Range-Wide Aerodynamic Optimization of Darrieus Vertical Axis Wind Turbines Using CFD and Surrogate Models

Abstract

The depletion of fossil fuel resources and the growing need for sustainable energy solutions have increased interest in vertical axis wind turbines (VAWTs), which offer advantages in urban and variable-wind environments but often exhibit limited performance at low tip speed ratios (TSRs). This study optimizes VAWT aerodynamic behavior across a wide TSR range by varying three geometric parameters: maximum thickness position ($a/b$), relative thickness (m), and pitch angle ($\beta$). A two-dimensional computational fluid dynamics (CFD) framework, combined with the Metamodel of Optimal Prognosis (MOP), was used to build surrogate models, perform sensitivity analyses, and identify optimal profiles through gradient-based optimization of the integrated $C_p$–$\lambda$ curve. The Joukowsky transformation was employed for efficient geometric parameterization while maintaining aerodynamic adaptability. The optimized airfoils consistently outperformed the baseline NACA 0021, yielding up to a 14.4% improvement at $\lambda =2.64$ and an average increase of 10.7% across all evaluated TSRs. Flow-field analysis confirmed reduced separation, smoother pressure gradients, and enhanced torque generation. Overall, the proposed methodology provides a robust and computationally efficient framework for multi-TSR optimization, integrating Joukowsky-based parameterization with surrogate modeling to improve VAWT performance under diverse operating conditions.

1. Introduction

The growing global energy demand and the depletion of fossil fuel reserves have accelerated the transition toward renewable energy sources. Wind energy has emerged as one of the most promising alternatives due to its sustainability, wide availability, and reduced environmental impact [1]. While Horizontal Axis Wind Turbines (HAWTs) dominate large-scale wind farms because of their efficiency and scalability [2], Vertical Axis Wind Turbines (VAWTs) offer distinct advantages for urban and decentralized applications. Their omnidirectional wind capture, simple mechanical structure, and lower noise generation make them well suited for turbulent and highly variable environments [3,4]. Recent reviews further highlight the renewed relevance of VAWTs in distributed renewable energy systems and microgrid integration [5,6].

Among VAWTs, the H-Darrieus configuration—comprising straight blades mounted on a vertical shaft—combines structural simplicity with mechanical durability. Numerical and experimental studies have progressively established reliable CFD practices for simulating these turbines. Castelli et al. [7] developed a CFD-based prediction model validated against experimental measurements, while Trivellato and Castelli [8] refined time-step selection criteria using the Courant–Friedrichs–Lewy (CFL) condition for rotating grids. Rezaeiha et al. [9] demonstrated the strong influence of pitch angle on power performance and startup behavior, and Hansen et al. [10] expanded modeling to turbine pairs, revealing significant array effects. Miller et al. [11] further showed that solidity strongly governs overall performance trends. Collectively, these studies confirm CFD as a robust design tool, but they also reveal that most optimization efforts have focused on single operating conditions or on limited parameter variations.

Airfoil optimization has also been extensively investigated. Zhang et al. [12] employed the Joukowsky transformation to efficiently control maximum thickness position ($a/b$) and relative thickness (m), reducing dimensionality without compromising aerodynamic flexibility. Studies by Fiedler and Tullis [13] and Zhao et al. [14] confirmed the importance of pitch angle ($\beta$) in mitigating dynamic stall and improving torque generation, particularly at low TSRs. However, most existing works evaluate optimization at a single TSR, leaving unresolved how turbines can be simultaneously optimized across multiple TSRs to enhance robustness under variable wind conditions.

Recent contributions point toward the need for more holistic optimization approaches. Fertahi et al. [15] critically assessed CFD modeling discrepancies for Darrieus turbines and emphasized the importance of reliable validation. Chen et al. [16] introduced advanced machine-learning-based parameterization strategies, while Ahmad et al. [17] proposed a hybrid Double-Darrieus optimization framework. Tirandaz et al. [18] demonstrated that airfoil shape selection and TSR coupling significantly influence turbine performance, and Ullah et al. [19] explored morphing-blade strategies to better adapt turbines to variable operating conditions. Collectively, these studies highlight the lack of integrated frameworks that combine robust parameterization, surrogate modeling, and multi-TSR optimization.

Despite these advances, current VAWT optimization studies still exhibit key limitations. Most optimization efforts are conducted at a single TSR, which restricts robustness under variable wind conditions. Other studies rely on high-dimensional shape parameterizations [11], require extensive experimental datasets, or do not integrate sensitivity analysis with surrogate modeling, making the design process computationally intensive and difficult to generalize. Furthermore, no prior study has combined Joukowsky-based parameterization with a surrogate-assisted framework to quantify parameter influence and optimize performance across multiple TSRs simultaneously. This gap is critical because real VAWTs operate across a broad TSR range and require profiles that remain aerodynamically stable under fluctuating inflow conditions.

To address this gap, the present study introduces an integrated multi-TSR optimization methodology that couples Joukowsky parameterization, variance-based sensitivity analysis, and the Metamodel of Optimal Prognosis (MOP). The proposed framework enables low-dimensional yet flexible geometric control, efficient exploration of the design space, and fast gradient-based optimization. Using two-dimensional CFD as the validation backbone, the method maximizes the integrated $C_p$–$\lambda$ curve over the operational range $\lambda =2.33$–$3.09$. The main scientific contribution of this work lies in demonstrating that a compact three-parameter Joukowsky representation, combined with surrogate modeling, can yield airfoil designs optimized simultaneously across multiple TSRs—providing a computationally efficient and scalable pathway for advancing VAWT aerodynamic design.

2. Methodology

2.1. H-Darrieus Turbine Configuration and Performance Metrics

The H-Darrieus turbine (Figure 1) is a widely adopted configuration of Vertical Axis Wind Turbines (VAWTs), characterized by straight blades rotating around a vertical shaft. Its ability to capture wind from any direction without the need for a yaw mechanism makes it particularly suitable for operating in turbulent and highly variable wind environments. The baseline turbine examined in this study consists of three straight blades employing a NACA 0021 airfoil.

As the turbine rotates, each blade experiences a continuously varying angle of attack (AoA), leading to complex aerodynamic interactions. The aerodynamic forces acting on the blades can be decomposed into lift ($F_L$) and drag ($F_D$), which in turn can be expressed as tangential ($F_T$) and normal ($F_N$) components. The tangential component governs torque production, while the normal component contributes to the overall blade loading.

Two non-dimensional parameters are commonly used to characterize VAWT performance:

• Tip Speed Ratio (TSR, $\lambda$): the ratio of blade tip speed to free-stream velocity.
• Solidity ($\sigma$): a measure of blade area density around the rotor.

Performance evaluation relies on the torque coefficient ($C_t$) and the power coefficient ($C_P$), which quantify aerodynamic efficiency throughout this work. Their analytical definitions and derivations are provided in Appendix A.

The main geometrical and operational characteristics of the turbine are summarized in

Table 1. Main geometrical features of the H-Darrieus turbine.

Characteristics	Turbine
Number of blades, n	3
Diameter, D [m]	1.03
Height, H [m]	1.00
Swept area, A [m2]	1.03
Solidity, $\sigma$	0.50
Airfoil	NACA 0021
Airfoil chord, C [mm]	85.8
Tip speed ratio, $\lambda$	2.64
Free-stream velocity [m/s]	9.0
Rotational speed [rad/s]	46.14
Max. thickness position	30%
Profile thickness	21%

.

This turbine configuration forms the baseline of all CFD simulations and serves as the reference for the subsequent optimization stages.

