Numerical Investigation of Hybrid Darrieus/Savonius Vertical Axis Wind Turbine Subjected to Turbulent Airflows

Abstract

The present work investigated numerically turbulent airflows over a hybrid Darrieus/Savonius vertical axis wind turbine. Firstly, the isolated turbines were validated in comparison to previous studies from the literature. Later, new recommendations were obtained for the simulation of a hybrid turbine subject to turbulent airflow. The numerical simulations consisted of the solution of time-averaged equations of mass and momentum in x and y directions using the finite volume method, available in the commercial code Ansys Fluent (version 2022 R1). For closure of turbulence, the k − ω SST (Shear Stress Transport) model was employed. For lower magnitudes of tip speed ratio (TSR), the hybrid turbine improved the power coefficient (C_P) compared to the Darrieus turbine (e.g., by 70% at TSR = 0.75), thereby demonstrating the self-starting capability of the hybrid configuration. Unexpectedly, at the optimal TSR = 1.5, the hybrid turbine performed about 6.5% better than the Darrieus turbine, indicating that the balance between the additional power generated by the Savonius rotor and losses caused by flow disturbances in the hybrid configuration was positive. As a novelty, results highlighted the role of each rotor (Darrieus and Savonius) for the performance of the hybrid turbine by comparing it with isolated Darrieus and Savonius turbines under the same conditions.

1. Introduction

In recent decades, extreme climate events have intensified as a result of environmental changes, leading to substantial material damage, human and economic losses, and increased health risks [1,2]. Accordingly, growing scientific and societal efforts have been directed toward understanding the drivers of climate imbalance and identifying effective mitigation strategies. A prominent approach to alleviating climate change and maintaining the trend on global energy demand is the diversification of the energy matrix by expanding the share of renewable sources, such as wind, solar, and wave energy, and reducing dependence on fossil fuels. According to the Low Emissions Scenario technical report by [3], it is projected that by 2050, around 80% of global energy consumption will originate from renewable sources. Moreover, wind energy alone is expected to expand up to eightfold relative to its current generation capacity [3].

Among renewable sources, wind energy has been one of the most important alternatives [4]. Consequently, research aimed at improving wind energy exploration remains of considerable importance. Wind turbines perform a main role in this direction, being categorized into two main distinct categories: horizontal axis wind turbines (HAWTs) and vertical axis wind turbines (VAWTs) [5,6]. Although HAWTs dominate large-scale wind farms due to their high efficiency, VAWTs present several advantages, including design simplicity, the absence of a yaw mechanism, ease of maintenance, and the ability to operate under irregular wind conditions, which makes them particularly suitable for urban applications [7]. Moreover, floating VAWTs also presented advantages in comparison with HAWTs, such as higher stability, lower aerodynamic wakes, and easier operation and maintenance, which can be important for the implementation in offshore wind farms [8]. VAWTs can be classified into two subtypes: lift-driven Darrieus turbines and drag-driven Savonius turbines. While Savonius turbines are more suitable for operation at low tip-speed ratios (TSRs), the Darrieus is more efficient at high TSRs [6,9]. Another characteristic of the Darrieus turbine is the difficulty for self-start at low wind velocities [5], which is an advantage for the Savonius turbine [6].

The hybridization of turbines arises from the necessity of benefiting the wide range in magnitudes of velocity of the wind. Therefore, one of the ways to minimize the operation difficulties of some VAWTs consists in the combination of different types such as the Darrieus and Savonius turbines. However, the characterization of the main parameters over turbine performance is necessary for better comprehension and new developments of the device [10].

Considering the importance of the hybridization of the turbines, the Darrieus/Savonius turbine has been recently investigated in the literature. For example, Liang et al. [11] performed a numerical investigation of a hybrid Darrieus with three aerodynamic profiles and a Savonius with two blades. Authors used a k − ε model in RANS (Reynolds-Averaged Navier–Stokes) to approach for closure of turbulence. Moreover, the influence of the attachment angle (α), number of blades for the Darrieus rotor (N), and radius ratio of the two types of rotor (RR = R_Sav/R_Dar) over the power coefficient were investigated. The study showed that self-starting performance, under the investigated conditions, can have a significant improvement by adding the Savonius turbine to the Darrieus turbine, and costing only a small compensation on the power efficiency when geometric parameters are optimized. Later, Asadi and Hassanzadeh [12] performed a similar numerical investigation considering other geometrical parameters, e.g., number of Darrieus blades of N_D = 2, a different profile for the Darrieus blade (NACA 0018), and overlap ratio of the Savonius turbine (s/c = 0.2). Moreover, the closure of turbulence was performed with the k − ω SST (Shear Stress Transport) model. Authors noticed that independent of the free-wind velocity, the hybrid rotor with an attachment angle of α = 45° presented a maximum power coefficient for rotors with TSR = 1.5 and 2.5, and α = 0° led to a better performance for TSR = 3.5. Chegini et al. [6] investigated numerically a hybrid Darrieus/Savonius turbine to enhance the self-starting of the Darrieus turbine and proposed the inclusion of deflectors placed in front of and alongside the hybrid turbine, seeking to increase its efficiency. The modeling of turbulence was also performed with the k − ω SST model. Authors observed that coupling the Darrieus and Savonius turbines benefited the power coefficient (C_P) by around 27% for the lowest investigated TSR = 1.45. However, when TSR is augmented, the hybrid turbine performance decreases due to limitations in aerodynamic performance of the Savonius turbine. The insertion of deflectors proved to be a good strategy for solving this problem, since at the optimum TSR = 2.6, the use of deflectors increased the C_P by around 30% by means of increasing wind energy density at the upwind and leeward regions of the turbine. Afterwards, Redchyts et al. [13] performed a numerical analysis of a hybrid Darrieus and Savonius turbine using a specialized package of computational aerodynamics based on the finite volume method (FVM) and treating the turbulence with the one-equation Strain-Adaptive Linear Spalart–Allmaras (SALSA) model. This study focused on the comprehension of the generation of vortex structures and their relationship with the torque generated in the turbine. Results indicated that the Savonius turbine contributed only a small percent of the total torque produced by the installation, being the main contribution achieved by the Darrieus turbine, mainly in the windward section of the trajectory. Moreover, the interaction between the vortices generated by the Savonius rotor and the blades of the Darrieus rotor caused a strong decrease in the torque coefficient in the leeward part of the trajectory. Recently, Arrieta-Gomez et al. [14] conducted thirty-six numerical experiments to investigate the effects of radius ratio (RR), coupling angle (α), and TSR on the turbine performance. Computational fluid dynamics (CFD) simulations to obtain the turbine torque were performed with the commercial code ANSYS Fluent 2024 R2, based on the FVM. Moreover, the k − ε RNG (Renormalization Group) model was employed for closure of time-averaged momentum equations. To investigate the sensitivity and interaction among the parameters, statistical analysis and five regression models were developed and evaluated using the Akaike Information Criterion (AIC) and Bayesian Information Criterion (BIC), which are statistical criteria to predict new data based on maximum likelihood estimation. Results indicated that the parameter RR had the highest influence on turbine efficiency. It was also noticed in the literature, recent contributions about the distribution of arrangements of hybrid Darrieus/Savonius turbines and optimization of a pair of hybrid turbines investigating various parameters of the arrangement, such as turbine spacing, configuration angle, direction of rotation, relative height, and pitch angle [15,16]. Despite these important investigations, there are no geometrical configurations defined yet for the turbine, as well as computational parameters used in all simulations. To illustrate this situation,

