Integrated Approach to Aerodynamic Optimization of Darrieus Wind Turbine Based on the Taguchi Method and Computational Fluid Dynamics (CFD)

Abstract

This paper presents a numerical study of the optimization of the geometric parameters of a four-bladed Darrieus vertical-axis wind turbine (VAWT) with a NACA 0021 aerodynamic profile. The aim of the study was to increase the aerodynamic efficiency of the turbine by selecting optimal values of the rotor diameter and blade chord length. The Taguchi method using an orthogonal array was used as an optimization method, which reduced the number of necessary calculations from 77 to 20 while maintaining the reliability of the analysis. CFD modelling was performed in the ANSYS 2022 R2 Fluent software environment based on a two-dimensional non-stationary model, including a full rotor revolution and an analysis of the steady-state mode for the twentieth cycle. As a result of the analysis, the optimal parameters were determined: rotor diameter D = 3 m and chord length c = 0.4 m. Additionally, for the selected configuration, the numerical model was validated by constructing the dependence of the power coefficient Cp on the tip speed ratio λ in the range from 0.2 to 2.8. The maximum value of Cp was 0.35 at λ = 2.2, which is an increase of ~64% compared to the least efficient rotation mode in the considered range of λ. The obtained results allow us to conclude that the Taguchi method can be used in combination with CFD modelling for fast and accurate optimization of the aerodynamic parameters of low-power wind turbines.

1. Introduction

Concerns about global warming, depletion of fossil fuel reserves, and stricter environmental regulations in the global energy market have increased the demand for renewable energy sources [1].

In the context of the global energy transition, the need for efficient and reliable solutions for renewable energy generation is increasing [2]. Wind energy plays a key role in this process, but existing wind turbine designs often have limitations in efficiency and application conditions [3].

The existing Darrieus wind turbine designs, despite a number of advantages such as independence from wind direction and compactness, have certain limitations in terms of wind flow energy conversion efficiency [4]. The main problems are related to uneven torque, relatively low wind energy utilization factor, and difficulty of self-starting at low wind speeds [5].

Modern wind power is based on two main types of turbines, differing in the orientation of the rotation axis [4,6,7]. Horizontal-axis turbines (HAWT) are structures where the rotor blades are connected to a horizontal shaft that transmits rotation to the electric generator. The peculiarity of this type is the need for constant orientation to the wind direction, which is ensured by special sensors and a positioning system [8].

An alternative is vertical axis-wind turbines (VAWT), the design of which involves connecting the blades to a vertical shaft [9]. Despite being less common, these units are becoming increasingly popular due to technological improvements and a number of design advantages. However, their efficiency is somewhat limited by operating conditions—operation in the surface layer with turbulent air flows affects the output power [4,10].

Vertical-axis wind turbines (VAWTs) offer a number of significant advantages over horizontal-axis turbines (HAWTs), making them attractive for certain applications [11].

One of the key advantages of VAWTs is their structural simplicity and cost-effectiveness. The absence of the need for a wind orientation system significantly reduces both production and operating costs [12]. The unified design of the blade aerodynamic profiles also helps optimize production costs. The placement of the main mechanical components—the generator, transmission, and braking system—close to the ground surface significantly simplifies the maintenance of the installation [13].

From an environmental and safety perspective, vertical-axis turbines demonstrate advantages in the form of reduced noise levels and reduced risk to birds [14]. An important technical characteristic is their ability to operate effectively in turbulent and gusty wind conditions, which expands the possibilities of their application in various climatic conditions. Theoretically, studies show that VAWTs are capable of achieving a power coefficient comparable to HAWTs, which opens up prospects for their wider application in the energy sector [15,16].

The relevance of this study in the field of modeling and optimization of the Darrieus VAWT is due to several key factors of modern energy and technological development. The following paragraphs present an analysis of key research achievements in the field of studying and improving Darrieus turbines. The most significant works that have made a significant contribution to understanding the principles of operation and optimization of this type of wind turbines are considered.

Akhlaghi et al. conducted a numerical study on the effect of geometric parameters on the performance of the Darrieus VAWT using CFD and DMST methods. They found that increasing the number of blades improved the initial torque, but at high TSR, the efficiency decreased due to the flow interaction between the blades. Increasing the chord length to 0.09 m increased the power coefficient by 23% at low TSR, but the efficiency decreased at high rotation speeds. Introducing a 60° helical angle reduced the amplitude of torque oscillations, improving operational stability, and using a J-shaped blade profile increased the power coefficient by 19% at initial TSR, improving the self-starting of the turbine [17].

Yusri et al. conducted a 2D simulation of an H-shaped Darrieus turbine in ANSYS Fluent to evaluate the turbine performance at different wind speeds. They found that the maximum torque (0.9365 N m) and power (19.6658 W) were achieved at TSR = 0.9, after which the performance decreased. The turbine exhibited poor self-starting performance at low TSR due to the dominance of drag over lift. Using the k-ω SST turbulence model, the flow behavior was accurately predicted and the design parameters were optimized [18].

Wilberforce et al. performed a performance analysis of VAWTs, which showed the influence of mesh density on turbine blade design, concluding that wind speed is directly related to stator design [19].