The tip speed ratio (TSR) is a key parameter governing the aerodynamic regime of Darrieus turbines. It is defined as:

$$\lambda ={\displaystyle \frac{\omega R}{U_{\infty }}},$$

(1)

where $\omega$ is the rotational speed, R is the rotor radius, and $U_{\infty }$ is the free-stream velocity.

In this study, $\omega$ was selected based on the experimental operating conditions reported by Castelli et al. [7] and was verified by matching the reference value $\lambda =2.64$ at $U_{\infty }=9\mathrm{m}/\mathrm{s}$. This ensures consistency between the CFD baseline configuration and the validation dataset.

The tip speed ratio fundamentally shapes the aerodynamic behavior of VAWTs. At low TSRs ($\lambda <2$), blades encounter large angles of attack, triggering extensive flow separation, intermittent reattachment, and pronounced dynamic stall. As TSR increases, the effective AoA decreases, reducing stall severity and stabilizing the torque waveform.

TSR is also closely related to rotor solidity ($\sigma$). Higher-solidity turbines typically operate at lower optimal TSR due to elevated aerodynamic loading, as demonstrated experimentally by Miller et al. [20]. Consequently, performance curves generally exhibit a well-defined optimal TSR range where the power coefficient reaches its maximum.

For the present turbine ($\sigma =0.50$), previous studies [9,11,21] indicate that peak aerodynamic efficiency occurs near $\lambda \approx 2.3$–$3.1$. This aligns with the operational region targeted in the present optimization, ensuring that the optimized airfoil geometries are relevant to the turbine’s practical performance envelope.

2.2. Governing Equations and Turbulence Modeling

The aerodynamic flow around the H-Darrieus turbine was simulated using the unsteady incompressible Reynolds-Averaged Navier–Stokes (URANS) equations. These equations enforce the conservation of mass and momentum and are widely used in wind turbine aerodynamics because they provide a good compromise between accuracy and computational cost [22,23,24].

The governing equations consist of the continuity equation:

$${\displaystyle \frac{\partial \left(\rho u_i\right)}{\partial x_i}}=0,$$

(2)

and the momentum equation:

$${\displaystyle \frac{\partial \left(\rho u_i\right)}{\partial t}}+{\displaystyle \frac{\partial \left(\rho u_i{u}_j\right)}{\partial x_j}}=-{\displaystyle \frac{\partial p}{\partial x_i}}+{\displaystyle \frac{\partial {\tau }_{ij}}{\partial x_j}}+S_M,$$

(3)

where $u_i$ are the velocity components, $\rho$ is the fluid density, p is the static pressure, ${\tau }_{ij}$ is the viscous stress tensor, and $S_M$ represents momentum source terms. The stress tensor is expressed as:

$${\tau }_{ij}=\mu \left[\left({\displaystyle \frac{\partial u_i}{\partial x_j}}+{\displaystyle \frac{\partial u_j}{\partial x_i}}\right)-{\displaystyle \frac{2}{3}}{\delta }_{ij}{\displaystyle \frac{\partial u_k}{\partial x_k}}\right],$$

(4)

where $\mu$ is the dynamic viscosity and ${\delta }_{ij}$ is the Kronecker delta.

To model turbulence, the Shear Stress Transport (SST) k–$\omega$ model was employed. This model blends the near-wall accuracy of the standard k–$\omega$ formulation with the robustness of the k–$\epsilon$ model in the free-stream, making it particularly suitable for flows with separation and adverse pressure gradients, such as those encountered in VAWTs [25,26,27].

The transport equation for turbulent kinetic energy (k) is:

$${\displaystyle \frac{\partial \left(\rho k\right)}{\partial t}}+{\displaystyle \frac{\partial \left(\rho k{u}_j\right)}{\partial x_j}}={\displaystyle \frac{\partial }{\partial x_j}}\left[{\Gamma }_k{\displaystyle \frac{\partial k}{\partial x_j}}\right]+G_k-Y_k+S_k+G_b,$$

(5)

and for the specific dissipation rate ($\omega$):

$${\displaystyle \frac{\partial \left(\rho \omega \right)}{\partial t}}+{\displaystyle \frac{\partial \left(\rho \omega u_j\right)}{\partial x_j}}={\displaystyle \frac{\partial }{\partial x_j}}\left[{\Gamma }_{\omega }{\displaystyle \frac{\partial \omega }{\partial x_j}}\right]+G_{\omega }-Y_{\omega }+S_{\omega }+G_{\omega b}.$$

(6)

Here, $G_k$ and $G_{\omega }$ are production terms, $Y_k$ and $Y_{\omega }$ are dissipation terms, ${\Gamma }_k$ and ${\Gamma }_{\omega }$ denote effective diffusivities, $S_k$ and $S_{\omega }$ are user-defined source terms, and $G_b$ and $G_{\omega b}$ represent buoyancy effects.

The SST k–$\omega$ model was selected due to its proven reliability in predicting separated and unsteady flows, which are prevalent in vertical axis wind turbines [28,29].

All simulations were carried out under the following assumptions:

• Two-dimensional URANS formulation.
• Incompressible, unsteady, turbulent flow.
• No transition model was used; laminar-to-turbulent transition was handled through turbulence closure.
• Three-dimensional effects such as tip vortices and end losses were neglected to reduce computational cost, following the justification of prior 2D VAWT studies [2,30].

A summary of the numerical setup is provided in

Table 2. Summary of simulation settings.

Characteristics	Turbine
Fluid properties	Incompressible
Turbulence model	k–$\omega$ SST
Solver	Pressure-based
Gradient	Least Square Cell-Based
Momentum	Second Order Upwind
Turbulent Kinetic Energy	Second Order Upwind
Specific Dissipation Rate	Second Order Upwind
Pressure–velocity coupling	Coupled
Maximum residuals	$1\times {10}^{-6}$

. Convergence was ensured by maintaining scaled residuals below ${10}^{-6}$ and by confirming stabilization of the power coefficient ($C_P$) over several turbine revolutions.

Rotating–mesh CFD methodologies similar to those employed here have been successfully validated in recent high-fidelity VAWT studies, such as the work of Kariman et al. [31], who demonstrated the suitability of sliding-mesh techniques for resolving unsteady blade–wake interactions.

2.3. Computational Domain and Mesh

Two-dimensional CFD simulations were employed to balance computational efficiency with predictive accuracy. Although 2D modeling neglects certain three-dimensional effects—such as tip vortices, spanwise flow, and axial variations in angle of attack—it has been shown to provide reliable aerodynamic trends for preliminary design and optimization. The absence of tip-vortex modeling may introduce slight overpredictions of performance, particularly for turbines with low height-to-diameter ratios [2,30]. Nevertheless, 2D simulations remain a standard and practical approach for early-stage VAWT analysis [7,10,32], while higher-fidelity 3D modeling or tip-loss corrections can be incorporated at significantly higher computational cost.

The computational domain dimensions were selected following established guidelines for bluff-body aerodynamics and VAWT simulations [22,30,32]. The domain extended $5D$ upstream, $25D$ downstream, and $10D$ above and below the rotor centerline, providing sufficient space for wake development, vortex convection, and flow recovery (Figure 2).

Tip-speed ratios below $\lambda <2.0$ were intentionally excluded from the CFD campaign. This region corresponds to the self-starting phase of a Darrieus turbine, dominated by large-scale separation, intermittent reattachment, and strongly three-dimensional effects. As demonstrated by Bianchini et al. [2] and Rezaeiha et al. [24], 2D URANS simulations tend to mispredict starting torque and stall onset in this regime, making them unsuitable for quantitative assessment. Since the purpose of the present work is to optimize the power-producing region, the analysis focuses on $\lambda =2.33$–$3.09$, where 2D CFD has been widely validated and provides accurate cycle-averaged aerodynamic performance.