Table 1. Examples of configurations and geometric parameters used in the literature for hybrid Darrieus/Savonius turbines.

Parameter	Values
	Liang et al. [11]	Asadi and Hassanzadeh [12]	Chegini et al. [6]	Redchyts et al. [13]	Arrieta-Gomez et al. [14]
Darrieus diameter (DD)	1500 mm	1000 mm	1030 mm	7200 mm	900 mm
Darrieus profile	NACA 0012	NACA 0018	NACA 0021	NACA 0020	NACA 0018
Darrieus blade chord (C)	220 mm	220 mm	858 mm	708 mm	200 mm
Number of Darrieus blades (ND)	1, 2, 3, 4	2	3	2	2
Ratio Darrieus/Savonius radius (DD/DR)	0.2, 0.25, 0.33	0.2	0.1942	0.125	0.2, 0.5, 0.8
Savonius profile	Semicircle	Semicircle	Semicircle	Semicircle	Semicircle
Number of Savonius blades (NS)	2	2	2	2	2
Overlap ratio (s/c)	0.1	0.2	0.03	215 mm	0 mm
Separation ratio (a/c)	NI *	NI *	NI *	0 mm	0 mm
Thickness of Savonius blade (e)	NI *	0.63 mm	NI *	10 mm	NI *
Attachment angles (α)	0°	0°, 45°, 90°	30°	0°	0°, 45°, 90°, 135°
Wind speed (V∞)	2 m/s	5, 10 m/s	9 m/s	13 m/s	1 m/s
Tip-Speed ratio (TSR)	0.5–3.75	1.5, 2.5, 3.5	1.45–3.2	3.0 (Darrieus), 0.583 (Savonius)	1.5, 2.0, 2.5
Working fluid	Air	Air	Air	Air	Water
Computational method	FVM	FVM	FVM	FVM	FVM
Closure turbulence modeling	k – ε	k – ω SST	k – ω SST	SALSA	k – ε RNG
Advection interpolation function	NI *	2nd-order upwind	NI *	upwind Rogers-Kwak	NI *
Pressure-velocity treatment	NI *	SIMPLE	SIMPLE	Artificial compressibility model	NI *
Turbine rotation model	NI *	Sliding mesh	Sliding mesh	Sliding mesh	NI *

* NI—Not informed.

shows some configurations and geometrical and computational parameters used in different numerical works from the literature.

In addition to the studies related to hybrid Darrieus/Savonius turbines, several important recent contributions have been performed to improve Savonius or Darrieus turbines, e.g., use of deflectors to improve the power coefficient of a pair of Savonius turbines [17], use of aerodynamic profiles for Savonius blades [18], use of double Darrieus turbines [19], and use of the Savonius turbine as a power take-off of an oscillating water column wave energy device [20]. More important insights about improvements in Savonius turbines, Darrieus turbines, and hybrid Darrieus/Savonius turbines are presented in the review works of [10,21,22].

The main objective of the present work is to investigate the influence of a hybrid Darrieus/Savonius turbine on power performance, as well as to identify changes in its behavior compared to an isolated Darrieus configuration under the same airflow conditions. To better understand the contribution of the Savonius rotor in the hybrid setup, simulations were also performed for an isolated Savonius turbine under the same conditions. Furthermore, to gain insight into the local behavior of hybrid and Darrieus turbines, the moment and drag coefficients of each blade were monitored as a function of the rotation angle, which has not been previously investigated in the literature. Another important aspect, which represents a secondary objective, concerns the limited exploration of different computational parameters (e.g., discretization scheme or pressure-coupling treatment) and their influence on turbine performance in the literature. Therefore, distinct numerical parameters for the discretization scheme were tested here, and other parameters, such as pressure–velocity treatment, residuals, meshes, residuals, number of iterations, and time-step are clearly defined to provide recommendations regarding simulation procedures for this type of problem. The best parameters were then applied in the simulation of the hybrid Darrieus/Savonius turbine, which was compared to the isolated Darrieus and Savonius configurations. To the best of the authors’ knowledge, the investigation performed here was not previously performed in the literature.

The remainder of this paper is organized as follows: Section 2 presents the problem description, including the computational domain, boundary conditions, and the performance indicators of the turbines. Section 3 details the mathematical and numerical model, describing the governing equations, the simulation parameters, the mesh parameters employed, and the ANSYS Fluent setup. Section 4 presents the results and discussion, including the influence of the discretization schemes, numerical model validation, mesh independence study, and turbine behavior and performance indicators for the different cases. Finally, Section 5 summarizes the main conclusions of the research.

2. Problem Description

In this section, the description of the computational domain, the number of cases studied, and the performance parameters investigated are detailed.