Research on straight-bladed vertical-axis wind turbines (VAWTs) highlights the influence of key design parameters such as chord length, rotor diameter, blade height, and blade number on the aerodynamic efficiency of turbines [20]. Researchers have found that symmetrical blade profiles such as NACA 0021 are popular for small turbines with power ratings below 10 kW due to their ability to operate in low wind speed conditions typical of urban and coastal environments [21].

Other studies focus on the dimensionless parameter to describe the ratio of the chord length to the circumference of the rotor, showing that the optimal ratio varies between 8–13% depending on the fixed rotor diameter or blade chord [22].

Isataev et al. [23] investigated the improvement of Darrieus rotor performance with asymmetric blades under low wind speed conditions (3–15 m/s). Wind tunnel experiments and numerical simulations using ANSYS Fluent showed that the addition of horizontal plates increased the torque by 18–22% at wind speeds of 3–6 m/s, enhancing self-starting ability and operational stability. The plates reduced vortex losses and increased the power coefficient (Cp), making the design promising for regions with low wind energy potential, although full-scale field testing is still required.

Eltayeb et al. compared three-, four-, and five-bladed configurations based on the NACA 2412 profile using URANS equations and the SST k-ω turbulence model. The results showed that the three-bladed configuration achieved the highest power coefficient (Cp), reaching 0.471 at a tip speed ratio (TSR) of 2.5, while the four-bladed and five-bladed configurations achieved Cp values of 0.372 and 0.207, respectively. However, at low TSR values (λ < 1), which are critical for self-starting conditions, the four-bladed design demonstrated a 15–20% higher initial torque, which is essential for overcoming the self-starting issue [24].

Another aspect highlighted by Zhuang et al. emphasizes that four-bladed rotors provide a more uniform torque distribution compared to three-bladed ones, reducing the amplitude of torque fluctuations by 30%. This significantly decreases vibrations and enhances the structural durability of the turbine, which is particularly important for turbulent flow conditions typical of urban environments where VAWTs are planned to be deployed [25].

Chen et al. [26] studied the use of slotted guide flaps (SGF) on the NACA 0021 airfoil for VAWTs, finding that SGFs reduce vortex intensity and drag while improving the lift-to-drag ratio. At higher TSR values (TSR > 2.5), SGFs outperformed passive guide flaps (PGF), increasing rotor efficiency by 13.9% at a TSR of 2.62.

Despite the importance of the conducted research, previous studies have focused on two- or three-bladed Darrieus wind turbines. In contrast, the present study focuses on a modified four-bladed Darrieus turbine, in which the blades are symmetrically distributed around the circumference and grouped into two rigidly connected pairs. This design contributes to improved torque stability and a more uniform distribution of aerodynamic loads, especially under turbulent wind conditions. Furthermore, the design is protected by a national patent of the Republic of Kazakhstan [27], authored by the participants of this study, highlighting its originality and practical significance. An additional feature of the configuration is the presence of two independent rotation shafts, each of which can be connected to a separate electric generator. This approach not only increases the overall system efficiency but also enables redundancy or flexible load distribution modes under variable wind conditions when, as in this paper, a four-bladed wind turbine model is considered. The design considered in this paper is a modified vertical-axis Darrieus wind turbine with four blades located symmetrically around the central axis (Figure 1) [27]. This installation consists of two independent pairs of blades rigidly fixed to each other by bearings. This allows us to consider the operation of one of the units separately, for example, the effect of the air flow on blades 1 and 4, which set the central shaft in motion.

The main objective of the study is to select rotor parameters such as diameter and chord length that provide the best balance between power coefficient and aerodynamic losses. To achieve this goal, numerical modeling is used in the ANSYS 2022 R2 Fluent software environment, based on a non-stationary two-dimensional formulation with a turbulent SST k–ω model that describes the flow near the walls and shear zones well. The calculated data are used to evaluate the efficiency of various configurations using the Taguchi method, which allows for identifying the optimal combination of parameters and determining the sensitivity of the characteristics to changes in geometry.

This article presents a detailed analysis of the aerodynamic characteristics of the innovative design and an assessment of its performance based on numerical modeling. Namely, the determination of optimal geometric parameters that contribute to an increase in the power coefficient while reducing aerodynamic losses. A description of the methodology, basic equations, convergence criteria, and the Taguchi method are given in Section 2. The results of modeling 20 combinations of blade diameter and chord length, as well as an analysis of the distribution of velocity and pressure contours, are given in Section 3. Section 4 displays the main results of the analysis.

2. Methodology

2.1. Model Geometry

The geometry of the model is two-dimensional (2D) (Figure 2) and consists of four NACA 0021 aerodynamic profiles with different geometric parameters, which is shown in

Table 1. Geometric parameters of the model.

Parameter	Meaning
Aerodynamic profile	NACA 0021
Rotor diameter D, m	1.5–4
Chord length c, m	0.2–0.8
Number of blades	4

. The NACA 0021 airfoil was selected for this study for several reasons: it is symmetric and has proven its effectiveness in vertical-axis wind turbines operating under varying angles of attack; it demonstrates resistance to dynamic stall; it is widely referenced in the literature, which allows for meaningful comparison with previous studies; and it shows good performance at low to medium wind speeds, aligning with the operating conditions of small-scale VAWTs considered in this research.

Although 3D CFD modeling provides a more detailed understanding of the air flow around the rotor, 2D modeling remains more attractive for studying the aerodynamic characteristics of VAWTs due to its lower computational cost, simplified mesh, and sufficient accuracy of results.

