63,428
Quadrilateral Mesh	72,360
Maximum $Y^{+}$	1.30
Average $Y^{+}$	0.86
Minimum $Y^{+}$	0.54
Skewness	0.05–0.20
Aspect Ratio	1.0–4.0
Orthogonal Quality	0.88–1.00

Figure 3. Time series of the dimensionless distance to the wall ($Y^{+}$).

3.3. Numerical Formulation

This study examines the shape of a vertical turbine using a pressure-based solver with a transient time formulation. The simulation lasts 40 s and is conducted at a flow velocity of 0.30 m/s and an angular velocity range of 0.4 rad/s to 3.8 rad/s. The average value of the output parameters is determined in the steady-state for at least 5 half-cycles. The details of this research are presented in Table 4. Note that second-order schemes are applied since the error in the second order avoids convergence issues in the transient formulation. This scheme provides higher accuracy and lower numerical diffusion than first-order schemes, while maintaining greater numerical stability. This approach enables a more accurate prediction of turbine loads and wake characteristics.

Table 4. CFD simulations—numerical schemes.

Parameter	Description
Turbulence Model	SST k-$\omega$
Solver	Pressure-Based
Time	Transient
2D Space	Planar
Simulation Time	40 s
Flow Velocity	0.30 m/s
Angular Velocity	0.4–3.8 rad/s
Rotor Revolutions	5–48

The SST k-$\omega$ model was selected because it is a noteworthy alternative to the CFD study of vertical turbines, since this model is computationally cheap, robust, and handles separation better than k-$ϵ$ turbulence models. Moreover, it achieves good accuracy in predicting the power and torque coefficients [60]. It is worth mentioning that the CFD simulations assume rigid turbine blades, which means no deformation effects were considered in the analysis to estimate the turbine loads.

3.4. Blade Improvement Methodology

The blade improvement methodology adopted in this study was based on a CFD-driven optimization framework combining parametric analysis, response surface modeling, and single-objective optimization. The primary objective was to improve the hydrodynamic efficiency of the Vertical Axis Autorotation Current Turbine (VAACT) by reducing the resisting torque generated by the returning blade, which had previously been identified as one of the main sources of performance loss in S-shaped drag-based rotors [61]. The overall blade improvement workflow adopted in this study is illustrated in Figure 4.

Figure 4. CFD-based blade improvement framework adopted for the VAACT.

In the first step, the blade geometry was parameterized using two nondimensional design variables: the maximum camber (M) and the position of maximum camber (P). These parameters are illustrated in Figure 1a and Figure 4. These parameters were selected because they directly influence the pressure distribution around the blade, the vortex-shedding process, and consequently the balance between driving and resisting torque during a rotor revolution.

The turbine profile was generated by the camber line equation of a NACA four-digit profile. This is expressed in Equation (3):

$$y_{mean}={\displaystyle \frac{D}{2}}·\left\{\begin{array}{cc}{\displaystyle \frac{M}{P^2}}·\left(2·P·x-x^2\right),\hfill & 0\le x<P\hfill \\ {\displaystyle \frac{M}{{\left(1-P\right)}^2}}·(1-2·P·x-x^2)\hfill & P\le x<1\hfill \end{array}\right.$$

(3)

Then, the upper and lower faces are generated as follows,

$$\begin{array}{cc}\hfill y_{upper}& =y_{mean}+{\displaystyle \frac{t}{2}}\hfill \end{array}$$

(4)

$$\begin{array}{cc}\hfill y_{lower}& =y_{mean}-{\displaystyle \frac{t}{2}}\hfill \end{array}$$

(5)

where $y_{mean}$ is the camber line equation; $y_{upper}$ and $y_{lower}$ are the upper and lower faces, respectively; t is the blade thickness; and $x\in [0;1]$ is the normalized distance from the center of rotation to the turbine tip.

Design of experiments (DOEs) was adopted to investigate the effects of the design variables on the power coefficient. Three variables were considered: the tip-speed ratio (TSR), maximum camber (M), and camber position (P). The TSR was varied from 0.2 to 1.9 with an increment of 0.1, M was varied from 0.15 to 0.85 with an increment of 0.10, and P was varied from 0.5 to 0.9 with an increment of 0.2. The combination of these levels resulted in a total of 432 CFD design points, corresponding to 18, 8, and 3 levels for TSR, M, and P, respectively.