2.1. Computational Domain and Studied Cases

The computational domain adopted in this study is shown in Figure 1. It is divided into two regions: a rotational zone (gray) and a stationary zone (outside the rotational zone). The diameter of the rotational region is defined to ensure that the simulation results are not influenced by domain boundaries. This value was set to approximately 1.5 times the turbine diameter, and all rotors are modeled to rotate counterclockwise under a constant angular velocity ($\dot{\theta }$) imposed within the rotating domain to simulate a stable flow condition around the turbine.

The computational domain is defined by a total length of L = 26D, a height of H = 12D, and an upstream length of L_I = 12D. In the simulations, the Darrieus and Savonius diameters were D_D = 800 mm and D_S = 320 mm, respectively, including the hybrid configuration simulations, corresponding to a radius ratio RR = 0.4. These values were defined based on previous studies by [23,24,25]. The rotational zones with turbine dimensions are illustrated in Figure 1b–d. It is worth noting that the no-slip boundary condition is imposed at the blades, i.e., u = v = 0 m/s, related to the rotational domain. The geometric parameters of the Savonius turbine were set as: overlap a = 0 mm, blade thickness e = 1.0 mm, blade chord c = 168.0 mm, and overlap s = 16 mm. The Darrieus rotor consists of three NACA 0018 blades offset by 120°, and a thick airfoil profile known for its higher lift at low rotational speeds. The blade chord length was set to C = 200 mm, with a fixed angle of attack φ = 0°. For the hybrid configuration, as the simulations are conducted in a two-dimensional domain, the Darrieus and Savonius turbines are assumed to have the same height.

Concerning the remaining boundary conditions, at the exit of the domain, it is considered null gauge pressure (P_g = 0 Pa), the symmetry boundary condition is assumed for the upper and lower surfaces of the domain, and a constant velocity is imposed at the inlet of the domain. The boundary conditions adopted at the inlet in all simulations are shown in

Table 2. Boundary conditions and parameters of simulation.

Parameter	Savonius	Darries/Hybrid
$V_{\infty } [m/s]$	8.00	8.00
IT [%]	1.00	1.00
$R{e}_D=\rho V_{\infty }{D}_{S,D}/\mu$	1.73 × 105	4.38 × 105
$\mathrm{Dynamic} \mathrm{viscosity} \mu \left[Pa·s\right]$	1.7804 × 10−5
Density ρ [kg/m3]	1.225
$TSR=\dot{\theta }R/V_{\infty }$	0.20; 0.30; 0.40; 0.50; 0.60; 0.70; 0.80; 1.00	0.50; 0.75; 1.00; 1.25; 1.50; 1.75; 2.00; 2.50
$\dot{\theta }$ [rad/s]	10; 15; 20; 25; 30; 40; 50	10; 15; 20; 25; 30; 35; 40; 50
Time of simulation [s]	3.5
Time for statistical analysis [s]	$1.75 \le t \le 3.55$

, which also presents the main parameters adopted in the different simulations. The parameters used in the hybrid configuration are defined based on the Darrieus characteristics. Despite the same rotational velocity being imposed for comparison among hybrid, Darrieus, and Savonius configurations, the TSR is different for the same angular velocity due to the different dimensions of the radius of the Savonius turbine and the other configurations. The turbulent intensity presented in Table 2 was defined by:

$$\mathrm{IT}\left(\%\right)=100{\displaystyle \frac{\sqrt{\overline{{u^{\prime }}^2}}}{V_{\infty }}}$$

(1)

where $\overline{u}$ represents the fluctuation averaged field (Reynolds stress), and V_∞ represents the free-stream horizontal component of the wind flow velocity [26].

The power coefficient obtained in the present study was validated against the experimental results of [5] for the Darrieus configuration, while the Savonius results were compared with the results reported in [23,24,27]. In the present work, a total of 37 simulations were performed for the following investigations:

(i)

4 simulations for investigation of the influence of discretization schemes on the performance of the Savonius turbine (validation/verification case);

(ii)

2 simulations for investigation of the influence of discretization schemes on the performance of the Darrieus turbine (validation/verification case);

(iii)

8 simulations for validation/verification of the Darrieus turbine (used in comparison with the hybrid configuration);

(iv)

7 simulations for validation/verification of the Savonius turbine with Re_DS = 8.67 × 10⁵ and TSR = 0.50; 0.75; 1.00; 1.25; 1.50; 1.75, and 2.00;

(v)

8 simulations for new recommendations for hybrid turbines;

(vi)

8 simulations for the Savonius turbine with Re_DS and TSRs presented in Table 1 allowing the comparison with the hybrid configuration.

2.2. Turbine Performance Parameters

To assess the aerodynamic performance of the turbine, various parameters can be employed. The power coefficient ($C_P$) and moment coefficient $\left(C_m\right)$ are the commonly utilized parameters [6]. The power coefficient is a non-dimensional parameter that evaluates the turbine efficiency, defined as the ratio between the power extracted by the rotor and the available power that could be harnessed. The C_P is calculated as follows [28]:

$$C_{p }= {\displaystyle \frac{2P}{{\rho }_{\infty }{A}_s{V}_{\infty }^3}}$$

(2)

where $P$ $(\mathrm{N} \mathrm{m}/\mathrm{s})$ represents the turbine’s generated power, $\rho (\mathrm{kg}/{\mathrm{m}}^3)$ is the free stream density, $A_s$ $\left({\mathrm{m}}^2\right)$ is the rotor’s swept area, and $V_{\infty }$ $(\mathrm{m}/\mathrm{s})$ is the free stream velocity. The turbine power is assessed by the product of generated torque on the rotor and angular velocity [28].

$$P=T_{turb}\dot{\theta }$$

(3)

where $T_{turb}$ (Nm) is the generated torque, and $\dot{\theta }$ ($\mathrm{rad} {\mathrm{s}}^{-1})$ is the angular velocity. The moment coefficient represents the ratio between the torque generated by the turbine and the torque exerted by the undisturbed flow. The $C_m$ is calculated as follows

$$C_{m }={\displaystyle \frac{2 T_{turb}}{\rho A_s{V}_{\infty }^2{R}_{S,D}}}$$

(4)

where $R_{S,D}$ (m) is the rotor’s radius. The tip speed ratio $\left(TSR\right)$ is a non-dimensional parameter widely employed to characterize the operating condition of a kinetic turbine. It is defined as the ratio between the tangential velocity and the free stream wind velocity. The TSR is defined as follows:

$$TSR={\displaystyle \frac{R_{S,D}\dot{\theta }}{V_{\infty }}}$$

(5)

The moment coefficient and the power coefficient are related through the $TSR$. This relationship can be expressed as

$$C_P=TSR·{ C}_m$$

(6)

3. Mathematical and Numerical Modeling

In this section, the mathematical and numerical procedures employed to simulate the fluid flow are presented. The governing equations, turbulence model, and computational setup are detailed to provide a complete overview of the modeling approach.

