To investigate power losses of a Darrieus–Savonius combined vertical axis wind turbine (hybrid VAWT) associated with the interaction between blades and wake, it is crucial to understand the flow phenomena around the turbine. This study presents a two-dimensional numerical analysis of vortex dynamics for a hybrid VAWT. The integration of a Savonius rotor in the hybrid VAWT improves self-starting capability but introduces vortices that cause transient load fluctuations on the Darrieus blades. This study attempts to characterize the flow features around the hybrid VAWT and correlate them with the Darrieus blade force variation in one revolution. Results demonstrate the capability of numerical modeling in handling a wide range of operational conditions: the relevant position of Savonius and Darrieus blades (attachment angle γ = 0 ° 90 °) and Savonius' tip speed ratio λS (0.2–0.8, varied Savonius' rotational speed). The torque increase in the Darrieus blade in hybrid VAWT (compared to a single Darrieus rotor) due to the appearance of the vortex shedding from the advanced Savonius blade is independent of the attachment angle and tip speed ratio. Apart from start-up and power performances of the hybrid VAWT, the most rapid force fluctuation is identified when the Darrieus blade interacts with Savonius' wake at γ = 0 ° and λ S = 0.8, which is considered undesirable. Furthermore, attachment angles of 60 ° and 90 ° exhibit better power coefficients compared to those of 0 ° and 30 ° for the hybrid VAWT. This study contributes to a comprehensive understanding of flow dynamics in hybrid VAWTs, revealing the correlation between torque variation and vortex development.

Vertical axis wind turbines (VAWTs) are not sensitive to the wind direction. Therefore, they can operate without active control. This great feature makes this turbine attractive to use in urban areas, where the wind direction changes continuously. They are categorized into two types: lift-driven Darrieus and drag-driven Savonius. There are abundant studies on the two types of turbines, and some notable results are listed in Table I, showing the type of the rotor, minimum (and maximum) operational tip speed ratio λmin (and λmax), and corresponding peak power performance c p max. Savonius rotor has a good self-starting performance.1–3 However, it has a low power efficiency. Darrieus rotor can achieve a maximum power coefficient of up to 0.3–0.5.1,4–7 However, it cannot self-start in practice.

TABLE I.

Operational conditions of Darrieus and Savonius rotors.

Author VAWT λmin λmax c p max
Hosseini and Goudarzi1   3-bladed Darrieus  0.48 
Castelli et al.4   3-bladed Darrieus  1.43  2.7  0.31 
Wang et al.5   2-, 3-, and 4-bladed Darrieus  0.8  4  ∼0.4 
Gosselin et al.6   3-bladed Darrieus  1.8  5.2  0.36 
Shao et al.7   2-bladed Darrieus  7  0.38 
Hosseini and Goudarzi1   Bach-type Savonius  0.6(self-start)  1  0.27 
Wekesa et al.2   Three-bladed Savonius  0.2(self-start)  ∼1.3  0.18 
Fatahian et al.3   Two-bladed vented Savonius  0.4(self-start)  0.275 
Author VAWT λmin λmax c p max
Hosseini and Goudarzi1   3-bladed Darrieus  0.48 
Castelli et al.4   3-bladed Darrieus  1.43  2.7  0.31 
Wang et al.5   2-, 3-, and 4-bladed Darrieus  0.8  4  ∼0.4 
Gosselin et al.6   3-bladed Darrieus  1.8  5.2  0.36 
Shao et al.7   2-bladed Darrieus  7  0.38 
Hosseini and Goudarzi1   Bach-type Savonius  0.6(self-start)  1  0.27 
Wekesa et al.2   Three-bladed Savonius  0.2(self-start)  ∼1.3  0.18 
Fatahian et al.3   Two-bladed vented Savonius  0.4(self-start)  0.275 