For each combination of ($M,P,TSR$), the blade geometry was generated in Rhinoceros 3D [62] and further processed in ANSYS Design Modeler [63]. Computational meshes were subsequently generated using ANSYS Meshing [64] and employed in transient two-dimensional CFD simulations performed in ANSYS Fluent [55]. The hydrodynamic loads acting on the turbine were calculated by solving the Reynold-Averaged Navier–Stokes equations using the SST (k-$\omega$) turbulence model.

The lift and drag forces were determined by the net force, shown in Equation (6). It is obtained by normal pressure ($p\mathbf{n}$) and viscous ($\tau$) contributions integration onto the turbine profile S,

$$\mathbf{F}={\int }_S\left(-p\mathbf{n}+\tau \right)dS$$

(6)

where $\mathbf{n}$ is the normal vector. Then, the lift and drag forces are defined perpendicularly and parallel to the flow so that $\mathbf{F}=D\widehat{\mathbf{i}}+L\widehat{\mathbf{j}}$.

On the other hand, the torque is given by Equation (7) as a function of the position vector $\mathbf{r}=\left(x,y,0\right)$ and the net force $\mathbf{F}=\left(F_D,F_L,0\right)$ for a two-dimensional approach:

$$\mathbf{T}=\underset{\mathrm{General}\mathrm{Formula}}{\underbrace{{\int }_S\mathbf{r}\times d\mathbf{F}}}=\underset{2\mathrm{D}\mathrm{Case}}{\underbrace{{\int }_S(x·F_L-y·F_D)dS}}$$

(7)

The power coefficient was determined by Equation (2), where the power output $P_{PTO}$ is the turbine torque times the rotation speed ($T·\omega$). Equation (8) shows the expression to calculate the lift, drag, and torque coefficients:

$$C_L={\displaystyle \frac{F_L}{\frac{1}{2}\rho A{U}_{\infty }^2}},C_D={\displaystyle \frac{F_D}{\frac{1}{2}\rho A{U}_{\infty }^2}},C_T={\displaystyle \frac{T}{\frac{1}{2}\rho AD{U}_{\infty }^2}}$$

(8)

The resulting torque and power output were used to determine the power coefficient ($C_P$), expressed in Equation (9), which was adopted as the performance metric throughout the single-objective optimization process. This procedure generated a database relating the turbine efficiency to the blade geometry and operating tip-speed ratio.

$${\overline{C}}_P={\displaystyle \frac{\overline{T}·\overline{\omega }}{\frac{1}{2}·\rho ·A·U_{\infty }^3}}$$

(9)

where $\overline{T}$ is the hydrodynamic torque, $\rho$ is the fluid density, A is the frontal area, and $U_{\infty }$ is freestream velocity.

The CFD database was subsequently used to construct a response surface model comprising 32 coefficients. A polynomial regression was fitted to the CFD data to express the power coefficient as a nonlinear function of maximum camber, camber position, and tip-speed ratio. The resulting model provided an analytical representation of the turbine performance, significantly reducing the computational cost associated with repeatedly evaluating candidate designs during optimization.

The response surface was then employed as the objective function of the optimization procedure. The optimization problem was formulated to maximize the power coefficient while respecting the prescribed geometric bounds. A single-objective genetic algorithm (GA) was employed to maximize the predicted turbine efficiency using the MATLAB 2024b ‘ga’ solver. The optimization involved three variables: the maximum camber ratio (M), the camber position (P), and the tip-speed ratio ($TSR$).

A population size of 200 individuals was adopted, with stochastic uniform selection and a scattered crossover operator using a crossover fraction of 0.80. A customized mutation operator was implemented with a mutation probability of 0.05. An elite count of 3 individuals was retained at each generation. The maximum number of generations was set to 300, while the function tolerance was ${10}^{-6}$ and the maximum number of stall generations was set to 100. The optimization converged after 59 generations, requiring 11,826 objective-function evaluations.

This optimization algorithm searched the design space using the response surface approximation to identify the combination of blade parameters and operating conditions that maximized turbine performance. Finally, the hydrodynamic mechanisms responsible for the performance improvement were investigated through analyses of lift, drag, torque, vorticity, and pressure fields and compared with the results achieved for the baseline turbine geometry (S-shaped VAACT profile). Table 5 summarizes the simulation matrix considered in the parametric analysis.

Table 5. Simulation matrix for the parametric analysis.