3.1. Governing Equations

For a mathematical modeling of turbulent, transient, and incompressible flow within a two-dimensional domain, the conservation of mass and balance of momentum equations are solved. The URANS approach is adopted to solve the momentum equation in terms of mean values. The mass conservation and momentum equations in $x$ and $y$ directions were defined as follows [29]:

$${\displaystyle \frac{\partial \overline{u}}{\partial x}}+{\displaystyle \frac{\partial \overline{v}}{\partial y}}=0$$

(7)

$$\rho \left[{\displaystyle \frac{\partial \overline{u}}{\partial t}}+\overline{u}{\displaystyle \frac{\partial \overline{u}}{\partial x}}+\overline{v}{\displaystyle \frac{\partial \overline{u}}{\partial y}}\right]=-{\displaystyle \frac{\partial \overline{p}}{\partial x}}+\left(\mu +{\mu }_t\right)\left({\displaystyle \frac{{\partial }^2\overline{u}}{\partial x^2}}+{\displaystyle \frac{{\partial }^2\overline{u}}{\partial y^2}}\right)$$

(8)

$$\rho \left[{\displaystyle \frac{\partial \overline{v}}{\partial t}}+\overline{u}{\displaystyle \frac{\partial \overline{v}}{\partial x}}+\overline{v}{\displaystyle \frac{\partial \overline{v}}{\partial y}}\right]=-{\displaystyle \frac{\partial \overline{p}}{\partial y}}+\left(\mu +{\mu }_t\right)\left({\displaystyle \frac{{\partial }^2\overline{v}}{\partial x^2}}+{\displaystyle \frac{{\partial }^2\overline{v}}{\partial y^2}}\right)$$

(9)

where $\overline{u}$ and $\overline{v}$ represents the mean velocity components in the $x$- and $y$-directions, respectively; $\overline{p}$ is the time-averaged pressure; $\mu$ denotes the dynamic viscosity; and ${\mu }_t$ is the turbulent viscosity, introduced through the closure process by applying the temporal averaging operator to the advective terms in Equations (8) and (9).

To solve ${\mu }_t$, the k – ω SST (Shear Stress Transport) turbulence model was employed, which accurately captures the external flow in vortex shedding regions [30,31]. This model provides a gradual transition in the calculation of ${\mu }_t$, using the k – ω formulation for inner boundary layers (near-wall regions) and the k – ε model for regions farther from the turbine, where the turbulent flow is more isotropic. According to [31,32], the turbulent viscosity is defined as follows:

$${\mu }_t={\displaystyle \frac{\overline{\rho }{\alpha }_1{k}_T}{max\left({\alpha }_1{\omega }_T,S_T{F}_2\right)}}$$

(10)

The turbulent viscosity is obtained from the solution of the turbulent kinetic energy transport equations (k) and specific dissipation rate ($\omega$) given by the following [31]:

$${\displaystyle \frac{\partial \left(\rho k\right)}{\partial t}}+{\displaystyle \frac{\partial \left({\rho \overline{u}}_{i}k\right)}{\partial x_i}}={\tilde{P}}_k-{ \beta }^{*}\rho k\omega +{\displaystyle \frac{\partial }{\partial x_i}}\left[\left(\mu +{\sigma }_k{\mu }_t\right){\displaystyle \frac{\partial k}{\partial x_i}}\right]$$

(11)

$${\displaystyle \frac{\partial \left(\rho \omega \right)}{\partial t}}+{\displaystyle \frac{\partial \left(\rho {\overline{u}}_i\omega \right)}{\partial x_i}}=\alpha \rho S^2-\beta \rho {\omega }^2+{\displaystyle \frac{\partial }{\partial x_i}}\left[\left(\mu +{\sigma }_{\omega }{\mu }_t\right){\displaystyle \frac{\partial \omega }{\partial x_i}}\right]+2\left(1-F_1\right)\rho {\sigma }_{\omega 2}{\displaystyle \frac{1}{\omega }}{\displaystyle \frac{\partial k}{\partial x_i}}{\displaystyle \frac{\partial \omega }{\partial x_i}}$$

(12)

where ${\tilde{P}}_k$ is turbulent kinetic energy production (a function that avoids turbulence generation in stagnant zones), and $S$ is the invariant measure of the strain rate. The constants ${\beta }^{*}= 0.09$, ${\alpha }_{1 }= 5/9$, ${\beta }_1 = 0.075$, ${\sigma }_{k1} = 0.85$, ${\sigma }_{\omega 1} = 0.5$, ${\alpha }_2 = 0.44$, ${\beta }_2 = 0.0828$, ${\sigma }_{k2} = 1$, and ${\sigma }_{\omega 2} = 0.856$ are standard model constants according to [31,32].