3D calculations require significant resources in terms of time and hardware, while 2D modeling allows for faster numerical experiments, which is especially important when analyzing a wide range of parameters. In addition, the complexity of the 3D mesh can lead to numerical errors and increased memory consumption, while a 2D model can use a denser mesh with lower computational costs, which increases the accuracy of local blade aerodynamic analysis. As shown in studies [28,29], the 2D CFD method gives fairly accurate results for estimating lift and drag coefficients, angles of attack, and VAWT aerodynamic efficiency, while the influence of three-dimensional effects such as blade tip vortices and secondary flows, although important, is not critical in the main plane of rotor rotation. In addition, 2D calculations allow for rapid testing of various geometric parameters such as diameter, chord length, and installation angle, accelerating the design optimization process, especially in the early stages of design, making them more effective in this study.

2.2. Initial and Boundary Conditions of the Model

Torque coefficient C_m and power coefficient C_p are the main aerodynamic parameters characterizing the efficiency of a vertical-axis wind turbine. The torque coefficient Cm reflects the ability of the rotor to generate torque under the action of aerodynamic forces, and the power coefficient Cp determines the proportion of the kinetic energy of the wind converted into useful mechanical energy. The torque coefficient and power coefficient of the vertical axis wind turbine are calculated from Equations (1) and (2), respectively.

$$C_m={\displaystyle \frac{2T}{\rho AR{U}^2}},$$

(1)

$$C_P={\displaystyle \frac{2Tw}{\rho AR{U}^3}}$$

(2)

where $A$—swept area (m²), $R$—turbine radius (m), $U$—wind speed (m/s), $\rho$—air density (kg/m³), $T$—torque (N×m), $w$—angular velocity of the turbine (rad/s). From Equations (1) and (2) it follows that

$$C_P={\displaystyle \frac{Rw}{U}}{C}_m$$

(3)

where ${\displaystyle \frac{Rw}{U}}=\lambda$—tip speed ratio.

The flow is incompressible and isothermal; therefore, the equations for continuity and impulse can be expressed as follows:

$$\nabla \times \overrightarrow{v}=0,$$

(4)

$${\displaystyle \frac{\partial \overrightarrow{v}}{\partial t}}+\overrightarrow{v}\times \nabla \times \overrightarrow{v}=-{\displaystyle \frac{1}{\rho }}\nabla P+\overrightarrow{F}+v{\nabla }^2\overrightarrow{v}.$$

(5)

Equation (5) can be further derived to obtain the Reynolds-averaged Navier–Stokes (RANS) equation. Using the Reynolds-averaging method, the turbulent fluctuations are averaged and decomposed into fluctuation and mean parts to reflect the influence of turbulent flow [30]. As a result, the RANS equation becomes as follows:

$${\displaystyle \frac{\partial \left(\overrightarrow{\rho v}\right)}{\partial t}}+\nabla \times \left(\rho \overrightarrow{v}\overrightarrow{v}\right)=-\nabla P+\overrightarrow{F}+\nabla \left(\mu \nabla \overrightarrow{v}\right)+\nabla \times \left(-\rho {\overrightarrow{v}}^{\prime }{\overrightarrow{v}}^{\prime }\right).$$

(6)

The last term $\left(-\rho {\overrightarrow{v}}^{\prime }{\overrightarrow{v}}^{\prime }\right)$ represents a turbulent fluctuation [31], which can be modeled using turbulence models shear stress transfer (SST) k-ω.

In this study, the k-ω SST turbulence model [32] was used to solve the governing equations of turbulent flow. The k-ω SST turbulence model has the advantages of the k-ω model for boundary layer flow and the advantages of the k-ε model in free flow, where ε is the dissipation rate of turbulent kinetic energy [33].

The transport equation for the k-ω SST turbulence model is:

$${\displaystyle \frac{\partial \left(\rho k\right)}{\partial t}}+{\displaystyle \frac{\partial \left(\rho u_{j}k\right)}{\partial x_j}}={\tau }_{ij}{\displaystyle \frac{\partial u_j}{\partial x_j}}-{\beta }^{*}k\omega +{\displaystyle \frac{\partial }{\partial t}}\left[\rho \left(\nu +{\sigma }_k{\nu }_T\right){\displaystyle \frac{\partial k}{\partial x_j}}\right]$$

(7)

$${\displaystyle \frac{\partial \left(\rho \omega \right)}{\partial t}}+{\displaystyle \frac{\partial \left(\rho u_j\omega \right)}{\partial t}}={\displaystyle \frac{\gamma }{{\nu }_T}}{\tau }_{ij}{\displaystyle \frac{\partial u_j}{\partial x_j}}-\beta {\omega }^2+{\displaystyle \frac{\partial }{\partial x_j}}\left[\rho \left(\nu +{\sigma }_{\omega }{\nu }_T\right){\displaystyle \frac{\partial \omega }{\partial x_j}}\right]+2\left(1-F_1\right){\displaystyle \frac{\rho {\sigma }_{\omega 2}}{\omega }}{\displaystyle \frac{\partial k}{\partial x_j}}{\displaystyle \frac{\partial \omega }{\partial x_j}}$$

(8)

where $k$—turbulent kinetic energy, $\omega$—specific frequency of turbulent energy dissipation, $u_j$—component of the velocity vector along the axis $x_j$, $x_j$—spatial coordinate, ${\tau }_{ij}$—Reynolds stress, $\nu$—molecular viscosity, ${\nu }_T$—turbulent viscosity, ${\sigma }_k$, ${\sigma }_{\omega }$, ${\sigma }_{\omega 2}$- empirical constants of the SST model, ${\beta }^{*}$, $\beta$, $\gamma$—empirical coefficients (model constants), $F_1$—the blending function used in the SST model to combine the k−ω and k−ε models.