Darrieus and Savonius rotors have their advantages and disadvantages. The Darrieus–Savonius combined vertical axis wind turbine (hybrid VAWT) combines the advantages and disadvantages of both types of turbines and offers some unique strengths and weaknesses. The sketch of the hybrid VAWT is depicted in Fig. 1. For their strengths, the hybrid VAWT can produce more torque for self-starting compared to a Darrieus turbine, beneficial from the existence of Savonius.1,8–10 In addition, it has a wide operation range and can generate power at either low or high inflow speeds, making it well-suited for locations with variable inflow conditions.11,12 For their weaknesses, the hybrid VAWT generates less power per unit of rotor area compared to a Darrieus turbine.1,8–12 The power and start-up performances of hybrid VAWT are affected by the wake introduced by the inner Savonius. The change in the wake patterns of hybrid VAWT has limited the amount of wind energy that can be captured and converted into mechanical energy. To study the effect of these limitations on the hybrid VAWT, engineers and researchers have investigated flow physics around the hybrid VAWT and developed different designs to improve the power performance of the hybrid VAWT.

FIG. 1.

Sketch of hybrid VAWT (Left: 3 D structure, right: top view).

FIG. 1.

Sketch of hybrid VAWT (Left: 3 D structure, right: top view).

Close modal

The effect of different parameters on the performance of the hybrid VAWT has been investigated in many studies. Roshan et al.13 investigated the effects of the arc angle of the Savonius blades Φ, the overlap ratio ε = e R S, and the Savonius blade curvature α = R 1 R 2 on the performance of the turbine. In each of the three variables, ε = 0.25 , α = 0.25, and Φ = 150 ° were proposed in order to optimize the self-starting performance and extend the operational range of the hybrid VAWT. In the study of Asadi and Hassanzadeh,14 the performance of a hybrid VAWT constructed by a two-bladed Darrieus rotor as the external rotor and a two-bladed Savonius rotor as the internal rotor was assessed numerically. It was demonstrated that the tip speed ratio of hybrid VAWT λhybrid strongly influenced the choice of an appropriate attachment angle γ (relevant position of Savonius and Darrieus blades). The power coefficient of the hybrid VAWT with an attachment angle of 45° was the optimal case for λ hybrid = 1.5 and 2.5. However, their study did not discuss the flow fields in detail. Kyozuka15 conducted a series of experiments to study the hydrodynamic characteristics of a Darrieus–Savonius rotor. They found that the attachment angle was important in the starting performances of the hybrid VAWT. However, the power coefficient was decreased to 70% of the separate Darrieus VAWT. Liang8 studied the effects of the radius ratio and attachment angle on the performance of a hybrid VAWT.

A few studies have been conducted to understand the blade–vortex interaction around a single VAWT. Bangga16 investigated the effects of airfoil thickness on the dynamic stall characteristics of high-solidity VAWTs. The results showed that the leading-edge vortex bubble was the main cause of the power loss for high-solidity VAWTs, and the leading-edge vortex strength and radius were reduced with increasing airfoil thickness. It also reported that the interaction between the blade and wake discouraged perfect periodicity when the turbine operates at a small tip speed ratio. Wang17 investigated the power performance of vertical axis wind turbines with different airfoil shapes and analyzed the interaction between vortex and wind turbine blades.

The blade–vortex interaction in the hybrid VAWT is more complex compared to the single VAWT. This complexity arises due to the presence of the Savonius rotor and its interaction with the Darrieus blades.9,18–20 The challenge of optimizing the hybrid VAWT requires a parametric study, e.g., the radius ratio, the attachment angle, and the tip speed ratio of the hybrid VAWT. Several studies have already been published to evaluate the effects of these parameters on the aerodynamic force of the hybrid VAWT. Liu et al.21 conducted a systematic study about the effects of the moment of inertia and radius ratio on the self-starting capability and power efficiency. Results showed that the hybrid VAWT with a small-size modified Savonius rotor ( R D R S = 3.5) can generate a higher maximum power coefficient ( c p max = 0.41) compared to the hybrid VAWT with a large-size Savonius rotor ( R D R S = 2.33 , c p max = 0.375). Kumar18 optimized the Savonius rotor diameter to maintain the power coefficient of the Darrieus rotor in the hybrid VAWT. The interaction between the closed configuration Savonius rotor and the flow resulted in the formation of von Karman vortices. These vortices then interacted with the Darrieus blades, causing flow detachment. Through their analysis, it has been determined that an optimal diametrical ratio of 3 led to the highest power coefficient for the Darrieus rotor. By integrating a compact Savonius rotor within the hybrid VAWT, one can harness its self-starting capabilities while mitigating potential flow structure interactions that might affect turbine performance. Saini and Saini22 numerically studied the effect of radius ratio and attachment angle on the performance of the hybrid hydrokinetic turbine. The pressure and velocity contours were analyzed for hybrid VAWT at unvaried conditions R D R S = 5 and γ = 30 °. The power and start-up performances were discussed for different operating conditions ( U = 0.5–2.5 m/s, λ = 2–3.6, γ = 0 ° 90 ° , R D R S = 3–5). Results showed that the power coefficient decreased as the radius ratio increased and the maximum starting torque was found at R D R S = 5 , γ = 30 ° and 60 °. The aforementioned investigations drive us to conceptualize a hybrid VAWT with a reduced-size Savonius rotor.