The blending functions $F_1$ and $F_2$ are defined by [31], which serve to adjust the solution between k – ω and k – ε models, given by the following:

$$F_1=tanh\left\{{\left\{min\left[max\left({\displaystyle \frac{\sqrt{k}}{{\beta }^{*}\omega y}},{\displaystyle \frac{500v}{y^2\omega }}\right),{\displaystyle \frac{4\rho {\sigma }_{\omega 2}k}{C{D}_{kw}{y}^2}}\right]\right\}}^4\right\}$$

(13)

$$F_2=tanh\left\{{\left[max\left({\displaystyle \frac{2\sqrt{k}}{{\beta }^{*}\omega y}},{\displaystyle \frac{500v}{y^2\omega }}\right)\right]}^2\right\}$$

(14)

where $C{D}_{kw}$ is given by the following:

$${CD}_{kw}=max\left(2\rho {\sigma }_{\omega 2}{\displaystyle \frac{1}{\omega }}{\displaystyle \frac{\partial k}{\partial x_i}}{\displaystyle \frac{\partial \omega }{\partial x_i}},{10}^{-10}\right)$$

(15)

3.2. Numerical Procedures and Parameters

The Finite Volume Method (FVM) is employed as a discretization procedure. More precisely, it is implemented in the commercial code ANSYS Fluent 2022 R1 [30,33]. Simulations were run on a desktop computer equipped with an Intel^® Core^TM i7-5820K CPU 3.30 GHZ (six cores), memory Desktop Gamer DDR4 with 16 GB of RAM, a motherboard LGA 2011-V3 Intel GA-X99-Gaming, and a hard disk 2TB Western Digital Green SataIII 64 MB. Each simulation required approximately 30 h of processing time.

Figure 2a–c shows the spatial discretization employed for hybrid, Savonius, and Darrieus rotors, respectively. The sliding mesh approach was adopted in the interface between the static zone and the rotational region to prescribe the rotational motion of the rotors. This method was commonly used in the literature to simulate the turbine’s motion [4,9,10,12,26,27]. In the rotational motion, the nodes move as a rigid body within the dynamic mesh zone, while the cell zones are coupled through non-conformal interfaces. As the mesh evolves in time, these interfaces are updated to the new relative positions of each zone [30]. The mesh quality parameters are presented in

Table 3. Details of the meshes employed in the present study.

Parameter	Savonius	Darrieus	Hybrid
Finite volumes	261,832	147,496	335,989
Finite volume in rotational zone	189,543	77,988	266,481
Inflation number in the near-wall regions	48	48	48
Inflation growth factor	1.1	1.1	1.1
Maximum y+	5.400	2.453	3.118
Minimum y+	0.023	0.213	0.015
Averaged y+	0.0036	0.898	0.502
Minimum orthogonal quality	0.069	0.234	0.031
Maximum skewness	0.722	0.693	0.695
Maximum aspect ratio	24.609	55.464	55.464

.

Mesh refinement is applied near the turbine walls, where higher velocity gradients are expected. The mesh near the walls is refined to ensure values of the dimensionless wall distance $y^{+} \le 1$. As can be seen in Table 3, the maximum y⁺ is superior to 1, but localized to a few points. In general, the y⁺ is much lower than unity, showing adequate refinement in the near-wall region. This parameter is given as follows [29,31]:

$$y^{+}={\displaystyle \frac{y\sqrt{{\tau }_w\rho }}{\mu }}$$

(16)

where ${\tau }_w$ represents the wall shear stress (Pa).

For pressure and velocity coupling, the Semi-Implicity Method for Pressure-Linked Equations (SIMPLE) algorithm is employed. The influence of advective terms is investigated for Savonius and Darrieus configurations. These results are confronted with the literature data from [5,23,27]. The first-order implicit scheme is used for the spatial discretization of the transient formulation.

The convergence criterion is set to 10⁻⁶. A total of 2000 time-steps are simulated for each case, with a time step size of $\Delta t = 0.00175 \mathrm{s}$, resulting in a total simulation time of 3.5 s. The maximum number of iterations per time step is set to 180. These parameters are the same previously used in the work of dos Santos et al. [24].

4. Results and Discussion

This section presents a study about the influence of discretization schemes used in RANS and k − ω SST equations on the power coefficient, the verification/validation of isolated Darrieus and Savonius turbines, and the performance assessment of the hybrid turbine configuration in comparison with the Darrieus and Savonius conventional turbines. The results are organized to highlight the power coefficient $\left(C_P\right)$ as a performance indicator for the torque behavior of each blade of the hybrid and Darrieus configurations, and flow velocity and total pressure fields to illustrate the flow behavior.

4.1. Influence of Discretization Scheme

This section, is focused on the investigation of the influence of distinct discretization schemes on the power coefficients for one TSR of Savonius and Darrieus turbines investigated in isolated form, and compared with previous results from the literature. This initial simplified investigation is performed since the adopted discretization schemes have not been widely described in the literature, as is the spatial discretization method (where most works use the FVM), the turbulence closure modeling (where k – ω SST is the most recommended modeling for free shear turbulent basin flows, including the flows over rotational turbines), and the pressure-velocity coupling scheme (where SIMPLE has been the most adopted) [4,9,34].

Firstly, the turbulent airflow over a Savonius turbine with Re_DS = 867,000 and TSR = 1.0 was simulated, and the results of C_P were compared with the previous experimental and numerical results of [27] and [23], respectively.

Table 4. Comparison of power coefficients (C_P) obtained with different discretization schemes for one TSR of Savonius and Darrieus turbines.

Parameter/Work	Savonius	Darrieus
$R{e}_D=\rho V_{\infty }{D}_{S,D}/\mu$	867,000	438,200
TSR	1.00	1.00
CP—first-order upwind	0.233	0.158
CP—second-order upwind	0.227	0.209
CP—MUSCL	0.259	----
CP—Hybrid upwind scheme	0.257	----
Sheldahl et al. (1977) [27]	0.228	----
Akwa et al. (2012) [23]	0.251	----
Elkhoury et al. (2015) [5]	----	0.157

illustrates the power coefficients obtained with the following discretization schemes: first-order upwind, second-order upwind, MUSCL (Monotonic Upstream-centered Scheme for Conservation Laws), and hybrid first- and second-order upwind, available in the software FLUENT 2022 R1 [30,33]. Results of the comparison are also presented in Table 4. As can be seen, the results obtained with first- and second-order upwind led to differences of only 2.2% and 0.4%, respectively, in comparison with the experimental predictions of [27]. Therefore, these two discretization schemes were tested in the simulation of an isolated Darrieus turbine with three blades, profile NACA 0018, Re_DD = 438,200 and TSR = 1.00.