2.4. Boundary Conditions and Solver Settings

A uniform inflow velocity of $U_{\infty }=9\mathrm{m}/\mathrm{s}$ was applied at the inlet with a turbulence intensity of $6.3\%$, matching the experimental conditions of Castelli et al. [7]. The outlet boundary was assigned a fixed static pressure of $0\mathrm{Pa}$, while the upper and lower boundaries were treated as symmetry planes. Blade surfaces were modeled using a no-slip condition.

The inner circular domain was rotated using a sliding-mesh approach with rigid-body rotation at angular velocity $\omega =\lambda U_{\infty }/R$, enabling accurate representation of unsteady blade–wake interactions and dynamic stall. Time advancement employed a second-order implicit scheme. The timestep was defined using an azimuthal increment of $\Delta \theta =1^{\circ }$, corresponding to $\Delta t=\Delta \theta /\omega$. This value was verified through the timestep sensitivity study presented in Section 2.6.2.

Turbulence was modeled with the SST k–$\omega$ model, incorporating Menter’s shear-stress–transport blending and curvature correction. The near-wall mesh resolution satisfied $y^{+}\approx 0.1$, enabling full integration to the wall without wall functions. Convergence was enforced by reducing residuals below ${10}^{-6}$ at every timestep.

2.5. Geometry Mesh Details

An unstructured mesh was adopted to accommodate the curved airfoil geometry. The mesh was divided into two zones: a rotating inner domain containing the blades and a stationary outer domain. Boundary-layer resolution was ensured by inserting 30 prism layers with a growth ratio of 1.2 and a first-layer height consistent with $y^{+}\approx 0.1$, suitable for the SST k–$\omega$ model [33].

The final mesh employed a hybrid topology, consisting of unstructured triangular elements in the far field and a body-fitted, structured O-grid–like refinement around each blade within the rotating region. This structure provides high resolution in the stagnation zone, along the leading and trailing edges, and in the near wake, while maintaining mesh flexibility in the outer field. The rotating inner domain radius was set to $1D$, with an additional refinement ring extending to $2D$ to resolve early wake vortices. Curvature-based refinement captured steep pressure gradients and vortex shedding at the leading and trailing edges.

The complete mesh contained approximately $2.24\times {10}^5$ elements, with $7.9\times {10}^4$ cells in the rotating domain and $1.45\times {10}^5$ in the stationary region. This distribution ensured high spatial resolution in regions of strong unsteadiness while keeping overall computational cost manageable.

Mesh quality was assessed using orthogonal quality and skewness metrics. High orthogonal quality (>0.95 on average) and low skewness (<0.02 on average) confirmed mesh suitability for resolving the unsteady aerodynamic field around the blades. Figure 3 shows the mesh detail around the airfoil.

Mesh statistics are summarized in

Table 3. Meshing metrics for orthogonal quality and skewness.

Metric	Min.	Max.	Average	Std. Dev.
Orthogonal Quality	0.11345	0.99457	0.95897	$5.345\times {10}^{-2}$
Skewness	$2.1423\times {10}^{-2}$	0.98231	$1.1134\times {10}^{-2}$	$2.035\times {10}^{-2}$

. To ensure numerical reliability, grid independence was verified using the Grid Convergence Index (GCI), which demonstrated that increasing mesh density produced negligible variations (<0.2%) in the predicted power coefficient. This refinement procedure is consistent with modern CFD best practices for rotating-blade turbines and aligns with the methodology of Kariman et al. [31], who emphasized mesh-quality control and discretization verification for rotating hydrodynamic turbines.

2.6. Numerical Verification: Revolutions and Azimuthal Increment

Before conducting the optimization study, the CFD setup was verified to ensure numerical reliability. This verification involved evaluating (i) the number of revolutions required to reach periodic steady state and (ii) the azimuthal increment required to adequately resolve unsteady blade loading. These checks are standard practice in VAWT CFD simulations and ensure the consistency of the results presented in Section 4 [2,8,34,35].

2.6.1. Revolutions to Periodicity

A reference simulation using the NACA 0021 airfoil at $\lambda =2.64$ was run for 90 revolutions. Figure 4 shows that the cycle-averaged power coefficient ($C_P$) stabilizes after approximately 20 revolutions, with less than 0.6% variation up to revolution 90. This confirms that 20 revolutions are sufficient to achieve periodic steady state, in agreement with thresholds reported in previous VAWT studies [34,35].

2.6.2. Azimuthal Increment Sensitivity

A timestep sensitivity analysis was performed using six azimuthal increments: ${10}^{\circ }$, $5^{\circ }$, $2.5^{\circ }$, $1^{\circ }$, $0.5^{\circ }$, and $0.{25}^{\circ }$. As shown in Figure 5, coarse increments overpredict $C_P$ and fail to capture unsteady loading, whereas the three finest increments produce nearly identical convergence curves. Thus, $\Delta \theta =1^{\circ }$ offers an optimal balance between accuracy and computational cost, consistent with the recommendations of Rezaeiha et al. [9].

These verified numerical settings were adopted for all CFD simulations in Section 4, ensuring that the performance comparisons and optimization results rest on a stable and consistent computational foundation.

2.7. Mesh Independence Test and Numerical Validation

A mesh independence test was conducted using three systematically refined meshes: coarse ($1.19\times {10}^5$ cells), medium ($2.24\times {10}^5$ cells), and fine ($3.95\times {10}^5$ cells). Refinement targeted the near-wall resolution and reduced element size around the airfoil surface, while maintaining consistent domain topology. The cycle-averaged power coefficient $\overline{C_P}$ was selected as the reference quantity for the Grid Convergence Index (GCI) assessment, following Celik et al. [36].

The medium mesh deviated by less than 0.19% relative to the fine mesh, indicating that further refinement would yield negligible improvements. Consequently, the medium-resolution mesh was used throughout the study. The characteristic cell size and the full GCI calculation procedure are presented in Appendix B.

The numerical results are summarized in

Table 4. Details of the mesh sensitivity analysis and discretization errors.

Parameter	Value
$N_1$	395,145
$N_2$	224,220
$N_3$	119,665
$r_{21}$	1.328
$r_{32}$	1.369
$\overline{C_{P1}}$	3.0170
$\overline{C_{P2}}$	3.0227
$\overline{C_{P3}}$	3.1557
${\mathrm{GCI}}^{32}$	0.1903%

. The computed discretization error between the medium and fine meshes (${\mathrm{GCI}}^{32}$) was $0.1903\%$, confirming that the medium mesh provides a sufficiently accurate and computationally efficient solution.

The numerical setup was further validated against the experimental results of Castelli et al. [7], who tested a three-bladed H-Darrieus turbine with a NACA 0021 airfoil over $\lambda =1.44$–$3.29$ at a freestream velocity of 9 m/s. A turbulence intensity of 6.3% was applied at the inlet, matching the experimental conditions. Although Belabes and Paraschivoiu [32] recommended slightly lower values for similar geometries, sensitivity tests showed that 6.3% produced the best agreement with Castelli’s dynamic stall characteristics and wake mixing. This intensity therefore represents an appropriate trade-off between realism and validation accuracy.