Parameter	Symbol	Value
Turbine Diameter	D	0.3 m
Thickness	t	3 mm
Reynold Number	$Re$	90,000
Tip-Speed Ratio	$TSR$	0.2–1.9
Maximum Camber	M	5–90%
Camber Position	P	50–90%

3.5. Blockage Correction Methodology

This investigation considers a blockage correction methodology based on the method proposed by Alexander et al. [65] and Alexander & Holownia [66]. This method and others (Pope & Harper [67], Bahaj [68], Maskell [69], Werle [70], Barnsley & Wellicome [71], and Houlsby et al. [72]) were analyzed by Jeong et al. [73] and Ross & Polagye [74] for lift-driven devices and axial rotors.

Alexander [65,66] and Maskell [69] are methodologies designed for drag-driven devices such as Savonius turbines. Furthermore, they [65,66] extended the Maskell [69] method to a Savonius turbine [73], suggesting it is the best option in this application. Alexander’s method proposes the ratio between the corrected velocity and the freestream velocity is [65,66,73]

$${\left({\displaystyle \frac{U_c}{U_{\infty }}}\right)}^2={\displaystyle \frac{1}{1-mϵ}}\therefore U_c={\displaystyle \frac{U_{\infty }}{{\left(1-mϵ\right)}^{1/2}}}$$

(10)

where $U_c$ is the corrected speed; $U_{\infty }$ is the freestream velocity; $ϵ=0.21$ is the blockage ratio; and m is a semi-empirical term used to correct the blockage effect. $m\approx 2.08$ at $ϵ=0.21$ [73].

By Equation (10), the corrected power coefficient ($C_{P,c}$) and tip-speed ratio ($TS{R}_c$) are, respectively,

$$\begin{array}{cc}\hfill C_{P,c}& ={\displaystyle \frac{P}{\frac{1}{2}\rho A{U}_c^3}}\hfill \\ \hfill & =C_{P,ϵ}·{\left(1-mϵ\right)}^{3/2}=0.42·C_{P,ϵ}\hfill \end{array}$$

(11)

$$\begin{array}{cc}\hfill TS{R}_c& ={\displaystyle \frac{\omega R}{U_c}}\hfill \\ \hfill & =TS{R}_{ϵ}·{\left(1-mϵ\right)}^{1/2}=0.75·TS{R}_{ϵ}\hfill \end{array}$$

(12)

$C_{P,ϵ}$ and $TS{R}_{ϵ}$ represent the performance properties determined by the $ϵ=0.21$ blockage condition.

4. Results

4.1. Model Verification

4.1.1. Domain

The sensitivity study for the domain is made by changing the upstream ($L_U$) and downstream ($L_D$) lengths of the control volume. To achieve a blockage ratio ($ϵ=D/W$) of $21\%$ for the width, it is maintained at $W=1.4\mathrm{m}$, the condition at the test facility used to validate the numerical model.

This analysis enables the study of the power coefficient ($C_P$) on the turbine by creating three domains: large (1), medium (2), and small (3). Their dimensions are provided in Table 6. It should be noted that the domain width is 1.4 m, and the rotating region diameter ($D_r$) is 0.35 m. The time-averaged power coefficient for the three domains is also presented, assuming that it is determined by Equation (9). The sensitivity test was carried out at a freestream velocity of $0.25\mathrm{m}/\mathrm{s}$ and a rotation speed of $\overline{\omega }=1.25\mathrm{rad}/\mathrm{s}$, resulting in a tip-speed ratio of $\overline{TSR}=0.75$.

Table 6. Domain dimensions and results—domain sensitivity test.

#	LU [m]	LD [m]	TSR [–]	${\overline{\mathbf{C}}}_{\mathbf{P}}$ [–]
1	3.0	6.0	0.75	32.92%
2	2.1	4.2	0.75	32.81%
3	1.5	3.0	0.75	32.87%

It is worth noting that the changes in the simulations between the medium–large and small–medium domains were ${\epsilon }_{D^{*},21}=-0.11\%$ and ${\epsilon }_{D^{*},32}=0.05\%$, respectively. The domain had oscillatory convergence, as the convergence ratio was negative ($R_{D^{*}}=-2.00$). The small domain approach was used in the simulations due to its time-saving and cost-effective nature compared to running cases using large or medium domains [75].