For this new investigation, the first- and second-order upwind interpolation functions led to deviations of 0.6% and 33% in comparison to predictions obtained in the literature by [5]. One possible explanation for this behavior is the greater difficulty in achieving convergence with the second-order upwind interpolation function, due to its dependence on a larger number of adjacent finite volumes. The use of a first-order upwind scheme for some turbine simulations may yield better results than a higher-order scheme due to its numerical diffusion. A second-order scheme is theoretically more precise, but it is also highly sensitive to mesh quality and can generate non-physical oscillations and instabilities. The first-order scheme’s diffusion may act as a numerical stabilizer, smoothing out instabilities and, in some cases, providing a more robust and physically representative solution that better matches experimental data. It is worth noting that the study was limited to applying a single discretization scheme throughout each entire simulation. Therefore, based on these comparisons, the first-order upwind is employed in the validation/verification step, along with new recommendations for hybrid turbines.

4.2. Validation/Verification of Numerical Model

In this section, the validation/verification is performed considering the influence of TSR on the time-averaged power coefficient ($\overline{C_P}$) for Darrieus and Savonius turbines simulated separately. Once validated, this methodology was then applied to simulate a hybrid turbine configuration, generating its C_P curve as a function of TSR.

Figure 3 illustrates the curve $\overline{C_P}$ = f(TSR) for the Darrieus turbine with Re_DD = 438,200, being the results compared with those obtained numerically in the work of Elkhoury et al. [5]. It can be observed that there is a very good agreement between the present results and those obtained by Elkhoury et al. [5], with some distinct magnitudes of $\overline{C_P}$ for TSRs = 0.75 and 1.5, where the present code underestimated and overestimated the magnitudes of C_p, respectively. Another noted difference is in the optimum point of the TSR, which reached TSR = 1.5 with the present code, while the results of Elkhoury et al. [5] estimated TSR = 1.25. Despite these local differences, considering the complexity of the turbulent flows over rotational aerodynamic profiles, it can be assumed that the predictions are similar.

Figure 4 shows the effect of TSR over $\overline{C_P}$ for the Savonius turbine at turbulent air flows with Re_DS = 867,000. The results are compared with the experimental results of Sheldahl et al. [27], and numerical achievements of Akwa et al. [23], and dos Santos et al. [24]. The trend of the $\overline{C_P}$ = f(TSR) obtained with the numerical methods, including the present method, is like that predicted by the experimental measures. In the region 0.75 ≤ TSR ≤ 1.25, which is the optimal operation region of the Savonius turbine, the present code led to differences inferior to 3.0% in comparison with other numerical/experimental predictions. For TSR = 2.0, the negative magnitude of $\overline{C_P}$ indicates that the turbine supplies energy to the surrounding airflow instead of obtaining energy from it. At this TSR, it can be noticed that the present method and the method employed by Akwa et al. [23] underestimated the negative magnitude of $\overline{C_P}$ in comparison with the experimental results of Sheldahl et al. [27]. Due to this behavior, this point is not considered in the hybrid turbine investigation, limiting the TSR_S to 1.0.

Overall, considering the difficulties of simulating turbulent flows over complex rotational geometries, as found in the present work, the results obtained with the present numerical method are in agreement with that previously available in the literature for isolated Savonius and Darrieus turbines. Therefore, it is assumed that the present computational methodology for turbine simulation is suitable for the new recommendations obtained with the hybrid Darrieus/Savonius turbine.

4.3. Behavior of Hybrid Darrieus/Savonius Turbine

Before obtaining results about the behavior of the hybrid turbine, a mesh sensitivity analysis was conducted to ensure that the numerical results are independent of the mesh resolution. The hybrid turbine configuration was selected for this analysis, as it represents the most complex mesh and encompasses the other configurations (isolated Savonius and Darrieus configurations). The mesh convergence analysis was performed for TSR = 0.50. The Grid Convergence Index (GCI) method, combined with Richardson extrapolation, was applied following the procedure described by Meana-Fernández [35].

Three meshes with varying refinement in the cross-streamwise direction were considered.

Table 5. Parameters of the mesh independence study for the three computational meshes.

Mesh	Mesh 1	Mesh 2	Mesh 3
Cells in streamwise direction	300	300	300
Cells in cross-streamwise direction	130	65	32
Grid increasing ratio (cross-streamwise direction)	1.1	1.1	1.1
Number of volumes	334,296	286,889	253,559
$\overline{C_P}$$\mathrm{results} \mathrm{for} 1.75\le t\le 3.5$ s	0.04634	0.04609	0.04160
TSR	0.50	0.50	0.50

summarizes the grid characteristics near the airfoil walls and the total number of volumes for each mesh. Once the mesh independence has been confirmed, the finest mesh is used for the subsequent simulations.

Figure 5 shows the variation in the average power coefficient $\overline{C_P}$ as a function of the grid characteristic size (h). The points are fitted using a polynomial interpolation (dashed blue line). The curve exhibits an asymptotic behavior at $\overline{C_P}$ = 0.0463, indicated by the Richardson extrapolation point (red circle) in the figure. This confirms that the finest mesh (Mesh 1) provides results sufficiently independent of the mesh resolution.

The hybrid Darrieus/Savonius turbine combines the two designs to complement the strengths of both isolated turbines (Darrieus and Savonius). The Savonius component provides the necessary torque for self-starting and efficient operation at low wind speeds, effectively overcoming the Darrieus rotor’s characteristic low performance in these conditions. To analyze the impact and interaction between the turbines, the Savonius, Darrieus, and hybrid turbine configurations were individually simulated. The transient behavior of C_P for each configuration is shown in Figure 6a–d. All turbines illustrated in Figure 6a–d were rotated at constant angular velocities of 10, 20, 25, and 50 rad/s, respectively. These constant angular velocities resulted in different tip speed ratios (TSR) for the Savonius and Darrieus as well as hybrid designs, due to their respective diameters.