Figure 6 compares the simulated and experimental $\overline{C_P}$ values across a range of TSRs. Good agreement is obtained in the mid-$\lambda$ region, with modest overprediction at low and high $\lambda$ due to well-known 2D model limitations, such as the omission of blade tip losses and spanwise effects.

Experimental data were available only for the baseline NACA 0021 airfoil; therefore, the optimized geometries were validated through numerical cross-checks. Specifically, surrogate model predictions were compared with independent CFD simulations not included in the training dataset, yielding discrepancies consistently below 5% across the evaluated TSR range. This cross-validation supports the robustness of the surrogate-assisted optimization methodology, even in the absence of direct experimental measurements for the optimized airfoils.

3. Optimization Framework and Surrogate Modeling

This section presents the optimization methodology used to enhance VAWT aerodynamic performance across multiple operating conditions. The proposed framework consists of four key stages: (i) parameter selection and geometric definition, (ii) design of experiments (DoE), (iii) surrogate modeling using the Metamodel of Optimal Prognosis (MOP), and (iv) gradient-based optimization of the integrated $C_P$–$\lambda$ curve. Figure 7 summarizes the complete workflow, from shape parameterization to performance validation.

3.1. Parameterization and Input Variables

The selection of the three optimization parameters ($a/b$, m, and $\beta$) was made to achieve a balance between aerodynamic expressiveness and computational efficiency. Large-dimensional parameterizations—such as the 12-variable configuration used by Ma et al. [11]—increase design flexibility but reduce the reliability of surrogate models due to sparse sampling. In contrast, the Joukowsky geometric parameters ($a/b$ and m) have been shown by Zhang et al. [12] to capture the dominant aerodynamic effects on camber distribution, thickness, and leading-edge curvature using only two variables.

The pitch angle ($\beta$) was included because of its strong influence on effective angle-of-attack modulation and dynamic stall behavior [9,13]. Together, these parameters constitute a compact but physically meaningful design space capable of representing the primary aerodynamic mechanisms relevant to H-Darrieus turbines. Only a limited number of prior studies have combined low-dimensional parameterization with multi-TSR optimization, underscoring the novelty and necessity of the present approach.

3.1.1. Airfoil Generation via the Joukowsky Transformation

The Joukowsky transformation was adopted to define the turbine blade shape. This classical conformal mapping technique transforms a circle in the complex plane into an airfoil-like profile through a small set of geometric parameters. Compared with traditional NACA four-digit formulations, the Joukowsky method provides continuous and smooth variations in thickness and camber using only two input parameters, making it ideal for surrogate-based optimization workflows.

The standard form of the transformation is:

$$z=\zeta +{\displaystyle \frac{C_J^2}{\zeta }},$$

(7)

where $C_J=2(a-m)$, with a and m defining the circle geometry in the $\zeta$-plane. By translating, rotating, and stretching this circle, symmetric airfoils are obtained in the physical z-plane. Figure 8 illustrates the mapping process.

Airfoil contour points are defined in polar form in the $\zeta$-plane and mapped into $(X,Y)$ coordinates on the physical plane through Equation (7). The complete formulation is provided in Appendix C.

The two geometric input parameters are:

• Relative thickness (m)—controls overall airfoil thickness.
• Maximum thickness position ($a/b$)—controls the chordwise location of maximum thickness.

Parameter bounds were selected based on the ranges found in previous optimization studies. Zhang et al. [12] demonstrated that $a/b$ values between 0.95 and 1.09 reproduce realistic thickness distributions, while $m=0.03$–$0.053$ corresponds to typical symmetric-section thicknesses for small to medium-scale VAWTs. Figure 9 shows how variations in $a/b$ shift the thickness peak toward the leading or trailing edge.

3.1.2. Pitch Angle

The pitch angle ($\beta$) was included as a third input variable owing to its strong aerodynamic influence. Even small preset pitch angles can significantly modify the instantaneous angle of attack, thereby affecting torque symmetry, dynamic stall intensity, and startup characteristics.

Fiedler and Tullis [13] reported peak performance at $\beta =-3.9^{\circ }$ in wind tunnel experiments, whereas Rezaeiha et al. [9] observed up to a 6.6% improvement in $C_P$ at $\beta =-2^{\circ }$ using URANS simulations. Based on these findings, the pitch angle was bounded between $-7^{\circ }$ and $4^{\circ }$ to encompass the most commonly reported effective values.

In this study, $\beta$ is treated as a uniform fixed angle applied symmetrically to all blades, as illustrated in Figure 10.

The complete set of optimization variables is therefore:

• $a/b$—maximum thickness position;
• m—relative thickness;
• $\beta$—fixed pitch angle.

These variables define the design space explored in the subsequent DoE, surrogate modeling, and optimization procedures.

The selection of these three parameters reflects their direct aerodynamic significance and their proven effectiveness in prior optimization studies. The Joukowsky parameters ($a/b$, m) control the features most strongly associated with dynamic stall, pressure recovery, and overall turbine performance, while avoiding the high dimensionality of spline-based or multi-parameter NACA formulations [11,12]. The pitch angle $\beta$ is a critical operational variable that directly influences effective angle of attack and torque production, as demonstrated experimentally and numerically in [9,13]. Together, this reduced yet physically expressive parameter set enables efficient surrogate modeling with minimal loss of aerodynamic flexibility, aligning with the need for compact and robust optimization frameworks highlighted in recent VAWT literature [15,18].

3.2. Design of Experiments and Sensitivity Analysis

A well-constructed Design of Experiments (DoE) is essential for obtaining a reliable global sensitivity analysis of the design space. Although Monte Carlo Simulation (MCS) [38] is a classical stochastic approach, it can suffer from sample clustering and uneven domain coverage when the number of samples is limited, which reduces exploration quality and may bias surrogate training.

To address this limitation, Latin Hypercube Sampling (LHS) was adopted. LHS is a stratified sampling method that ensures improved uniformity of sample distribution without substantially increasing the number of CFD evaluations. Depending on solver runtime and computational constraints, a sample size between 80 and 120 points was used, which proved sufficient for accurate variance-based sensitivity assessment across the three input variables.

The resulting dataset was used to compute the Coefficient of Importance (CoI) for each input parameter, quantifying their contribution to variability in the objective function (see Appendix D.2 for full formulation). Only the variables with the highest CoI were retained for the optimization step to minimize dimensionality, reduce overfitting risk, and ensure the robustness of the surrogate modeling framework.

3.3. Surrogate Modeling via Metamodel of Optimal Prognosis (MOP)

Following the sensitivity analysis, surrogate models of the CFD response surface were constructed using the Metamodel of Optimal Prognosis (MOP) [39,40]. The MOP framework significantly reduces computational cost while retaining high predictive fidelity. It does so by automatically testing several surrogate formulations and selecting the one that maximizes statistical performance based on metrics such as the Coefficient of Determination (CoD) and the Coefficient of Prognosis (CoP) (see Appendix D.1).

Surrogate modeling is crucial because exhaustive CFD-based evaluations of the entire design space are prohibitive. Each URANS simulation involves hundreds of iterations per revolution and requires multiple revolutions to reach periodic steady state, making direct optimization computationally infeasible. The MOP framework provides an efficient alternative by ensuring rapid function evaluations during the optimization stage while maintaining high accuracy.