In this paper, the ANSYS Fluent CFD package is used to simulate the flow around the VAWT. ANSYS Fluent automatically calculates F₁ and switches the model between modes, ensuring accuracy at walls and stability in free flow. It also uses the default coefficients above.

ANSYS Fluent software simulates the fluid flow around the vertical axis of a four-bladed wind turbine with a NACA 0021 airfoil. In the 2D flow analysis, the computational domain is divided into two domains: fixed and rotating [34]. A 2D model VAWT was considered within a rectangular area of size 20D × 60D, as shown in Figure 3. The parameters of the calculation domain are shown in

Table 2. Parameters of computational domain.

Parameter	Meaning
Rotor diameter	D
Rotating area diameter	1.5D
Distances from the rotor center to the velocity inlet	20D
Distances from the rotor center to the pressure outlet	40D
Distances from the rotor center to the symmetry walls	10D

. In this configuration, the ratio of rotor height to the total height of the computational domain is less than 5%, which falls within the acceptable blockage ratio limits where the blockage effect on the flow is considered negligible. Thus, the influence of the domain boundaries on the simulation results is minimized, as confirmed by the stability of the computations and the achievement of convergence by the 20th rotation.

The dimensions of the computational domain of the simulation are based on the work of Guoqiang Tong et al. [22], which are sufficient to model the flow around the VAWT, and ensure that the space around the turbines in the leeward region is sufficient to characterize the flow behavior in the VAWT system; such a 2D model has been widely used in the literature to model the performance of VAWT instead of a complex 3D model without losing generality in the accuracy of numerical simulation.

Model boundary conditions at the inlet, the flow velocity, turbulent kinetic energy, and specific dissipation frequency are specified. At the outlet (pressure outlet), zero static pressure is specified and gradients from the internal cells for k and ω are used. On the blade surface (wall), the no-slip condition (U = 0) is applied, with k = 0 and ω calculated automatically by wall functions. The stationary and rotating domains are connected via a sliding interface, ensuring conservation of both mass and momentum. Each case was simulated for at least 20 revolutions, and the average data value of the last rotation cycle was used for subsequent processing. In addition, the simulation was considered converged when the data error over two revolutions fell below 1% throughout the simulation. The time step was set to 0.0016 s with 40 iterations at each step, corresponding to a rotor rotation angle of approximately 1° per time step. The input parameters of the model are shown in

Table 3. Input parameters of model.

Parameter	Meaning
Wind speed, U	8 m/s
Rotation frequency, $\upsilon$	100 rpm
Air density, ρ	1.23 kg/m3
Molecular viscosity, $\nu$	1.789 × 10−5 m2/s
Empirical coefficients (model constants), ${\beta }^{*}$, $\beta$, $\gamma$	0.09; 0.075; 0.5532 [32]
Switching function, $F_1$	varies from 0 to 1 [35]
Empirical constants of the SST model, ${\sigma }_k$,${\sigma }_{\omega }$, ${\sigma }_{\omega 2}$	0.85; 0.5; 0.856 [32]

.

2.3. Grid Independence Study

An unstructured mesh was used for both the stationary and rotating regions. In addition, mesh refinement was performed near the interface boundary and the blades. The structured mesh around the blade was constructed in such a way as to provide the coefficient values y⁺ = 1. Since the mesh density and quality can have a significant impact on the CFD calculation results, a mesh convergence study was conducted to determine the effect of mesh resolution on the calculation result of the torque coefficient C_m.

The mesh distribution for a typical four-bladed NACA 0021 case is shown in Figure 4.

The cell growth factor was 1.05, and the maximum cell size was less than the profile chord length. The grid structure is focused on the vertical-axis wind turbine (VAWT) region, with inflation (boundary) grid layers used around the airfoil and in the rotating region. These are necessary to improve the accuracy of the flow simulation in the boundary layer adjacent to the airfoil surface, as well as to correctly account for shear stresses and velocity gradients near the walls in the rotating region. This is especially important for an adequate description of turbulence and viscous effects. The thickness of the first layer was set to 10⁻⁴ m, and the growth rate in the inflation grids was 1.1. As part of the study, grid independence was tested on one of the typical geometries, with a rotor diameter of 3 m and a chord length of 0.4 m. A mesh independence study was conducted to verify the accuracy of the solution using three levels of global mesh refinement: coarse, medium, and fine meshes. Both the boundary layer and external regions of the domain were proportionally refined with a maximum growth rate of 1.05 to facilitate smooth transitions between mesh sizes. The power coefficient and the torque coefficient, averaged over a single blade, served as the primary metrics for evaluating mesh sensitivity. It was determined that the discrepancy between the medium and fine meshes was less than 2%, confirming the mesh independence of the solution. Comprehensive results are presented in

Table 4. Grid independence study.