These studies have explored various aspects related to flow structures and turbine performance individually. There exists a gap in our understanding regarding the variation of blade force in response to vortex dynamics. In a word, a comprehensive discussion on the correlation between flow structures and turbine performances has not been thoroughly addressed yet. To address this gap, further studies are required to conduct in-depth analysis that correlate flow structures with specific turbine performance metrics. It involves identifying and quantifying the key flow features, such as vortices, wake interactions, turbulence patterns, and their direct influence on critical turbine parameters, such as power output and torque. This study is an extended work in terms of investigating the correlation between blade–vortex interaction and blade force variation.

The characterization and correlation of the vortex dynamics and blade torque play an important role in the design phase of the hybrid VAWT. In addition, the study of vortex dynamics might be useful for future studies on the engineering model of the hybrid VAWT. A parametric study of hybrid VAWT is conducted to better understand the flow mechanisms and performances of the hybrid VAWT. The presence of the Savonius rotor affects not only the rotor performance but also the downstream wake pattern. Their correlations are explored based on the Savonius-related parameters: attachment angle (position of Savonius blade relative to Darrieus blade) and Savonius' tip speed ratios (varied Savonius' rotation speed) in the current work.

Over the past few years, the growth in computational capabilities has paved the path for tackling the computation of VAWTs using computational fluid dynamics (CFD). However, these simulations require an expensive computational investment, resulting in this method still impractical for design objectives. As a result, researchers have turned to simplified two-dimensional (2 D) models, which significantly reduce computational demands. The two-dimensional (2 D) blade-resolved simulations can capture the primary vortex dynamics and yield comparable trends in quantitative results for VAWTs (e.g., streamwise velocity and blade force).23–25 Fatahian et al.3 investigated the flow dynamics and rotor performance of a vented Savonius using 2 D unsteady simulation in ANSYS Fluent. The numerical results of the rotor aerodynamic performance exhibited similar trends to the experimental results.26 Li et al.27 studied the wave–turbine interaction of a Savonius hydrokinetic turbine using 2 D simulations in STAR-CCM+. The simulated wave evolution was validated against the theoretical and experimental values. Results indicated that the turbine diameter and blade number will affect the turbine efficiency. Vigneswaran and VishnuKumar28 explored the effect of co-flow jet (CFJ) velocity, the injection height, and the injection mass flow rate on the aerodynamic coefficient of a 2 D co-flow jet (CFJ) airfoil in ANSYS Fluent. Results showed that the jet velocity played a significant role in affecting the aerodynamic coefficient of CFJ airfoils. The aforementioned geometrical parameter (attachment angle) and operational parameter (Savonius' tip speed ratio) are all 2 D parameters. With the aim of characterizing the flow features of the hybrid VAWT and conducting a cost-effective parametric study, 2 D simulations are considered sufficient in this study.

This paper is structured as follows. The numerical procedure is presented in Sec. II. The effects of attachment angle and Savonius' tip speed ratio on the hybrid VAWT performances and correlation of vortex dynamics and blade torque are analyzed in Sec. III. The main results are concluded in Sec. IV, followed by the mesh independence study and meshing strategy of single rotors in  Appendixes A and  B.