The three surrogate formulations considered within the MOP framework were:

• Polynomial regression: A global approximation of the form:

  $$y\left(x_i\right)=\widehat{y_i}\left(x_i\right)+{ϵ}_i=p^T\left(x_i\right)\theta +{ϵ}_i,$$

  (8)

  where $p\left(x_i\right)$ is the polynomial basis, $\theta$ is the regression coefficient vector, and ${ϵ}_i$ is the residual. Polynomial regression is simple, interpretable, and widely used in engineering design [41,42].
• Moving Least Squares (MLS): A local approximation technique expressed as:

  $$y\left(x_i\right)=p^T\left(x_i\right)\alpha \left(x_i\right),$$

  (9)

  where the coefficient vector $\alpha \left(x_i\right)$ varies with the evaluation point. This method offers improved capability for capturing local nonlinearities compared with global polynomial models [30].
• Kriging approximation: A hybrid interpolation model expressed as:

  $$y\left(x_i\right)=f\left(x_i\right)\theta +Z\left(x_i\right),$$

  (10)

  where $f\left(x_i\right)$ forms the global regression component, $\theta$ are its coefficients, and $Z\left(x_i\right)$ is a zero-mean Gaussian process with spatial correlation. Kriging exactly interpolates the training data and provides uncertainty estimates, making it particularly suitable for aerodynamic response surfaces [43,44].

Individual surrogate models were constructed for each tip speed ratio ($\lambda$) in the set $\{2.33,2.64,3.09\}$. Model performance was evaluated through both CoD and CoP. The Coefficient of Prognosis (CoP), proposed by Most and Will [39], is defined as:

$$CoP=1-{\displaystyle \frac{S{S}_e^{\mathrm{prediction}}}{S{S}_T}},$$

(11)

where $S{S}_e^{\mathrm{prediction}}$ is the prediction error sum of squares and $S{S}_T$ is the total response variance. A threshold of $CoP>0.90$ was used to ensure acceptable surrogate accuracy, following established best practices in surrogate-based aerodynamic optimization [45,46].

3.4. Optimization Statement and Method

The optimization problem in this study involves the three CFD input variables: the pitch angle ($\beta$), the Joukowsky shifted distance (m), and the major-to-minor axis ratio ($a/b$). The $a/b$ ratio controls the maximum thickness location along the chord, m regulates overall thickness, and $\beta$ adjusts the blade’s angular offset relative to the flow.

The lower and upper bounds of these variables are given in

Table 5. Lower and upper bounds for the input variables.

Input Variable	Lower Bound	Upper Bound
$a/b$	0.95	1.09
m	0.03	0.053
$\beta$	$-7$	4

.

The optimization objective is to maximize the integrated power coefficient across the operational tip speed ratios $\lambda =\{2.33,2.64,3.09\}$:

$$\mathrm{Find}X={\left[\begin{array}{c}{\displaystyle \frac{a}{b}},m,\beta \end{array}\right]}^T\mathrm{that}\mathrm{maximizes}{\mathsf{\int }}_{{\lambda }_0}^{{\lambda }_f}{C}_{P}d\lambda$$

(12)

Given the low dimensionality of the design space and the smoothness of the surrogate-generated response surfaces, gradient-based optimization methods were selected. These techniques offer rapid convergence and are well-suited for surrogate-assisted aerodynamic optimization workflows.

3.5. Optimization Process and Model Validation

CFD simulations were performed to validate the optimal control points predicted by the gradient-based solvers and to assess the consistency of the surrogate-based optimization framework. The optimization space consists of three input variables: the pitch angle ($\beta$) and the Joukowsky geometry parameters—the shifted distance (m) and the major-to-minor axis ratio ($a/b$). The objective is to maximize the integral of the average power coefficient ($\overline{C_P}$) across the selected tip speed ratios.

Gradient-based optimization was carried out using Newton’s method implemented via the Nonlinear Programming by Quadratic Lagrangian (NLPQL) strategy. NLPQL is well suited for low-dimensional nonlinear problems and efficiently handles both equality and inequality constraints [47,48]. This second-order approach enables fast convergence by exploiting local curvature information of the response surface.

For comparison, the Downhill Simplex algorithm [49,50] was also applied. Unlike gradient-based methods, the Simplex method is derivative-free and operates by evolving a set of $n+1$ vertices in an n-dimensional design space. At each iteration, the worst-performing vertex is replaced through reflection, expansion, contraction, or shrink steps until convergence criteria are satisfied. Using both methods provided an additional check on robustness and helped identify potential local optima.

To verify the reliability of the optimization results, each predicted optimum was subsequently evaluated using an independent CFD simulation. A consistency metric—the Margin of Error (MoE)—was used to quantify the deviation between the surrogate prediction $Z_i$ and the CFD-obtained value Z:

$$\mathrm{MoE}=\left|{\displaystyle \frac{Z-Z_i}{Z}}\right|.$$

(13)

Only solutions with $\mathrm{MoE}<5\%$ were accepted as valid optima. Points exceeding this threshold were discarded to ensure that the final selected profiles exhibit strong agreement between surrogate predictions and high-fidelity CFD evaluations. This filtering criterion ensured consistency, eliminated outliers, and increased confidence in the final optimization results.

3.6. Computational Cost

The Design of Experiments and surrogate modeling required a total of 300 CFD simulations (100 samples per tip speed ratio). Each 2D URANS simulation (20 revolutions, $\Delta \theta =1^{\circ }$, k–$\omega$ SST) required approximately 2.5 h of wall-clock time on a 32-core compute node, resulting in a total computational expenditure of nearly 750 h, or about 24,000 core-hours.

A brute-force optimization over the same three-dimensional design space and TSR range using 3D CFD would require several thousand simulations and exceed 150,000 core-hours. In contrast, the MOP-based surrogate framework enables evaluation and optimization using only the DoE sample set, reducing computational effort by more than an order of magnitude while maintaining excellent predictive fidelity (CoP > 0.99).

This substantial reduction in computational cost demonstrates the practicality and scalability of the surrogate-assisted optimization framework for VAWT aerodynamic design.

4. Results

4.1. Surface Construction and Meta-Model Performance

Following Latin Hypercube Sampling (LHS) of the design space, CFD simulations were performed to obtain the aerodynamic responses corresponding to each sampled point. These simulations yielded the velocity fields, pressure distributions, and cycle-averaged power coefficients across all three tip-speed ratios ($\lambda$). The CFD computations were carried out in ANSYS Fluent (2024 R1), while sensitivity analysis and surrogate-model construction were performed in OptiSlang within the ANSYS Workbench environment.

Table 6. Summary of meta-models created for $\lambda =2.33$.

Meta-Model Approximation	CoD	CoP
LRM (1st order)	0.7556	0.7705
LRM (2nd order)	0.9999	0.99996
MLS (1st order)	0.9743	0.9448
MLS (2nd order)	0.9852	0.9950
Kriging	0.9790	0.9953

,

Table 7. Summary of meta-models created for $\lambda =2.64$.

Meta-Model Approximation	CoD	CoP
LRM (1st order)	0.7820	0.7923
LRM (2nd order)	0.9999	0.9991
MLS (1st order)	0.9388	0.9267
MLS (2nd order)	0.9517	0.9740
Kriging	0.9504	0.9674

and

Table 8. Summary of meta-models created for $\lambda =3.09$.

Meta-Model Approximation	CoD	CoP
LRM (1st order)	0.7669	0.7824
LRM (2nd order)	0.9993	0.9996
MLS (1st order)	0.9799	0.9516
MLS (2nd order)	0.9848	0.9864
Kriging	0.9900	0.9970

summarize the performance of the surrogate models for $\lambda =2.33$, $2.64$, and $3.09$. Five approximation techniques were tested: first- and second-order Linear Regression Models (LRM), first- and second-order Moving Least Squares (MLS), and Kriging. Across all three operating conditions, the second-order LRM consistently achieved the highest Coefficient of Determination (CoD) and Coefficient of Prognosis (CoP), with values exceeding 0.9 in every case.