Mesh Level	Total Cells (×103)	Cm	Cp
Coarse	400	0.1718	0.3436
Medium	600	0.1534	0.3068
Fine	800	0.1503	0.3006

.

A sensitivity test was conducted to assess the impact of the mesh transition between the boundary layer and the domain mesh. The results showed that the absence of a noticeable transition in the initial mesh did not significantly affect the computational results, with deviations remaining within 1%, as illustrated in Figure 5.

To assess the accuracy of the numerical simulation and eliminate the influence of discretization on the calculation results, ten variants with different total numbers of elements were considered: 178566, 214127, 290642, 341081, 415934, 471592, 565187, 620011, 727322, and 809273. In all cases, the grid was densified in the rotor region, and inflation layers were used around the airfoil and in the rotating region to accurately resolve the boundary layer. The torque coefficient (Cm), representing the average torque coefficient for the four blades, was used as the key evaluation criterion, and the results are presented in Figure 6.

As can be seen from the results, with an increase in the number of cells from 178566 to 620011, the torque coefficient increases by an average of 10–15% between steps, which indicates the sensitivity of the model to the quality of the grid. However, starting from 620011 elements, the increments of Cm become less significant—less than 1%—which indicates the achievement of grid convergence and the independence of the results from further refinement of the grid.

Based on this analysis, the grid 620011 elements were chosen as optimal. We used the same mesh for the other configurations, since the topology and flow patterns were not very different. Conducting a mesh study separately for each geometry would have been redundant, since the simulation settings and physics of the problem were kept the same.

2.4. Taguchi Method

The Taguchi method is a powerful tool for optimizing parameters including wind turbine (WT) geometry [36]. In this method, the signal-to-noise ratio (S/N) is used to assess the qualitative nature of each system factor [37]. In the Taguchi method, there are three types of S/N ratio: nominal is better (NB), larger is better (LB), and smaller is better (SB).

In this paper, the aerodynamic characteristics obtained from CFD simulation were used as output parameters to calculate the signal-to-noise ratio (S/N) using the Taguchi method. The torque coefficient (C_m) was used to calculate the S/N by the “larger is better” criterion (S/N_LB), since the goal was to maximize the rotor efficiency. The drag coefficient (Cd), in turn, was used to calculate the S/N by the “smaller is better” criterion (S/N_SB), since reducing the drag is directly related to reducing energy losses and increasing the overall efficiency of the wind turbine.

$$S/N_{LB}=-10\log \left({\displaystyle \frac{1}{n}}{\displaystyle \sum _{i=1}^n\frac{1}{y_i^2}}\right)$$

(9)

$$S/N_{SB}=-10\log \left({\displaystyle \frac{1}{n}}{\displaystyle \sum _{i=1}^n{y}_i^2}\right)$$

(10)

In the initial formulation of the problem, for each rotor diameter value in the range from 1.5 m to 4.0 m with a step of 0.25 m (11 levels in total), 7 chord length options were considered, from 0.2 m to 0.8 m with a step of 0.1 m. Thus, the full factorial combination of parameters included 77 unique combinations.

To improve the efficiency of calculations and reduce the number of CFD calculations, while maintaining the reliability of the results, the Taguchi method was used. With its help, it was possible to optimize the experimental plan, reducing the number of combinations to 20 most informative options based on an orthogonal array. This made it possible to significantly reduce the computational load while maintaining the representativeness of the entire range of parameters studied.

To implement the Taguchi method in this work, an orthogonal array L20 was used, corresponding to a combination of two factors with different numbers of levels.

Table 5. Orthogonal array L20 for two factors: rotor diameter (D) and chord length (c).

Test No.	Rotor Diameter D, m	Chord Length c, m
1	1.5	0.2
2	1.5	0.4
3	1.5	0.6
4	1.5	0.8
5	2.25	0.2
6	2.25	0.4
7	2.25	0.6
8	2.25	0.8
9	3	0.2
10	3	0.4
11	3	0.6
12	3	0.8
13	3.75	0.2
14	3.75	0.4
15	3.75	0.6
16	3.75	0.8
17	4	0.2
18	4	0.4
19	4	0.6
20	4	0.8

illustrates the structure of the L20 orthogonal array for two factors: rotor diameter (D) and chord length (c).

The L20 (5 × 4) orthogonal array ensures a uniform and balanced distribution of parameter combinations, allowing only 20 calculations to be performed instead of the 77 required for a full factorial experiment. This approach preserves the statistical significance of the results and allows the influence of each parameter to be assessed independently of the others, eliminating cross-distortions.

3. Results and Discussions

3.1. Analysis of Velocity, Pressure and Torque Fields

Based on the numerical simulation, a comprehensive assessment of the influence of the blade chord length on the aerodynamic behavior of the Darrieus vertical-axis wind turbine was performed for different rotor diameters. The analysis included visualization of the velocity fields (Figure 7) and pressure (Figure 8), as well as an assessment of the change in the torque coefficient depending on the rotor rotation angle for the last, twentieth revolution (Figure 9, Figure 10, Figure 11, Figure 12 and Figure 13). The results are presented below, reflecting the flow characteristics, the nature of load distribution, and the dynamics of the generated torque coefficient for different values of the chord length and rotor diameter.