The Savonius turbine exhibited lower power coefficient (C_P) values compared to the other turbines within the chosen TSR range, except for higher TSR_D = 2.50, where TSR_D represents the TSR for Darrieus and hybrid configurations. For lower TSR magnitudes, Figure 6a, the presence of the Savonius rotor in the hybrid configuration altered the behavior of C_P as a function of time in comparison with the isolated Darrieus turbine, increasing the magnitude of C_P by at least a small amount. This effect happened even in a TSR_S = 0.20 (Savonius TSR), where its torque still had a lower magnitude than in regions of TSR_S~1.00. The overall behavior of the hybrid turbine appears to be primarily guided by the Darrieus design, but with a notable phase shift in TSR caused by the torque influence of the Savonius turbine. This shift, which is quantified by an angular value of nearly 0.13 rad (7.45 degrees), likely depends on the specific TSR and the overall hybrid turbine design. The relative positioning of the Savonius and Darrieus rotor blades, as well as the number of blades, diameters, and blade profiles, may be a significant factor contributing to this behavior.

As the TSR increases, the hybrid turbine’s behavior becomes even more dominated by the Darrieus turbine, see Figure 6b–d. In these scenarios, the aerodynamic forces acting on the Darrieus blades are the primary drivers of the turbine’s response. The specific choice of blade profiles can significantly influence this behavior. This is because the Darrieus turbine is particularly well-suited for high TSRs, which forces the hybrid turbine to adopt similar performance characteristics. These findings emphasize that a Darrieus-like blade design is crucial for the hybrid turbine, especially at higher TSRs. As the TSR increases to TSR_S = 1.00 and TSR_D = 2.50 (Figure 6d), a different behavior is observed. In this condition, the Savonius still produces a positive power coefficient (C_P = 0.233), while the Darrieus and hybrid configurations begin to show negative magnitudes around C_P~−0.2. Since the rotational velocity was imposed during the simulation, a negative C_P indicates that the turbine is putting energy into the fluid—a scenario that would not occur under real-world operating conditions. This divergent behavior between the Savonius, Darrieus, and hybrid configurations can be related to the operation of the Savonius turbine in a near optimum region of operation (TSR_S = 1.00) while the Darrieus turbine operates far from optimum conditions (TSR_D = 2.50). The results of Figure 6 also illustrate the importance of the ratio between the Darrieus and Savonius dimensions for the hybrid configuration in terms of the combination of TSR operation of the Darrieus and Savonius rotors. For the present configuration, the peak performance of the Savonius occurred only where the performance of the Darrieus turbine passed from the optimum TSR point.

To investigate more precisely the influence of the association between the Darrieus and Savonius turbines, the instantaneous momentum coefficient (C_m) and drag coefficient (C_d) as a function of the angle of rotation of the turbine’s blade (θ_b) (see the schematic representation of the blades number and angle of rotation in Figure 7) are illustrated in Figure 8 and Figure 9, respectively, for TSR_D = 0.5. More precisely, Figure 8a illustrates the C_m as a function of θ_b for the three blades of the isolated Darrieus turbine, where θ_b represents the angle position of each blade that occurs at different times. Figure 8b shows the C_m as a function of θ_b for the isolated Savonius turbine and the Savonius rotor in the hybrid configuration, and Figure 8c shows the C_m as a function of θ_b for each blade of the Darrieus hybrid turbine and a comparison with one of the isolated Darrieus turbines. Figure 7 illustrates a schematic representation of the blade number, the angle θ_b, and the sign of rotation of the turbine.

Figure 8a and Figure 9a show that only slight differences are observed for C_m and C_d as a function of θ_b, showing a kind of symmetry among the blades. It is also observed that there are peaks of C_m and C_d at each rotation of the turbine when each blade is near the upper position of the rotational region. Figure 8b and Figure 9b illustrate the comparison of C_m and C_d obtained for the isolated Savonius turbine and the same parameters measured in the Savonius rotor (with the same dimensions as the isolated Savonius case) inserted in the hybrid turbine. Results indicated that both C_m and C_d of the Savonius turbine suffered a step decrease from the isolated configuration to the hybrid one. A possible reason for this behavior is related to the perturbation of airflow after passing through the blades of the Darrieus rotor and impinging on the Savonius blades, see the velocity and pressure fields in Figure 10 and Figure 11, with a comparison between Darrieus and hybrid configurations. A Savonius turbine operates primarily due to the difference in drag force between its concave and convex surfaces. The addition of Darrieus blades can significantly alter these drag differences, which in turn affects the hybrid turbine’s performance. In Figure 8c and Figure 9c it can be observed that the inclusion of the Savonius turbine leads to changes in the angle at which the peak for both C_m and C_d happened in comparison to the isolated Darrieus turbine. More precisely, the peaks of the hybrid turbine changed 0.4 rad (22.9 degrees) later than the isolated Darrieus turbine.

To quantify the mean differences between the hybrid and Darrieus turbines, as well as the contribution of the Savonius rotor to the performance of the hybrid turbine,

Table 6. Averaged moment coefficient for TSR = 0.50 over a rotation.

Arrangement	$\mathbf{Moment} \mathbf{Coefficient} \overline{{\mathit{C}}_{\mathit{m}}}$
	Blade 01	Blade 02	Blade 03	Savonius Blade	Sum
Hybrid	0.02223	0.01028	0.01192	0.04879	0.09323
Darrieus	0.02324	0.02309	0.02404	-	0.07038
Savonius	-	-	-	0.31044	0.31044

and

Table 7. Averaged drag coefficient for TSR = 0.50.

Arrangement	$\mathbf{Drag} \mathbf{Coefficient} \overline{{\mathit{C}}_{\mathit{d}}}$
	Blade 01	Blade 02	Blade 03	Savonius Blade	Sum
Hybrid	0.26902	0.24956	0.26513	0.18839	0.97210
Darrieus	0.30569	0.29813	0.29909	-	0.90291
Savonius	-	-	-	0.92253	0.92253

show the $\overline{C_m}$ and $\overline{C_d}$ for the hybrid configuration, isolated Darrieus and Savonius turbines at TSR = 0.5. As can be seen, the $\overline{C_m}$ of the Darrieus rotor for the hybrid configuration increases in comparison with the isolated Darrieus configuration (32%). However, the contribution of the Savonius blades conducted the hybrid turbine to a better performance than the Darrieus configuration. It is also observed that there is a step difference between the Savonius rotor in the hybrid configuration and the isolated Savonius rotor. Despite the strong decrease in $\overline{C_m}$ for the Savonius rotor in the hybrid configuration, it helps the global performance of the hybrid turbine to be higher than the isolated Darrieus configuration.