These results demonstrate that the underlying CFD response surfaces are well represented by quadratic terms, consistent with their smooth variation with respect to $a/b$, m, and $\beta$. The superior performance of quadratic regression is in line with previous aerodynamic surrogate-modeling studies, where polynomial approximations have frequently outperformed more complex methods when the response surfaces exhibit regular curvature and monotonicity [29,35,39,44]. Therefore, the second-order LRM was selected as the primary surrogate model for all optimization stages due to its outstanding predictive performance and computational efficiency.

Table 9. Consolidated performance of second-order LRM surrogate models across multiple $\lambda$.

$\mathit{\lambda }$	Design Pts	Failed Pts	CoD	CoP	Highest $\overline{{\mathit{C}}_{\mathit{P}}}$	$\Delta \overline{{\mathit{C}}_{\mathit{P}}}$ [%]
2.33	97	3	0.99994	0.99996	0.3055	11.10
2.64	93	7	0.99992	0.99908	0.3512	14.36
3.09	95	5	0.99929	0.99964	0.3385	12.96

consolidates the performance of the selected second-order LRM models across all TSRs. For each $\lambda$, the table lists the number of design points, number of failed simulations, and the achieved CoD, CoP, and highest cycle-averaged power coefficient ($\overline{C_P}$). The $\Delta \overline{C_P}$ column reports the relative improvement over the baseline NACA 0021 profile.

Across the three operating conditions, the optimized designs achieved 11.1–14.4% improvement relative to the baseline. These results compare favorably with previously published optimization studies. Ma et al. [11] reported approximately 20% improvement using twelve geometric variables at significantly higher computational cost. Rezaeiha et al. [9] achieved 6–8% gains through pitch optimization only, while Trentin et al. [21] reported increases of roughly 9–12% for small-scale turbines. In contrast, the present study achieved comparable gains using only three parameters ($a/b$, m, $\beta$) within a multi-TSR optimization framework. This demonstrates both the efficiency and the aerodynamic relevance of Joukowsky-based parameterization combined with MOP surrogate modeling.

4.2. Coefficient of Importance and Sensitivity Results

A sensitivity analysis was carried out using the Coefficient of Importance (CoI) method in OptiSlang to quantify the relative influence of the three design variables—pitch angle ($\beta$), relative thickness (m), and maximum-thickness position ($a/b$)—on the cycle-averaged power coefficient ($\overline{C_P}$).

To facilitate the interpretation of the sensitivity trends identified by the CoI analysis, surrogate response surfaces were constructed using the second-order Linear Regression Model (LRM). Figure 11 and Figure 12 illustrate the resulting two- and three-dimensional surrogate response surfaces, which provide a geometric representation of the relationship between the design variables and the cycle-averaged power coefficient across the investigated tip speed ratios.

Figure 11 presents two-dimensional top-down projections of the surrogate response surfaces, highlighting the combined influence of pitch angle with relative thickness and maximum-thickness position. These views facilitate the identification of optimal design regions and clarify how secondary geometric parameters refine performance once the pitch angle is fixed. Complementarily, the three-dimensional response surfaces shown in Figure 12 provide additional insight into the curvature, smoothness, and global trends captured by the surrogate model.

Figure 13, Figure 14 and Figure 15 then present the CoI matrices for $\lambda =2.33$, $2.64$, and $3.09$, respectively, with the cumulative contributions of the design variables indicated by the red curve in each plot.

Across all operating conditions, $\beta$ emerges as the dominant parameter, contributing 76% for $\lambda =2.33$, 80.4% for $\lambda =2.64$, and 85% for $\lambda =3.09$. This progressive increase highlights that the pitch angle plays the primary role in defining the aerodynamic response and that its influence strengthens with increasing tip-speed ratio. In comparison, the geometric parameters m and $a/b$ exhibit secondary contributions and together account for less than 25% of the impact across the entire TSR range.

The physical origins of this trend lie in the direct relationship between $\beta$ and the instantaneous angle of attack experienced by the rotating blades. Given that VAWTs operate under highly unsteady inflow and rapid angular variations, small changes in $\beta$ significantly alter the timing and amplitude of angle-of-attack cycles. This, in turn, affects flow attachment on the upwind half-cycle, the severity of the adverse pressure gradient during the downwind motion, and the onset and extent of dynamic stall. Such mechanisms are consistent with the findings of Rezaeiha et al. [9], who demonstrated that even modest pitch adjustments can noticeably reduce separation and cyclic aerodynamic losses.

Additionally, the influence of $\beta$ interacts with the optimized geometry. As shown in Section 4.3, the optimized airfoil possesses a forward-shifted maximum thickness and a smoother camber distribution, which improve leading-edge suction and mitigate abrupt pressure gradients. These mechanisms, also reported by Ma et al. [11] and Buchner et al. [51], naturally enhance the effectiveness of $\beta$ as it controls the magnitude and phase of unsteady suction peaks.

The increasing dominance of pitch angle with higher TSRs agrees with previous optimization and parametric studies, including the surrogate-based analysis of Trentin et al. [21]. Thus, the present CoI results align well with the broader literature and reinforce the conclusion that $\beta$ is the primary driver of aerodynamic performance across the full TSR range considered.

4.3. Optimization and Validation of Best Designs

Using the high-accuracy surrogate meta-models described earlier, the optimization was performed using two approaches: Nonlinear Programming by Quadratic Lagrangian (NLPQL) and the Downhill Simplex algorithm. Both optimization routines were executed within OptiSlang with the objective of maximizing the cycle-averaged power coefficient ($\overline{C_P}$) integrated over the range $\lambda =2.33$–$3.09$, according to the formulation provided in Section 3.4.

The design variables—pitch angle ($\beta$), relative thickness (m), and maximum-thickness position ($a/b$)—were selected based on their ranked influence from the sensitivity analysis (Section 4.2). Each optimal design obtained from the surrogate models was then validated through independent CFD simulations. Model accuracy was assessed by computing the Margin of Error (MoE), with only solutions satisfying $\mathrm{MoE}<5\%$ considered acceptable. Both optimization methods satisfied this requirement, although NLPQL produced slightly closer agreement with the CFD results.

Table 10. Optimized design parameters obtained from NLPQL and Downhill Simplex algorithms. Predicted and simulated values of the integrated $\overline{C_P}$ differ by less than 0.4%, confirming the accuracy and reliability of the surrogate optimization framework.

Opt. Model	Optimal $\mathit{a}/\mathit{b}$	Optimal m	Optimal $\mathit{\beta }$ [°]	Predicted ${\displaystyle {\mathsf{\int }}_{{\mathit{\lambda }}_0}^{{\mathit{\lambda }}_{\mathit{f}}}\overline{{\mathit{C}}_{\mathit{P}}}}$	Simulated ${\displaystyle {\mathsf{\int }}_{{\mathit{\lambda }}_0}^{{\mathit{\lambda }}_{\mathit{f}}}\overline{{\mathit{C}}_{\mathit{P}}}}$	MoE [%]
NLPQL	1.054	0.0407	$-3.491$	0.2545	0.2540	0.2150
SIMPLEX	1.053	0.0408	$-3.491$	0.2532	0.2523	0.3560

summarizes the optimal configurations and the corresponding predicted and simulated values of the integrated $\overline{C_P}$. Both NLPQL and Simplex converged to nearly identical designs, differing by less than 0.4% between predicted and simulated performance. This validates the robustness and consistency of the surrogate-based optimization framework.