4.1.2. Grid

The grid sensitivity test involves three grids with a refinement ratio of approximately $\sqrt{2}$. The number of elements in each grid is provided in Table 7. The refinement ratio can be determined using Equation (13), which is calculated by taking the square root of the ratio between the number of elements in the medium–fine and coarse–medium grids. The value of n in the equation depends on the characteristics of the model. For example, $n=2$ for a two-dimensional model, and $n=3$ for a three-dimensional model.

$$r_G=\sqrt{{\displaystyle \frac{N_1}{N_2}}}\approx \sqrt{{\displaystyle \frac{N_2}{N_3}}}$$

(13)

Table 7. Grid setup and results—grid sensitivity test.

#	Grid	Elements	${\mathbf{r}}_{\mathbf{G}}$	${\mathbf{y}}_{\mathbf{Turb}.}^{+}$	${\mathbf{y}}_{\mathbf{Wall}}^{+}$	${\overline{\mathbf{C}}}_{\mathbf{P}}$
1	Fine	268,134	1.35	1.59	1.81	33.40%
2	Medium	147,064	1.37	1.47	2.26	32.76%
3	Coarse	77,914	-	1.74	3.27	31.96%

Table 7 contains information about the dimensionless distance to the wall ($y^{+}$) at the turbine and the control volume. The value is less than 6.0, indicating that the model can capture turbulent phenomena at all layers of the boundary layer. It is worth noting that the $y^{+}$ value at the turbine is less than 2.0, while it is less than 4.0 at the walls.

During the sensitivity test, changes were observed between medium–fine and coarse–medium grids when the power coefficient was adopted as a parameter. The changes resulted in a convergence ratio of $R_G=0.8$ with ${\epsilon }_{G,21}=-0.64\%$ and ${\epsilon }_{G,32}=-0.80\%$. After this, a monotonic convergence of the grid was observed. The medium grid was found to have a small error compared to the fine grid (less than 2.0%) and was chosen for the test.

4.1.3. Time-Step Size

The study aimed to analyze the effect of different time-step sizes on the power coefficient. Three time-step sizes were tested: 0.007 s, 0.010 s, and 0.014 s. Table 8 shows the power coefficient values for each time-step size. The analysis revealed that the power coefficient exhibited an oscillatory convergence concerning the time-step size.

Table 8. Time-step size and results—time-step sensitivity test.

#	Time-Step Size [s]	$\mathbf{TSR}[-]$	${\overline{\mathbf{C}}}_{\mathbf{P}}[-]$
1	0.007	0.75	32.92%
2	0.010	0.75	32.76%
3	0.014	0.75	32.81%

Specifically, the power coefficient for the medium time-step of $\Delta t_2=0.010\mathrm{s}$ was smaller than that for both $\Delta t_1=0.007\mathrm{s}$ and $\Delta t_3=0.014\mathrm{s}$. Based on these findings, a time-step size of 0.010 s was chosen for the simulations. The changes between medium–fine and coarse–medium are ${\epsilon }_{T,21}=-0.16\%$ and ${\epsilon }_{T,32}=0.05\%$, respectively. Thus, the convergence ratio is $R_T=-3.0$.

4.1.4. Simulation Uncertainty

A summary of the verification procedures was presented in Table 9. The expressions listed in the table are applied to determine the domain and the time-step uncertainties for oscillatory convergence. For grid convergence, Richardson extrapolation is used as described in Equation (14). To account for monotonic convergence, a safety factor of $F_S=1.25$ is applied to the expression. These procedures are following the ITTC guidelines [75].

$${\delta }_{R{E}_{i,1}}^{*\left(1\right)}={\displaystyle \frac{{\epsilon }_{i,21}}{r_i^{p_i}-1}}$$

(14)

where

$$p_i={\displaystyle \frac{ln\left({\epsilon }_{i,32}/{\epsilon }_{i,21}\right)}{ln\left(r_i\right)}}$$

(15)

Table 9. Summary of verification results and parameter uncertainties.

Test	${\mathbf{R}}_{\mathbf{i}}$	Convergence	Expression	${\mathbf{U}}_{\mathbf{i}}$
Domain	−2.0	Oscillatory	${\displaystyle \frac{1}{2}}·\left({\widehat{S}}_U-{\widehat{S}}_L\right)$	0.053%
Grid	0.8	Monotonic	$F_S·\left|{\delta }_{RE}^{*\left(1\right)}\right|$	3.206%
Time-Step	−3.0	Oscillatory	${\displaystyle \frac{1}{2}}·\left({\widehat{S}}_U-{\widehat{S}}_L\right)$	0.080%

Then, the simulation uncertainty can be determined as $U_{SN}=\sqrt{U_{D^{*}}^2+U_G^2+U_T^2}$ which provides $U_{SN}=3.208\%$.