The 2 D simulations are conducted in OpenFOAM.29 The hybrid VAWT consists of Savonius blades inside and Darrieus blades mounted on the same rotation axis. The Savonius rotor has two semicircle blades without an overlap and a gap distance, which is a relevant work to Ref. 23. The Darrieus blades are chosen as NACA0021, corresponding to the work in Ref. 30. The geometrical and operational parameters of the hybrid VAWT are shown in Table II, where λ S = ω S R S U and λ hybrid = λ D = ω D R D U .

TABLE II.

Detailed geometrical and operational information of the hybrid turbine.

Parameter Value/Specification Unit
Radius of inner Savonius RS  0.148 
Radius of outer Darrieus RD  0.74 
Tip speed ratio of inner Savonius λS  [0.2, 0.4, 0.6, 0.8]  ⋯ 
Tip speed ratio of outer Darrieus λD  [2.4, 2.9, 3.7, 4.0, 4.2]  ⋯ 
Tip speed ratio of hybrid VAWT λhybrid  [1.0, 2.4, 2.9, 3.7, 4.0, 4.2]  ⋯ 
Blade of inner Savonius  Semicircle  ⋯ 
Blade of outer Darrieus  NACA0021  ⋯ 
Chord length of outer Darrieus  0.075 
Attachment angle γ  [0, 30, 60, 90]  ° 
Parameter Value/Specification Unit
Radius of inner Savonius RS  0.148 
Radius of outer Darrieus RD  0.74 
Tip speed ratio of inner Savonius λS  [0.2, 0.4, 0.6, 0.8]  ⋯ 
Tip speed ratio of outer Darrieus λD  [2.4, 2.9, 3.7, 4.0, 4.2]  ⋯ 
Tip speed ratio of hybrid VAWT λhybrid  [1.0, 2.4, 2.9, 3.7, 4.0, 4.2]  ⋯ 
Blade of inner Savonius  Semicircle  ⋯ 
Blade of outer Darrieus  NACA0021  ⋯ 
Chord length of outer Darrieus  0.075 
Attachment angle γ  [0, 30, 60, 90]  ° 

The attachment angle γ and the tip speed ratio of Savonius λS are varied to conduct an in-depth study that characterizes the flow structure relevant to blade force variation. It is worth noting that λS is changed by varying its rotation speed instead of radius because the operational condition is more convenient to control than the geometrical condition during turbine operation.

The unsteady, incompressible Reynolds-averaged Navier–Stokes (URANS) equations are solved by PIMPLE [a combination of PISO (pressure-implicit split-operator) and SIMPLE (semi-implicit method for pressure-linked equations)] algorithm to improve the numerical accuracy upon multi sub-iterations. The governing equations include the conservation of mass and the conservation of momentum, which are expressed by Eqs. (1) and (2), respectively. ui and uj represent the velocity components in the i and j directions. The model of Eq. (2) reflects the fact that the flow around the rotating blade is turbulent, and closure of the URANS equations is achieved using the shear stress transport (SST) k-w turbulence model. This turbulence model can predict boundary layers under strong adverse pressure gradients and has been validated in the application of wind turbines with good results.31,32 It predicts accurate results for separation flows.33,34 The turbulent kinetic energy k and dissipation rate w are expressed as Eqs. (3) and (4),35,36 respectively. In Eqs. (3) and (4), the generation of the turbulent kinetic energy and dissipation rate are represented by Gw and Gk, respectively. Γk and Γw represent the effective diffusivity of k and w, respectively. Yw and Yk are the dissipation of w and k, respectively. Sk and Sw are source terms. Dw is the blending function for the standard k-ε and standard k-w model31,32
u i x i = 0 ,
(1)
ρ u i t + ρ ( u i u j ) x j = p x i + x j [ ( μ + μ t ) u i x j ] ,
(2)
( ρ k ) t + ( ρ k u i ) x i = x j ( Γ k k x j ) + G k Y k + S k ,
(3)
( ρ w ) t + ( ρ w u i ) x i = x j ( Γ w w x j ) + G w Y w + D w + S w .
(4)

The schematic of the computational domain and boundary conditions is depicted in Fig. 2. The mesh details of the computational domain are shown in Fig. 3. The inflow comes from left to right of the computational domain; the boundary conditions of the computational domain are determined as follows:

  • Inlet boundary: The inlet wind speed is equal to the freestream velocity.