The dominant effect of pitch angle revealed in this study is consistent with earlier experimental and numerical investigations. Fiedler and Tullis [13] reported that small preset pitch angles substantially influence stall onset and reversed-flow behavior in high-solidity H-rotors, while Zhao et al. [14] noted that variable-pitch systems, although potentially beneficial, often yield improvements restricted to narrow TSR ranges and introduce mechanical complexity and fatigue concerns. The present results provide a unifying explanation: pitch angle is indeed aerodynamically decisive, but its benefits are maximized when paired with an airfoil geometry that enhances pressure recovery and delays dynamic stall. By jointly optimizing fixed pitch and airfoil thickness distribution, the present approach captures the aerodynamic benefits typically associated with variable-pitch systems without their practical drawbacks.

Figure 16 compares the optimized airfoil shape with the baseline NACA 0021 profile. The optimized profile shows increased camber and a forward-shifted maximum-thickness location, features that collectively improve flow attachment, delay stall onset, and enhance the overall aerodynamic efficiency of the rotor.

4.4. Performance Comparison of Optimized Profiles

This section compares the aerodynamic performance of the optimized and baseline (NACA 0021) airfoil configurations across the three target tip-speed ratios ($\lambda =2.33$, $2.64$, and $3.09$). The analysis focuses on the instantaneous torque coefficient ($C_T$), as well as the surface-pressure distributions at azimuthal positions corresponding to minimum and maximum torque. Together, these quantities reveal the dynamic-stall behavior and flow-attachment characteristics responsible for the improvements obtained through optimization.

4.4.1. Performance at $\lambda =2.33$

Figure 17 shows the instantaneous torque coefficient $C_T$ for both airfoils. The optimized profile reduces the extent of negative-torque regions and increases the minimum $C_T$ value, thus providing more consistent energy extraction throughout the rotation. Such improvements are particularly valuable at lower TSRs, where Darrieus turbines are most susceptible to negative torque due to large angle-of-attack excursions.

Figure 18 and Figure 19 present pressure contours at the azimuthal positions associated with minimum and maximum torque. In the baseline configuration (panel a), pronounced low-pressure pockets form near the trailing edges of blades 2 and 3 (around $\theta \approx {120}^{\circ }$ and ${240}^{\circ }$), indicating flow separation and vortex shedding—phenomena known to reduce lift and increase cyclic losses [3,21]. The optimized airfoil (panel b) suppresses these structures, exhibiting smoother pressure recovery and a weaker wake. At peak torque, both profiles generate strong pressure differentials, but the optimized geometry maintains a more uniform suction-side distribution, leading to enhanced lift and reduced cyclic fluctuations.

4.4.2. Performance at $\lambda =2.64$

At $\lambda =2.64$, the benefits of the optimized profile become more pronounced. As shown in Figure 20, the baseline turbine exhibits negative torque dips, while the optimized design eliminates these instabilities and maintains a smoother, more symmetric torque curve.

Pressure contours in Figure 21 and Figure 22 further illustrate these trends. The NACA 0021 profile continues to exhibit trailing-edge separation zones consistent with early-stall behavior documented in experimental VAWT studies [3]. By contrast, the optimized airfoil maintains attached flow over a larger portion of the suction side, sustaining higher suction peaks and generating stronger torque. This behavior leads to improved aerodynamic loading with reduced cyclic variability.

4.4.3. Performance at $\lambda =3.09$

At the highest TSR considered, neither airfoil produces negative torque (Figure 23). However, the optimized profile consistently achieves slightly higher $C_T$ values throughout the entire revolution, confirming its robustness across different operating points.

Pressure contours in Figure 24 and Figure 25 indicate that the baseline profile still forms trailing-edge low-pressure zones, although weaker than at lower TSRs. The optimized airfoil suppresses most of these features, maintaining smoother pressure gradients and enhanced suction behavior, which contributes to its superior torque production.

4.4.4. Physical Mechanism for Stall Delay in the Optimized Profile

The enhanced performance of the optimized airfoil is closely linked to the forward shift of maximum thickness and the corresponding modification of surface-pressure gradients. By moving the thickness peak toward the leading edge, the suction peak is strengthened and the adverse pressure gradient is distributed more gradually along the mid-chord region. This reduces boundary-layer separation and delays the onset of dynamic stall—an essential improvement for VAWTs operating under large angle-of-attack oscillations.

Qin et al. [3] showed that dynamic-stall vortex intensity and timing are strongly governed by the leading-edge suction behavior and the distribution of the adverse pressure gradient. Similarly, Buchner et al. [51] reported that smoother pressure recovery reduces vortex shedding and stabilizes aerodynamic loading. For Darrieus turbines, Ma et al. [11] demonstrated that forward-thickness airfoils suppress mid-chord separation bubbles, thereby increasing $C_P$. These same mechanisms are evident in Figure 18, Figure 19, Figure 20, Figure 21, Figure 22, Figure 23, Figure 24 and Figure 25, where the optimized profile exhibits weaker low-pressure cores and smoother suction-side contours.

4.4.5. Cycle-Averaged Performance Across TSR

Figure 26 compares the cycle-averaged power coefficient $\overline{C_P}$ across the studied TSR range. The optimized airfoil consistently outperforms the baseline NACA 0021, achieving gains of approximately 11–14% across all operating points. The consistency of these improvements indicates that the optimized design is robust and effective beyond the specific TSR range used for optimization.

It is important to note that some discrepancy between CFD predictions and experimental data is expected due to the inherent limitations of two-dimensional VAWT simulations. The 2D model treats the blades as infinitely extruded, neglecting tip vortices and spanwise flow, which are known to reduce $C_P$ in real turbines. Similar overpredictions in 2D URANS simulations have been widely reported [2,34]. Therefore, the trends observed in Figure 6 reflect established modeling limitations rather than deficiencies in the optimization methodology.

The magnitude and nature of the improvements align with previous optimization studies. Rezaeiha et al. [9] reported a 6.6% improvement from pitch-angle optimization alone, while Ma et al. [11] obtained up to 20% gains using a high-dimensional geometric parameterization at a single TSR. In contrast, the present work achieves sustained 11–14% improvement across multiple TSRs using only three variables ($a/b$, m, $\beta$), demonstrating a favorable trade-off between dimensionality, robustness, and computational cost.

Overall, the optimized airfoil demonstrates clear and consistent gains over the baseline NACA 0021. The improvements arise from the synergistic effect of a tuned pitch angle ($\beta \approx -3.5^{\circ }$), which regulates instantaneous angle of attack and mitigates dynamic stall, and geometric adjustments introduced through the Joukowsky parameters ($a/b$ and m), which shift the maximum thickness forward and thicken the mid-chord region. Together, these modifications enhance leading-edge suction, smooth the adverse pressure gradient, suppress trailing-edge separation, and reduce negative-torque intervals, resulting in robust performance gains across all examined operating conditions.

4.5. Influence of Turbulence Intensity on the Optimized Airfoil

Additional two-dimensional CFD simulations were performed to evaluate how variations in inflow turbulence intensity (TI) affect the aerodynamic robustness of the optimized airfoil. Three representative TI values—3%, 8%, and 12%—were selected based on inflow conditions commonly reported for small-scale Darrieus VAWTs [2,32,34], complementing the baseline case at 6.3%.