4.2. Model Validation

The CFD validation followed the International Towing Tank Conference (ITTC) guidelines for uncertainty analysis in CFD verification and validation [75]. The error between the real data ($D^{\prime }$) and the simulation (S) was determined by Equation (16):

$$E=D^{\prime }-S$$

(16)

Meanwhile, the validation uncertainty ($U_V$) is achieved by Equation (17), where $U_{D^{\prime }}$ is the data uncertainty and $U_{SN}$ the simulation uncertainty:

$$U_V^2=U_{D^{\prime }}^2+U_{SN}^2$$

(17)

The experimental uncertainty ($U_{D^{\prime }}$) was determined from the tests accomplished at the current flume of the Waves and Currents Laboratory (LOC-COPPE/UFRJ) [30]. It has length of 22 m, a width of 1.4 m, a depth of 0.50 m, and flow velocities in the range 0.05–0.5 m/s. The tests are assumed two-dimensional since the free surface effects/deformations are negligible, and the model bottom does not produce vortex shedding at the gap found between the model tests and the test facility (see Figure 5).

Figure 5. Experimental setup—tests with S-shaped VAACT at LOC-COPPE/UFRJ.

The Qualisys Track Manager (QTM) was used to determine the kinematic properties of the model test, as shown in Figure 6. It measures the angular displacement ($\theta$) of the VAACT, enabling us to estimate both the rotation speed ($\omega$) and angular acceleration ($\dot{\omega }$) by applying an upwind scheme. The first-order schemes are described in Equation (18).

$$\begin{array}{cc}\hfill {\omega }_i& ={\displaystyle \frac{{\theta }_{i+i}-{\theta }_i}{t_{i+i}-t_i}}+\Delta \left(O\right)\hfill \\ \hfill {\dot{\omega }}_i& ={\displaystyle \frac{{\omega }_{i+1}-{\omega }_i}{t_{i+1}-t_i}}+\Delta \left(O\right)\hfill \end{array}$$

(18)

Figure 6. Measuring devices of the test: (a) Qualisys system; (b) Acoustic Doppler Velocimeter.

Figure 7 presents a diagram of the power take-off (PTO) system. The mechanical power take-off device uses the time derivative of the potential energy to evaluate power output.

Figure 7. Mechanical power take-off (PTO) system—weight lifting approach.

The power output can be calculated using the equation $P\left(t\right)=m·g·\dot{h}\left(t\right)$, where m is the mass lifted, g is the acceleration due to gravity, and $\dot{h}$ is the vertical velocity of the mass. Note that $\dot{h}$ can also be represented as $\dot{h}=\omega ·r_p$, where $r_p$ is the pulley radius attached to the vertical axis. The average efficiency of the turbine can be measured as follows:

$$\begin{array}{cc}\hfill \overline{\eta }& ={\displaystyle \frac{\overline{P}\left(t\right)}{\frac{1}{2}·\rho ·A·U_{\infty }^3}}\hfill \\ \hfill & ={\displaystyle \frac{m·g·r_p·\overline{\omega }\left(t\right)}{\frac{1}{2}·\rho ·A·U_{\infty }^3}}\hfill \end{array}$$

(19)

The equation $T^{PTO}=m·g·r_p$ leads to the PTO torque. To estimate the velocity components, an Acoustic Doppler Velocimeter (ADV) is placed 3 m upstream of the test section, as shown in Figure 6. This is a Nortek Vectrino Profiler [76]. A 100 Hz sampling rate is established in the measurements. The turbulence intensity at the test section ranges from 3.5% to 4.2%.