  • Outlet boundary: The given outlet pressure is the standard ambient pressure.

  • Wall boundary: No-slip rotating wall boundary conditions are used along the blade surfaces.

  • Slip plane: There is a sliding mesh interface between the moving and the stationary grids. CyclicAMI boundary condition is employed to couple conditions between a pair of patches that share the same outer bounds. This boundary condition allows the physical rotation of the rotation subdomain. And the force variation would be well evaluated.

FIG. 2.

Computational domain and boundary conditions (Not true to scale).

FIG. 2.

Computational domain and boundary conditions (Not true to scale).

Close modal
FIG. 3.

Mesh generation in the computational domain.

FIG. 3.

Mesh generation in the computational domain.

Close modal

A mesh independence study of a hybrid VAWT with two semicircular Savonius blades and two Darrieus blades is conducted to determine the appropriate cell size. Three mesh configurations (coarse, medium, and fine) are involved with cell numbers of 1.99 × 105, 1.79 × 105, and 1.40 × 105, respectively. The computational domain and the first row's cell height are identical to the single-rotor case, as shown in  Appendixes A and  B. The simulation residual and time step size are selected as 1 × 10−5 and 0.24°/step to be consistent with the single Savonius and Darrieus simulations, respectively. The values of maximum skewness of the three mesh configurations are 0.70, 0.64, and 0.67, respectively. The average dimensionless wall distance y+ is below one. The power coefficients from different mesh configurations and mesh details are shown in Table III. The variations of the power coefficient are depicted in Fig. 4. The power coefficients of the hybrid VAWT have a similar trend over the three mesh configurations. The fluctuations at θ = 60 ° 120 ° and θ = 240 ° 300 ° are attributed to the presence of Savonius' wake. This work will focus on torque fluctuation in response to the wake. The average power coefficients in Table III show that the medium mesh configuration yields reasonably accurate results with a 1.57% difference from the fine mesh configuration. To ensure cost-effective computations, a medium mesh is employed in the subsequent analysis.

TABLE III.

Power coefficient of hybrid VAWT from different mesh configurations, λ hybrid = 4.0 , U = 4.01 m/s, and γ = 0 °.

Mesh Fine Medium Coarse
Number of cells  1.99 × 105  1.68 × 105  1.40 × 105 
Darrieus blade discretization  102  76  66 
Savonius blade discretization  304  220  140 
Power coefficient cp  0.3753  0.3694 (−1.57%)  0.3651 (−2.72%) 
Mesh Fine Medium Coarse
Number of cells  1.99 × 105  1.68 × 105  1.40 × 105 
Darrieus blade discretization  102  76  66 
Savonius blade discretization  304  220  140 
Power coefficient cp  0.3753  0.3694 (−1.57%)  0.3651 (−2.72%) 
FIG. 4.

Power coefficient variation for hybrid VAWT with different mesh configurations, γ = 0 ° , R D R S = 5.0 , λ hybrid = 4.0.

FIG. 4.

Power coefficient variation for hybrid VAWT with different mesh configurations, γ = 0 ° , R D R S = 5.0 , λ hybrid = 4.0.

Close modal

A time independence study for the hybrid VAWT has been conducted to verify the stability of the simulation. Azimuthal increments d θ of 0.06 ° , 0.12 ° , 0.24 ° , 0.48 ° , 0.96 °, and 4.8 ° are applied, and the power coefficients cp are shown in Table IV. An Azimuthal increment of 4.8 ° tends to underestimate cp by 5.1% compared to 0.06 °. Among cases with d θ 0.96 °, a negligible difference in cp is observed. The comparison shows that d θ = 0.24 ° is a cost-effective choice in this study.

TABLE IV.

Power coefficient of hybrid VAWT from different azimuthal increments, λ hybrid = 4.0 , U = 4.01 m/s, and γ = 0 °.