At low TI (3%), the optimized profile exhibits slightly higher peak $C_p$ due to reduced inflow fluctuations, which promote more coherent blade loading. Conversely, higher TI levels (8–12%) introduce stronger turbulent mixing and small-scale velocity perturbations, causing a mild reduction in $C_p$ at elevated tip-speed ratios. These trends are consistent with prior observations that higher turbulence accelerates transition, enhances mixing, and modulates dynamic-stall behavior [3,9,34]. Importantly, across all TI levels, the optimized airfoil consistently outperforms the baseline NACA 0021, demonstrating that the performance gains achieved through the geometric and pitch-angle optimization are robust under realistic inflow variability.

Figure 27 presents the complete set of $C_p\left(\lambda \right)$ curves for all turbulence intensities.

4.6. Scientific Framing

The present findings complement existing aerodynamic optimization studies for Darrieus VAWTs and clarify how the proposed methodology fits within the broader research landscape. Most prior optimization efforts focus on a single tip-speed ratio [11,21] or employ high-dimensional geometric parameterizations that substantially increase computational cost. The combination of Joukowsky parameterization and surrogate modeling used in this work directly addresses the need for computationally efficient multi-TSR optimization, a gap highlighted in recent reviews of VAWT design strategies [5,6]. By reducing the design space to three physically meaningful variables, the framework enables efficient exploration while preserving sufficient geometric flexibility for aerodynamic improvements.

The incorporation of the Metamodel of Optimal Prognosis (MOP) further enhances the methodology by providing a highly accurate and interpretable surrogate with CoP values exceeding 0.99 across all TSRs. Although recent machine-learning–based shape-parameterization approaches can represent highly complex geometries [16,18], they typically require large training sets and may generate non-physical shapes if unconstrained. In contrast, the present method offers a practical and physically grounded alternative for early-stage design, balancing robustness, accuracy, and computational efficiency.

From a physical standpoint, the results advance current understanding of VAWT aerodynamics by demonstrating that pitch angle remains the dominant performance driver not only at isolated TSRs—an observation previously documented in [9,13]—but consistently across a broad operational range. The optimized airfoil, characterized by a forward-shifted maximum thickness and smoother pressure recovery, enhances stall-delay mechanisms and reduces negative-torque intervals. These aerodynamic benefits persist across all TSRs examined, highlighting the robustness of the optimized design under variable operating conditions, a key requirement for decentralized or urban wind-energy systems.

Overall, the combination of Joukowsky parameterization, surrogate modeling, and multi-TSR optimization yields a scalable methodology for VAWT design and offers new insight into how geometric and operational parameters jointly influence turbine performance.

4.7. Practical Significance of the Optimized Airfoil

The 11–14% increase in cycle-averaged power coefficient achieved by the optimized airfoil has direct practical implications for small- and medium-scale VAWTs. First, the reduction of negative-torque regions and the smoother instantaneous torque curve indicate a significant decrease in torque ripple. Lower cyclic loading reduces fatigue accumulation on the shaft, struts, and blade attachments, extending the turbine’s structural lifetime and enabling lighter support components.

Second, the improved $C_P$ implies that, for a given power output, the turbine can operate with a smaller rotor diameter or reduced blade chord, lowering material costs and simplifying installation—an important advantage for rooftop and urban deployments where spatial constraints are significant.

Third, the robustness of the improvement across multiple TSRs ensures higher efficiency under variable wind conditions, reducing sensitivity to gusts, shear, and turbulence. This stability translates into improved annual energy yield and more predictable performance in realistic operating environments.

Altogether, these benefits underscore that the proposed optimization approach not only advances aerodynamic understanding but also provides tangible advantages for the practical design and deployment of next-generation Darrieus VAWTs.

5. Conclusions

This study presented a computationally efficient multi-TSR optimization framework for Darrieus VAWT airfoils that integrates Joukowsky parameterization, variance-based sensitivity analysis, surrogate modeling via the Metamodel of Optimal Prognosis (MOP), and gradient-based optimization. The methodology successfully reduced the design space to three physically meaningful parameters—pitch angle ($\beta$), relative thickness (m), and maximum-thickness position ($a/b$)—while preserving the geometric flexibility necessary for aerodynamic performance enhancement.

The optimized profiles delivered a robust 11–14% improvement in the cycle-averaged power coefficient ($\overline{C_P}$) across the TSR range $\lambda =2.33$–3.09. These gains were verified through independent CFD simulations, showing excellent agreement with surrogate predictions (MoE < 5%). Detailed flow-physics analysis revealed that the optimized airfoil suppresses separation vortices, delays dynamic stall, and reduces negative-torque intervals—mechanisms directly linked to improved pressure recovery, enhanced torque generation, and smoother aerodynamic loading.

Overall, the results demonstrate that combining Joukowsky-based geometric control with multi-condition surrogate optimization is a powerful and computationally affordable strategy for improving VAWT aerodynamic performance.

5.1. Main Findings

• Multi-TSR optimization effectiveness: The proposed framework increased $\overline{C_P}$ by 11–14% relative to the NACA 0021 baseline, outperforming typical single-TSR optimization approaches found in the literature.
• Surrogate performance: Second-order polynomial regression (LRM²) consistently achieved CoD and CoP values greater than 0.99, outperforming MLS and Kriging due to the smooth quadratic nature of the aerodynamic response surfaces.
• Dominant design parameter: Pitch angle ($\beta$) contributed more than 76–85% of the variance in $\overline{C_P}$ across all TSRs, confirming its primary role in regulating effective angle of attack, delaying stall, and stabilizing instantaneous torque.
• Optimized airfoil physics: The optimized geometry exhibited a forward-shifted maximum thickness and slightly increased camber, which smoothed adverse pressure gradients, reduced separation bubbles, and weakened trailing-edge vortices—consistent with mechanisms reported by Buchner et al. [51].
• Validation robustness: Independent CFD evaluations confirmed the accuracy of the surrogate-based optima, with a maximum deviation of less than 0.4% in integrated $\overline{C_P}$.

5.2. Practical Implications

• Improved start-up and low-TSR behavior: Reduced negative-torque zones enhance startup torque and expand the range of usable operating conditions.
• Lower fatigue and smoother operation: The reduction in torque ripple decreases cyclic loading on the shaft, struts, and blade attachments, enabling lighter support structures and longer fatigue life.
• Design efficiency: The low-dimensional parameterization and high-fidelity surrogate models drastically reduce CFD evaluations, making the method ideal for early-stage aerodynamic design and rapid prototyping.
• Urban and distributed wind energy: Consistent performance across multiple TSRs enhances energy yield in turbulent and variable wind environments typical of decentralized installations.

5.3. Limitations

• 2D simulation constraints: The URANS framework neglects tip losses, spanwise flow, and three-dimensional vortex dynamics, which may cause slight overprediction of $C_P$.
• Structural considerations not included: The study did not evaluate stress distributions, fatigue loads, or aeroelastic effects.
• Surrogate model dependency: Surrogates were trained exclusively on 2D CFD data; no experimental validation of the optimized airfoil has yet been undertaken.
• Fixed-pitch assumption: Only fixed preset pitch angles were considered; variable-pitch strategies could be explored in future work.

5.4. Future Work

• Extend the optimization framework to three-dimensional CFD including dynamic-stall corrections and tip-loss modeling.
• Develop a coupled aero–structural optimization to assess fatigue, deflection, and material constraints.
• Perform wind-tunnel or field experiments to verify aerodynamic performance and validate 2D surrogate predictions.
• Investigate the use of hybrid parameterizations (e.g., Joukowsky + Bezier control points) to increase geometric flexibility while maintaining physical interpretability.
• Explore integration into urban wind-energy systems, assessing performance under turbulence, gusts, and yawed inflows.