The uncertainty quantification is determined by Equation (20).

$$U_D\left(y\right)=k_{95}·{\displaystyle \frac{{\sigma }_y}{\sqrt{N}}}$$

(20)

where ${\sigma }_y$ is the standard deviation of y and N denotes the sample length, typically comprising three samples. Assuming a Gaussian distribution, the coverage factor for a 95% confidence level is taken as $k_{95}=1.96$.

$${\sigma }_y=\sqrt{{\displaystyle \frac{{\sum }_{i=1}^N{\left(y_i-\overline{y}\right)}^2}{N-1}}}$$

(21)

The average value of a parameter y is given by Equation (22):

$$\overline{y}={\displaystyle \sum _{i=1}^N}{\displaystyle \frac{y_i}{N}}={\displaystyle \frac{y_1+y_2+\dots +y_{N-1}+y_N}{N}}$$

(22)

The validation results are presented in Table 10 and Figure 8. The absolute error, $\left|E\right|$, between the experimental and CFD results is lower than the validation uncertainty, $U_V$, for all five configurations considered. These results indicate good agreement between the experimental measurements and the CFD predictions.

Table 10. Validation of the numerical model using ITTC guidelines [75].

Case	D	${\mathit{U}}_{\mathit{D}}$	S	${\mathit{U}}_{\mathit{SN}}$	$\left|\mathit{E}\right|$	${\mathit{U}}_{\mathit{V}}$	Validated?
A1	22.80%	2.65%	23.93%	3.21%	1.13%	4.16%	Yes
A2	19.28%	1.19%	20.09%	3.21%	0.81%	3.42%	Yes
A3	14.84%	0.67%	14.78%	3.21%	0.06%	3.28%	Yes
A4	12.62%	0.58%	13.03%	3.21%	0.41%	3.26%	Yes
A5	10.19%	0.50%	10.56%	3.21%	0.37%	3.25%	Yes

Figure 8. Model validation—comparison between CFD and experimental models. The error bars mean the simulation and experimental uncertainties, respectively.

It is worth noting that the rotational speeds used in the numerical simulations match the experimentally obtained operating conditions and represent the turbine’s rotational response under the corresponding power take-off conditions. Therefore, the validated CFD model provides a reliable basis for predicting the hydrodynamic performance and energy-harvesting characteristics of the hydrokinetic turbine within the investigated operating range.

4.3. Blade Enhancement Technique

A screening analysis made by CFD simulations allowed for determining the surface response to estimate the power coefficient ($C_P$) behavior as a function of the maximum camber (M), the camber position (P), and the tip-speed ratio ($TSR$). Equation (23) provides this response surface,

$$\begin{array}{c}\hfill C_{P_H}(M,P,TSR)=h_0+h_1·M+h_2·P+h_3·TSR+h_4·M^2\\ \hfill +h_5·P^2+h_6·TS{R}^2+h_7·M^3+h_8·P^3+h_9·TS{R}^3+h_{10}·M^4\\ \hfill +h_{11}·P^4+h_{12}·TS{R}^4+h_{13}·M·P+h_{14}·M·TSR\\ \hfill +h_{15}·P·TSR+h_{16}·M·P·TSR+h_{17}·M^2·P\\ \hfill +h_{18}·M^2·TSR+h_{19}·P^2·M+h_{20}·P^2·TSR\\ \hfill +h_{21}·TS{R}^2·M+h_{22}·TS{R}^2·P+h_{23}·M^2·P^2\\ \hfill +h_{24}·M^2·TS{R}^2+h_{25}·P^2·TS{R}^2\\ \hfill +h_{26}·M^2·P·TSR+h_{27}·M·P^2·TSR\\ \hfill +h_{28}·M·P·TS{R}^2+h_{29}·M^3·P\\ \hfill +h_{30}·M^3·TSR+h_{31}·P^3·TSR\end{array}$$

(23)

where $h_0,h_1,\dots ,h_{31}$ are the unknown coefficients.

The unknown coefficients were firstly adjusted based on the M-$M\in \left[0.05;0.90\right]$, P-$P\in \left[0.50;0.90\right]$, and $TSR$-$TSR\in \left[0.20;1.90\right]$—ranges defined in the two-dimensional simulations with the turbine profiles. The adequacy of the polynomial representation was evaluated using the coefficient of determination, yielding $R^2\approx 0.95$. This result provides a suitable representation of the investigated design space (see Table 11).

Table 11. Power coefficient surface statistics.

Parameter	Value
$R^2$	0.94962
p-Value	$5.7\times {10}^{-276}$
Standard Error	0.00056

Table 12 shows the results found by applying the optimization method using the response surface presented in Equation (23). The maximum camber is $0.49·(D/2)$, the camber position is between $0.58·(D/2)$, and the tip-speed ratio is about $1.07$. The turbine’s power coefficient is around 42.05%. The development of the optimization method through the generations is illustrated in Figure 9. This regression achieves good adherence for the data obtained from the numerical simulations.