Azimuthal increment d θ 0.06 ° 0.12 ° 0.24 ° 0.48 ° 0.96 ° 4.8 °
Time step size dt (s)  4.83 × 10−5  9.66 × 10−5  1.932 × 10−4  3.864 × 10−4  7.728 × 10−4  3.864 × 10−3 
Power coefficient cp  0.3693  0.3693  0.3694  0.3693  0.3693  0.3504 
Azimuthal increment d θ 0.06 ° 0.12 ° 0.24 ° 0.48 ° 0.96 ° 4.8 °
Time step size dt (s)  4.83 × 10−5  9.66 × 10−5  1.932 × 10−4  3.864 × 10−4  7.728 × 10−4  3.864 × 10−3 
Power coefficient cp  0.3693  0.3693  0.3694  0.3693  0.3693  0.3504 

The sensitivity study based on the computational domain size has been conducted. The distance between the rotor and the domain outlet is varied from 18 R hybrid , 24 R hybrid to 30 R hybrid. The power coefficients of hybrid VAWT for computational domains with varied outlet distances are listed in Table V. The difference between cp for hybrid VAWT with an outlet distance of 18 R hybrid and 30 R hybrid is about 13.02%. This deviation drops to 0.43% for the hybrid VAWT with an outlet distance of 24 R hybrid. Therefore, the distance between the rotor and the domain outlet is chosen as 24 R hybrid.

TABLE V.

Power coefficient of hybrid VAWT from different distances between the rotor and domain outlet, λ hybrid = 4.0 , U = 4.01 m/s, and γ = 0 °.

Distance between the rotor and outlet 18 R hybrid 24 R hybrid 30 R hybrid
Power coefficient cp  0.3227 (−13.02%)  0.3694 (−0.43%)  0.3710 
Distance between the rotor and outlet 18 R hybrid 24 R hybrid 30 R hybrid
Power coefficient cp  0.3227 (−13.02%)  0.3694 (−0.43%)  0.3710 

Another sensitivity study examines the impact of varying numbers of revolutions on the average results. By systematically altering this parameter, we assess its influence on the power performances of the hybrid VAWT. This analysis provides insights into the stability and reliability of the findings, helping determine the optimal number of revolutions for obtaining consistent and representative results. The power coefficient variation in one cycle for different numbers of revolution is shown in Fig. 5. It is observed that the power coefficient tends to have an identical variation if more revolutions are taken into account. The mean power coefficient against different revolutions and different numbers of revolutions is depicted in Fig. 6. Due to the complex turbulence around the hybrid VAWT, the mean power coefficient of the single revolution fluctuates between 0.36 and 0.38. The mean power coefficient of several revolutions tends to converge at a magnitude of 0.37 when the 10th to 17th revolutions are averaged. In this work, eight revolutions (10th–17th) are used for the calculation of the mean power coefficient for the hybrid VAWT.

FIG. 5.

Power coefficient variation for hybrid VAWT in different numbers of revolution, γ = 0 ° , R D R S = 5.0 , λ hybrid = 4.0.

FIG. 5.

Power coefficient variation for hybrid VAWT in different numbers of revolution, γ = 0 ° , R D R S = 5.0 , λ hybrid = 4.0.

Close modal
FIG. 6.

Mean power coefficient for hybrid VAWT in different revolutions and different numbers of revolutions, γ = 0 ° , R D R S = 5.0 , λ hybrid = 4.0 (x axis for solid line: Total number of revolution considered for the mean power coefficient).

FIG. 6.

Mean power coefficient for hybrid VAWT in different revolutions and different numbers of revolutions, γ = 0 ° , R D R S = 5.0 , λ hybrid = 4.0 (x axis for solid line: Total number of revolution considered for the mean power coefficient).

Close modal

The start-up and power performances of a hybrid VAWT with a two-bladed Savonius and two-bladed Darrieus are estimated in this section. The effects of the attachment angle and the inner Savonius' tip speed ratio (varied Savonius' rotation speed) on the power performances of the hybrid VAWT are analyzed in this section. To further understand the flow physics of the hybrid VAWT, the Darrieus blade force variation is correlated with the wake patterns, which will provide insights into the hybrid VAWT design.

Power performances of single Savonius, single Darrieus, and hybrid VAWT with various tip speed ratios are shown in Fig. 7. The CFD simulations of Savonius have been validated against experimental results in Ref. 37. The numerical results of Darrieus are validated in  Appendix B. There is a lack of experimental investigation on the studied hybrid VAWT, but a similar numerical method and meshing strategy have been applied to the hybrid VAWT. The sensitivity study of the hybrid VAWT indicates that the simulation stability and convergence are sufficient. The single Savonius results from experiments are lower than those from CFD simulations. This can be attributed to the absence of 3 D effects (e.g., tip vortices), which are not discussed in this work.