Figure 7a–e show the velocity fields reflecting the effect of changing the rotor diameter (D = 1.5–4 m) and blade chord length (c = 0.2–0.8 m) on the flow structure around the Darrieus wind turbine. At smaller diameters (D = 1.5–2.25 m), the flow is characterized by pronounced circulation and local acceleration zones, especially at chords of 0.4 and 0.6 m, which indicates favorable torque generation. With an increase in D to 3–4 m, the flow becomes more uniform and “clean”, but the intensity of the accelerated zones decreases, which may indicate a drop in local pressure gradients and a decrease in the beneficial interaction between the flow and the blades. At c = 0.2 m, pronounced turbulence and flow separation are observed, and at c = 0.8 m, signs of rotor overload are observed, accompanied by a slowdown behind the blades and an increase in drag. The most stable and balanced flow with optimal energy concentration is observed at c = 0.4–0.6 m, regardless of the diameter, which confirms the universality of these values for the efficient operation of the wind turbine under the given conditions.

Figure 8a–e show the static pressure distribution fields for different combinations of the rotor diameter (from D = 1.5 m to D = 4.0 m) and chord length (from c = 0.2 m to c = 0.8 m). The analysis showed that at small diameters (D = 1.5–2.25 m), more pronounced pressure gradients are observed on the blade surface, especially at c = 0.4–0.6 m, which indicates active lift formation and effective interaction of the flow with the profile. As the diameter increases to D = 3–4 m, the pressure field becomes more uniform, the oscillation amplitude decreases, and the high- and low-pressure zones become less contrasting, which may indicate a decrease in the aerodynamic load and, as a consequence, a drop in efficiency. At c = 0.2 m, sharp local pressure jumps are recorded in all cases, which can lead to vibrations and unstable operation. At c = 0.8 m, the pressure at the inlet edge increases, the resistance increases, especially at small D. The most balanced and uniform pressure distribution is observed at c = 0.4–0.6 m for all diameters, which confirms the optimality of these chord values in the conditions of the range under study.

Figure 9 shows the change in the torque coefficient Cm depending on the rotor rotation angle θ for a diameter D = 1.5 m and different blade chord lengths. In Figure 9a, at c = 0.2 m, sharp torque fluctuations with peaks and dips are observed, which indicates unstable turbine operation and a significant influence of turbulent effects. In Figure 9b, at c = 0.4 m, the Cm curve takes an almost sinusoidal shape with a high amplitude and stable periodicity, which indicates a more efficient and balanced rotor operation. In Figure 9c, at c = 0.6 m, it demonstrates an even smoother and more stable torque distribution while maintaining a high average value, which may indicate approaching the optimal operating mode. In Figure 9d, at c = 0.8 m, the average torque level decreases, and the oscillations become less symmetrical, which indicates a drop in efficiency due to increased resistance. Thus, the most favorable conditions for torque generation are achieved with a chord length of 0.4–0.6 m.

Figure 10 shows the graphs of the change in the torque coefficient Cm depending on the rotor rotation angle θ for a rotor diameter D = 2.25 m and different chord lengths. In Figure 10a, at c = 0.2 m, the curve has a low amplitude, and Cm remains positive over the entire range, but the values are low, which indicates a weak torque. In Figure 10b, at c = 0.4 m, deep oscillations with high positive and negative peaks are observed, which indicates a strong influence of unstable aerodynamic effects and uneven load distribution. In Figure 10c, at c = 0.6 m, it shows a stabilized sinusoidal shape with high positive torque values and minimal negative surges, which can be interpreted as an efficient and balanced rotor operating mode. In Figure 10d, at c = 0.8 m, a moderate decrease in amplitude is observed, although the torque remains predominantly positive. Thus, with a diameter D = 2.25 m, the most effective in terms of value and stability of the moment is a chord length of 0.6 m, while at 0.4 m, undesirable instability with a possible reverse moment is observed.

Figure 11 shows the change in the torque coefficient Cm depending on the rotor rotation angle θ for a diameter D = 3 m and different blade chord lengths. For c = 0.2 m (Figure 11a), the curve has a clear sinusoidal shape with fairly high amplitudes, but there are sections where the torque becomes negative, which may indicate instability in certain phases of rotation. For c = 0.4 m (Figure 11b), the graph remains positive over the entire range, and the oscillations are moderate and smooth, which indicates stable and efficient operation of the rotor. On the graph with c = 0.6 m (Figure 11c), the Cm values become extremely high, but also fluctuate strongly, including significant negative sections, which indicates unstable operation and sudden overloads. For c = 0.8 m (Figure 11d), the curve again takes a regular shape, remaining in the positive region, but the amplitude decreases. In general, the most balanced result in terms of stability and efficiency is observed at c = 0.4 m, while at c = 0.6 m overloads and instability occur.

Figure 12 shows the graphs of the change in the torque coefficient Cm depending on the rotor rotation angle θ with a diameter D = 3.75 m and different chord lengths. At c = 0.2 m (Figure 12a), the torque remains positive over the entire interval, but the oscillations are small and weakly expressed, which may indicate insufficient aerodynamic activity. At c = 0.4 m (Figure 12b), the curve has a clear sinusoidal shape with a high amplitude and positive torque values, which indicates stable and efficient operation. At c = 0.6 m (Figure 12c), significant negative values of Cm appear, alternating with positive ones, which indicates unstable rotor operation and possible braking phases. With a further increase in the chord length to 0.8 m (Figure 12d), the torque coefficient almost completely goes into the negative region, reaching values below -1, which indicates inefficient and even braking operation of the wind turbine. Thus, at D = 3.75 m, the best characteristics are observed at c = 0.4 m, while values c ≥ 0.6 m led to a significant decrease in efficiency.