Table 12. Optimal results achieved by optimization procedure. Results provided considering a blockage ratio of $ϵ=21\%$.

Parameter	Value
M	0.4915
P	0.5809
$TSR$	1.0723
$C_P$	42.05%

Figure 9. Optimization generations for the blade optimization using polynomial regressions.

Figure 10 displays the efficiency surface as a function of the input parameters (M, P, and $TSR$) with the design point shown in red in the graph. The optimum blade geometry achieved by the response surface was then simulated to verify the model’s feasibility in predicting the turbine’s performance. The analysis shows the optimization limits led to an inaccurate power coefficient result since the global maximum was not close to the best turbine blades obtained in the parametric study, leading to a power coefficient of 37.6%.

Figure 10. Efficiency response surface obtained by CFD simulations. The red point refers to the optimum blade solution.

As a result, a new optimization was conducted, changing the lower and upper limits for maximum camber, camber position, and tip-speed ratio based on the parametric simulations. The inspection of the results made it possible to change the limits to regions at which the power coefficient is larger than 38%, leading to more realistic results. The updated results are summarized in Table 13.

Table 13. Optimal results achieved by the optimization method. Results provided considering a blockage ratio of $ϵ=21\%$.

Parameter	General	Updated
M limits	0.05–0.90	0.30–0.40
P limits	0.50–0.90	0.65–0.75
$TSR$ limits	0.20–1.90	0.9–1.3
M	0.4915	0.4000
P	0.5809	0.6500
$TSR$	1.07	1.06
$C_{P_{Optimum}}$	42.05%	40.37%
$C_{P_{CFD}}$	37.60%	40.68%
Relative error	10.58%	−0.78%

The lower and upper limits in the optimization algorithm were updated to achieve a high level of accuracy between the optimization model and the CFD simulations, with a relative error of less than 1%. The nonlinear coefficients used to build the nonlinear power coefficient curve with the “Updated” optimization method are provided in Appendix A.

5. Discussions

The hydrodynamic coefficients’ behavior in the time domain and as a function of the azimuth angle are provided in Figure 11 and Figure 12, respectively, comparing the performance of the optimized blade profile to the S-shaped profile. Note that both blade geometries are presented in Figure 13. Their power coefficients as a function of the tip-speed ratio are also depicted in Figure 14. The S-shaped profile induces a resisting torque, which causes a loss of performance between the angles of attack 120–150°, and 300–330°.

Figure 11. Time series of hydrodynamic coefficients—S-shaped VAACT and optimized turbine model.

Figure 12. Comparison of hydrodynamic coefficients from S-shaped VAACT and the optimized model proposed in this study by polar diagram for the: (a) Lift coefficient; (b) Drag coefficient; (c) Torque coefficient; (d) Power coefficient

Figure 13. Blade geometry—S-shaped VAACT and optimized profile.

Figure 14. (a) Power coefficient versus tip-speed ratio—S-shaped profile versus optimized profile; (b) optimized profile—power coefficient correction.

Regarding the lift and drag coefficients, no further changes can be observed in the polar diagram views. However, the time series shows the $C_L$ peaks are slightly more prominent at the S-shaped profile than at the optimized profile. Moreover, the drag coefficient for the S-shaped VAACT also shows high fluctuations. These behaviors are reflected in the performance in Table 14, where the power coefficient for bounded and unbounded fluids is described.

Table 14. Performance of S-shaped profile and optimized profile at bounded fluid ($ϵ=21\%$) and unbounded fluid.

Parameter	S-Shaped Profile	Optimized Profile
Maximum Camber—M	0.64	0.65
Camber Position—P	0.37	0.40
$TS{R}_{ϵ=21\%}$	1.06	1.06
$C_{P_{ϵ=21\%}}$	33.80%	40.68%
$TS{R}_c$	0.80	0.80
$C_{P,c}$	14.29%	17.19%

Figure 14 also ratifies the performance improvement comparing the power coefficient curve from the baseline S-shaped VAACT with the optimized profile proposed in the current study and the optimized turbine performance curves for the blocked and corrected coefficients, respectively. The figure shows that the optimized profile efficiency is enhanced for a tip-speed ratio around 1.0. The graph still provides the optimized profile, which presents greater performance than the baseline profile at $TSR\in [0.7;1.4]$. Their performance is nearly the same outside this $TSR$ interval ($TSR<0.7$ and $TSR>1.4$).