Figure 12 shows the relationship between the mean C_P and TSR for the Savonius, Darrieus, and hybrid turbines. It is reinforced here that, for the same angular velocity ($\dot{\theta }$) two distinct TSRs are obtained for the Darrieus and Savonius rotors due to difference in radius dimensions. This curve provides crucial insight into the power output characteristics of each turbine type. As expected, the Savonius turbine performs optimally at low TSRs (TSR_S = 0.8), while the Darrieus turbine is effective at higher TSR_D (TSR_D = 1.5). The inclusion of the Savonius rotor in the hybrid design positively influences performance at lower TSR magnitudes, specifically in the range 0.5 ≤ TSR_D ≤ 1.0, where the mean C_p is noticeably higher than that reached by the isolated Darrieus configuration. This behavior reinforces the previous observations from the literature that the Savonius turbine increases the torque of the turbine when associated with the Darrieus one. Unexpectedly, the hybrid turbine achieved a maximum $\overline{C_P}$ that is slightly higher than that of the Darrieus turbine alone (6.5%), a result not typically reported in the literature. While the magnitude of this improvement is small, it highlights a potential benefit of hybrid design. The performance of the hybrid turbine can be benefited by the additional power generated by the Savonius turbine and harmed by the disturbances of fluid flow generated by the Savonius turbine impinging the blades of the Darrieus turbine. For the optimal TSR_D = 0.75, this phenomenon was not as pronounced, perhaps due to the specific scale factor between the two turbine components (RR = 0.4). This suggests that the relative size and positioning of the Savonius and Darrieus sections may play a critical role in mitigating the aerodynamic interference. Further investigation into the optimal scale ratio could help to minimize these negative effects and improve the overall efficiency of hybrid vertical-axis wind turbines. It is worth noting that the operation conditions of the TSR magnitudes for each rotor (which are also influenced by the radius ratios of the turbine) can be an important aspect in this issue. Further research is necessary to fully understand this phenomenon and its potential applications, particularly regarding the optimal integration and scaling of the two turbine types to maximize overall efficiency.

Despite some simplifications employed in the present work as the two-dimensional domain and consequent hypothesis of the same height for Darrieus and Savonius rotors in hybrid configuration, the present study obtained promising theoretical recommendations for the application of VAWTs. For instance, results demonstrated that for lower TSRs, the Savonius turbine improved the self-starting of the hybrid turbine. Results of the comparison also indicated that there is a balance between the additional power generated by the Savonius turbine and the additional perturbation of turbulent flow caused by the insertion of the Savonius rotor into the Darrieus one. Considering the VAWTs have been recommended for urban areas and floating offshore wind farms, the use of a hybrid configuration can be a promising strategy to improve the energy generated in these farms.

5. Conclusions

The present work performed a numerical analysis of turbulent airflows over hybrid Darrieus/Savonius vertical axis wind turbines, obtaining new recommendations about numerical procedures for the simulation of this problem and new comparisons among hybrid, Darrieus and Savonius turbines, seeking to understand the influence of the association between Darrieus and Savonius turbines. The numerical simulation was based on the solution of time-averaged equations of mass and momentum in x and y directions using the finite volume method, available in the commercial code Ansys Fluent, and the k−ω SST to tackle the closure of turbulence. Moreover, the sliding mesh methodology was employed for rotational movement of the turbines.

Firstly, an investigation of the reliability of the present computational method for the simulation of rotational Darrieus and Savonius turbines subjected to turbulent flows was conducted. Results indicated that the use of the first-order upwind interpolation function for treatment of advective terms resuled in a accurate prediction of $\overline{C_P}$ for the simulation of Darrieus and Savonius turbines, with a difference of 0.3% and 2.2% in comparison with studies in the literature [5,23,27]. Despite the very good agreement for the Savonius turbine (0.4% in comparison with experimental results by Sheldahl et al. [27]) obtained with second-order upwind, important discrepancies were noticed for the Darrieus turbine (33% in comparison with the results by Elkhoury et al. [5]). In a second instance, the effect of TSR over $\overline{C_P}$ for isolated Darrieus and Savonius turbines were simulated, being obtained satisfactorily in agreement with previous findings in the literature considering the difficulties in the simulation of turbulent flows over rotational domains of complex turbine configurations.

Results of the hybrid turbine indicated that, for lower magnitudes of TSR, the hybrid turbine improved the $\overline{C_P}$ in comparison with the Darrieus turbine. For TSR_D = 0.50 and 0.75, for example, differences of 35% and 70% were obtained, respectively. This is a promising indication that the insertion of the Savonius turbine can benefit the self-start of the Darrieus turbine. These results agreed with previous findings in the literature. For the optimal TSR_D = 1.5, the hybrid turbine led to a performance nearly 6.5% superior to the Darrieus turbine, which was initially unexpected. The probable reason for this behavior is related to the additional contribution of the Savonius turbine for the power of the system to be higher than the reduction in power in the Darrieus turbine in the hybrid configuration due to new flow disturbances caused by the insertion of the Savonius turbine. This behavior was not previously obtained in the literature and requires further investigation. For TSR_D ≥ 1.75, the hybrid turbine obtained an inferior performance in comparison with an isolated Darrieus turbine, as previously described in the literature. Another important observation was the change in the instantaneous operation of the blades of the Darrieus turbines when the Savonius turbine was inserted. Examples are noticed in the change in angle of power peaks for blades of Darrieus turbines. Moreover, the Savonius behavior in a free stream condition and inserted in the Darrieus rotor was also changed.

For future studies, it is recommended to elaborate on the investigation of other geometrical parameters as different radius ratios of Darrieus/Savonius turbines, to try to make the highest performance operation of both rotors compatible in a similar region of TSR.