Figure 13 shows the graphs of the change in the torque coefficient Cm depending on the rotor rotation angle θ for a rotor diameter of D = 4 m and different chord lengths. For c = 0.2 m (Figure 13a), the torque remains stable and positive, but has a low amplitude, which indicates weak aerodynamic efficiency. For c = 0.4 m (Figure 13b), the graph shape becomes clear and sinusoidal, with pronounced peaks and positive C_m values, which indicates the most balanced and efficient operation of the rotor. As the chord increases to 0.6 m (Figure 13c), the torque goes into the negative region, which indicates braking and instability phases. For c = 0.8 m (Figure 13d), deep negative values down to −1.5 are observed, which indicates a critical decrease in efficiency and the predominance of the braking torque. Thus, at D = 4 m, the best rotor performance characteristics are observed at a chord length of c = 0.4 m, while values from 0.6 m and above lead to deterioration of aerodynamic properties.

The analysis of the change in the torque coefficient C_m depending on the rotor rotation angle showed a clear dependence of the aerodynamic efficiency of the wind turbine on the combination of the rotor diameter and the blade chord length. At small chord lengths (c = 0.2 m), the torque values remain positive, but low and weakly expressed, indicating a limited ability of the blades to generate useful aerodynamic force. With an increase in the chord to 0.4 m, most combinations demonstrate a stable, smooth, and predominantly positive behavior of the torque with a pronounced periodicity. This indicates the most balanced operation of the turbine, especially with diameters from 1.5 to 3.75 m. At c = 0.6 m, the torque coefficient increases in amplitude, but in many cases negative sections appear, especially at large diameters, indicating flow instability and possible braking phases. With a further increase in the chord to 0.8 m, the moment becomes predominantly negative, reaching extremely low values, which indicates a decrease in the efficiency of the rotor due to excessive aerodynamic resistance. Thus, based on the results obtained, it can be concluded that the optimal ratio of geometric parameters for stable and efficient operation of the Darrieus rotor is achieved with a chord length of about 0.4 m in a wide range of rotor diameters.

3.2. Evaluation of Aerodynamic Efficiency Using the Taguchi Method

The numerical simulation results (CFD) are presented in

Table 6. Results of numerical simulation.

N	D, m	c, m	Average Cm Value	Average Cd Value	Cm/Cd	${\left(\frac{\mathit{S}}{\mathit{N}}\right)}_{\mathit{L}\mathit{B}}$	${\left(\frac{\mathit{S}}{\mathit{N}}\right)}_{\mathit{S}\mathit{B}}$
1	1.5	0.2	0.050778932	0.069018	0.735735	−49.1447	16.98327126
2		0.4	0.062635	0.127467	0.491382	−36.9787	11.83366
3		0.6	0.081099	0.154793	0.523919	−22.9858	10.16032
4		0.8	0.047308	0.168392	0.28094	−28.8862	9.390334
5	2.25	0.2	0.06516	0.135341	0.481451	−26.7383	17.30525
6		0.4	0.066237811	0.17066	0.388127	−43	15.30317
7		0.6	0.105615	0.243987	0.432871	−23.2597	12.23492
8		0.8	0.051378	0.26912	0.190911	−47.0095	11.6817
9	3	0.2	0.108817	0.15878	0.685332	−44.7695	17.6038
10		0.4	0.179685	0.196306	0.915331	−15.7684	14.1405
11		0.6	0.110579	0.273009	0.405038	−27.2382	11.2576
12		0.8	0.094196	0.344421	0.273491	−47.8903	9.214197
13	3.75	0.2	0.11134	0.2901	0.383799	−19.0884	10.74493
14		0.4	0.102123	0.378801	0.269595	−26.5002	8.431534
15		0.6	−0.13392	0.379914	−0.3525	x	x
16		0.8	−0.4006	0.41975	−0.95438	x	x
17	4	0.2	0.132862	0.312407	0.425285	−17.5532	10.10126
18		0.4	0.129829	0.407197	0.318836	−59.9071	7.80367
19		0.6	−0.5074	0.410599	−1.23576	x	x
20		0.8	−1.04973	0.416503	−2.52034	x	x

. The table shows the combinations of rotor geometric parameters—diameter D and chord length c—as well as the averaged values of the torque coefficient C_m and the aerodynamic drag coefficient C_d obtained for the last, twentieth rotor revolution, after the system has reached steady state. The C_m/C_d ratio was also calculated for a quantitative assessment of the aerodynamic efficiency. In addition, the signal-to-noise ratio (S/N) values were determined using the Taguchi method: according to the criterion “larger is better” (S/N_LB) for the coefficient C_m, and according to the criterion “smaller is better” (S/N_SB) for the coefficient C_d. The obtained values allow an objective comparison of various combinations of geometric parameters and determination of the optimal rotor configurations.