The vorticity field around the S-shaped VAACT and the optimized profile are presented in Figure 15. The illustrations show that the vortex-shedding pattern has more intensity close to the S-shaped VAACT than the optimized blade geometry. Because the turbine’s vortex shedding is associated with rotor energy loss, this could explain the performance loss observed for the S-shaped profile relative to the new proposal.

Figure 15. Analysis of the vorticity field around the S-shaped VAACT (a,c,e,g) and the optimized blade geometry (b,d,f,h) for azimuth angles of $0^{\circ }$, ${45}^{\circ }$, ${90}^{\circ }$, and ${135}^{\circ }$, respectively.

On the other hand, the effect of blade optimization can be seen in Figure 15c,d. These figures display that the turbines do have a vortex generation at the returning blade that creates a resisting angular motion. This vortex presents a lower magnitude in the optimized blade than the classical S-shaped profile. This phenomenon contributes a resisting torque to the turbine. In Figure 15g,h, the vorticity field shows another advantage in the optimized blade geometry that consists of delayed vortex shedding. These figures show that the vortices are attached to the blade at ${135}^{\circ }$ azimuth angle. In contrast, the S-shaped profile presents an interaction between rotation and counter-rotating vortices that induces the vortex release.

The effect of the resisting torque in the turbine dynamics can also be seen in Figure 16 for the pressure coefficient around the rotors. Indeed, the blade shape at the tips has been a determining factor in the turbine dynamics. These illustrations allow us to conclude that the S-shaped VAACT has high positive and negative pressures at the returning blade, which causes the so-called resisting torque. On the other hand, the optimized blade geometry presents lower pressure values. This suggests the optimized profile achieves better hydrodynamic performance than the S-shaped VAACT.

Figure 16. Analysis of the pressure coefficient around the S-shaped VAACT (a,c,e,g) and the optimized blade geometry (b,d,f,h) for azimuth angles of $0^{\circ }$, ${45}^{\circ }$, ${90}^{\circ }$, and ${135}^{\circ }$, respectively.

6. Conclusions

This study presented a CFD-based methodology for improving the blade geometry of a Vertical Axis Autorotation Current Turbine (VAACT) under high-blockage conditions. The approach combined parametric analysis, response surface modeling, and single-objective optimization to identify a blade geometry capable of reducing the hydrodynamic losses associated with the returning blade. The main achievements and conclusions of the study are summarized below:

• Blade Geometry Optimization: The maximum camber (M), camber position (P), and tip-speed ratio ($TSR$) were identified as the main parameters governing the turbine performance within the investigated design space. The parametric analysis considered M = 5–90%, P = 50–90%, and $TSR$ = 0.2–1.9.
• Response Surface Model: A polynomial response surface was developed to represent the power coefficient as a function of M, P, and $TSR$. The model achieved a coefficient of determination of $R^2=0.95$, indicating good agreement with the CFD database.
• Optimized Blade Configuration: The updated optimization procedure identified $M=40\%$, $P=65\%$, and $TSR=1.06$ as the optimal design condition. The optimized geometry achieved a power coefficient of $C_P=40.68\%$ when evaluated directly using CFD at a blockage ratio of approximately $21\%$. They correspond to $TS{R}_c=0.80$ and $C_{P,c}=17.19\%$ after the blockage correction, respectively.
• Performance Improvement: At $TS{R}_{ϵ}=1.06$ ($TS{R}_c=0.80$) and a blockage ratio of $21\%$, the optimized profile achieved $C_{P,ϵ}=40.68\%$ ($C_{P,c}=17.19\%$), compared with $C_{P,ϵ}=33.80\%$ ($C_{P,c}=14.29\%$) for the conventional S-shaped VAACT profile, corresponding to a performance improvement of approximately $21\%$.
• Hydrodynamic Mechanism: The performance improvement is primarily associated with the reduction in the resisting torque generated by the returning blade. The optimized geometry also modifies the vortex-shedding and pressure-distribution patterns around the rotor, resulting in more favorable hydrodynamic behavior throughout the rotational cycle.

The results also demonstrated that combining parametric CFD analysis, response surface modeling, and single-objective optimization provides an effective methodology for improving the blade geometry and performance of high-blockage hydrokinetic turbines.