Figure 14 shows the average values of the torque coefficient C_m (blue columns) and the aerodynamic drag coefficient C_d (red columns) for all 20 combinations of geometric parameters generated according to the Taguchi orthogonal array. The abscissa axis shows the test numbers N corresponding to different combinations of rotor diameter and chord length.

The analysis of the graph shows that the optimal combination is observed for option No. 10, where the rotor diameter is D = 3 m and the chord length is c = 0.4 m. This configuration demonstrates the most favorable ratio between a high C_m value and a moderate C_d value, which indicates its high aerodynamic efficiency.

Figure 15 shows the dependence of the ratio of the torque coefficient C_m to the aerodynamic drag coefficient C_d on the combination number N, corresponding to different variants of geometric parameters formed on the basis of an orthogonal array. This characteristic reflects the aerodynamic efficiency of each configuration. The maximum value of the ratio Cm/Cd is observed with combination No. 10, which indicates the most effective combination of diameter and chord length among all the variants considered.

To quantitatively assess the influence of geometric parameters on the aerodynamic efficiency of the wind turbine, the signal-to-noise ratio (S/N) values were calculated using the Taguchi method in accordance with Equations (9) and (10). The torque coefficient C_m was used as an output parameter for the S/NLB criterion, since its maximization contributes to an increase in the useful work of the rotor. For the S/N_SB criterion, the aerodynamic drag coefficient C_d was used, since its reduction leads to a decrease in losses and an increase in the overall efficiency of the plant. Calculation of the S/N_LB and S/N_SB values allowed us to identify the most effective combinations of geometric parameters and evaluate their stability to external disturbances. The results of the analysis are presented in Figure 16. The highest value of the S/N_LB = −15.77 indicator, corresponding to the highest ratio Cm/CD = 0.948, was obtained with a configuration with a rotor diameter of D = 3 m and a chord length of c = 0.4 m. This indicates the most favorable relationship between the generated torque and aerodynamic drag. In addition, this configuration is characterized by a relatively high value of S/N_SB = 14.14, which indicates an acceptable level of drag compared to other combinations. The combination of these factors allows us to conclude that the option with D = 3 m and c = 0.4 m is optimal in terms of aerodynamic efficiency and resistance to disturbances, which is confirmed by the results of the Taguchi method.

3.3. Model Validation

To assess the reliability of the results of geometric parameter optimization and validation of the numerical model, a series of additional CFD calculations were performed for the configuration that showed the best aerodynamic characteristics according to the Taguchi method: rotor diameter D = 3 m, chord length c = 0.4 m, number of blades N = 4. Verification modeling was carried out at different values of the speed coefficient λ in the range from 0.2 to 2.8, with a step of 0.2. This range was chosen by analogy with the work of Guoqiang Tong et al. [22], where similar calculations were performed for a three-bladed wind turbine with a chord length of c = 0.3 m. Despite the difference in the number of blades, the use of a comparable range of λ allows an objective comparison of the nature of the change in the power coefficient Cp and a qualitative validation of the model.

Based on the values of the torque coefficient Cm obtained because of CFD modeling for the steady-state last revolution, using Equation (3), the dependence C_p = f(λ) was constructed, reflecting the efficiency of converting the kinetic energy of the flow into mechanical energy under different rotor rotation modes.

Figure 17 shows a comparison of the dependence of the power coefficient C_p on the speed coefficient λ for an optimized configuration of a vertical-axis wind turbine with four blades (rotor diameter D = 3 m, chord length c = 0.4 m) and a numerical model from the work of Guoqiang Tong et al. [22] for a three-bladed wind turbine with a similar diameter and chord length c = 0.3 m. Both curves demonstrate a characteristic dependence: with an increase in the speed coefficient, an increase in C_p is observed, reaching a maximum in the region of λ = 2.0–2.2, after which the efficiency decreases. The C_p values obtained in this work are lower by ~10% compared to the model by Guoqiang Tong et al. [22], which is due to the increase in total resistance with a four-blade layout. At the same time, the behavior of the curve as a whole remains consistent with the literature data, which confirms the physical correctness and realism of the performed CFD modeling.

4. Conclusions

Numerical modeling of 20 configurations of a four-bladed Darrieus wind turbine (NACA 0021 profile) showed that rotor geometry has a significant impact on the aerodynamic efficiency of the plant. For each diameter and chord length combination, the torque (C_m) and aerodynamic drag (C_d) coefficients were calculated in steady state. Based on these data, the signal-to-noise ratio values were determined using the Taguchi method.

It was found, as a result of calculations. that the configuration with a rotor diameter of 3 m and a chord length of 0.4 m provides the best aerodynamic indicators: S/N_LB = −15.77, S/N_SB = 14.14 and the ratio C_m/C_{D =} 0.948. This indicates a high efficiency of energy conversion at a relatively low level of resistance. The selected configuration was also tested for different values of the speed coefficient λ (from 0.2 to 2.8), which allowed us to plot the dependence C_p(λ). The maximum power C_p = 0.35 was achieved at λ = 2.2, which is 64% higher than the mode at λ = 0.2.

The obtained results demonstrate that the combination of the Taguchi method with CFD modeling allows to effectively solve the problems of optimizing the aerodynamic characteristics of the wind turbine. In the future, based on the optimized configuration, it is planned to develop and manufacture a prototype of the installation for experimental verification.