FIG. 7.

Power coefficient of single Savonius, single Darrieus, and hybrid VAWT.

FIG. 7.

Power coefficient of single Savonius, single Darrieus, and hybrid VAWT.

Close modal

As shown in Fig. 7, the single Savonius can operate at a tip speed ratio lower than 1, but the single Darrieus has a higher power output at high tip speed ratios. As studied in Refs. 18 and 21, reducing the size of Savonius would increase the maximum power performance of the hybrid VAWT. In Fig. 7, the optimum tip speed ratio of the single Darrieus is about five times that of the single Savonius. So, the radius of the outer Darrieus part is selected as five times that of the inner Savonius part in the baseline hybrid VAWT to gain optimum power for the combined configuration as well as a good start-up performance.

The comparison of the torque coefficients for the single Savonius and Savonius in hybrid VAWT is shown in Fig. 8. The variations of torque coefficient with the tip speed ratio have the same trend: torque coefficient decreases with the increasing tip speed ratio. However, the torque coefficient of Savonius in hybrid VAWT is lower than that of single Savonius due to the velocity deficit caused by the rotor rotating. The rotation of the hybrid VAWT leads to a high induction to the flow field, so the performance of the Savonius (or Darrieus) part in the hybrid VAWT is different from that of the single rotor with the same operational conditions. Since the Darrieus is the major contributor to the power generation in the hybrid VAWT, its tip speed ratio remains at optimal condition, and the effect of Savonius' tip speed ratio on the performance of the hybrid VAWT is studied. Savonius' tip speed ratio is varied by changing the rotation speed of Savonius. Their flow features and blade forces are analyzed in Sec. III B 2.

FIG. 8.

Torque coefficients of single Savonius and Savonius in hybrid VAWT.

FIG. 8.

Torque coefficients of single Savonius and Savonius in hybrid VAWT.

Close modal

As the Savonius blade position at static conditions influences the static torque,38 the static torque of the Savonius blade in hybrid VAWT is investigated under varied angular positions. The static torque coefficients of the single Savonius and Savonius in hybrid VAWT are compared in Fig. 9. It is observed that the Savonius in hybrid VAWT has a similar static torque variation to the single Savonius. The static torque is positive at most phase positions except 50 ° 69 °. So, the hybrid VAWT can self-start at most positions.

FIG. 9.

Static torque coefficient of single Savonius and Savonius in hybrid VAWT.

FIG. 9.

Static torque coefficient of single Savonius and Savonius in hybrid VAWT.

Close modal

Regarding the hybrid VAWT, advantages are taken from two types of single rotors in the way of starting up easily and maintaining as high power as a single Darrieus. However, the presence of Savonius in hybrid VAWT would suppress power generation.21 This section aims to find out the correlation between torque variation and vortex dynamics with varied attachment angle and Savonius' tip speed ratio (varied Savonius' rotation speed) in one revolution.

1. Effect of attachment angle

To understand the vortex dynamics effects on the blade forces, a detailed blade–vortex interaction study is conducted in this section. The interaction scheme between Savonius' wake and Darrieus blade is varied by the attachment angle. The power coefficients of hybrid VAWT with four attachment angles are depicted in Fig. 10 at three tip speed ratios. It is observed that hybrid VAWT with γ = 60 ° and 90 ° generates a relatively high power coefficient regardless of the tip speed ratio.

FIG. 10.

Effect of λhybrid on the power coefficient of hybrid VAWT, R D R S = 5.

FIG. 10.

Effect of λhybrid on the power coefficient of hybrid VAWT, R D R S = 5.

Close modal

Taking an example of λ hybrid = 4.0, the torque contributions of the Darrieus and Savonius blades in the hybrid VAWT are discussed below. The effect of the attachment angle on the torque generation of the Darrieus and Savonius blades is shown in Fig. 11. It is observed that the torque variation trend of the Savonius blade in hybrid VAWT shows very little variation in different attachment angles, while that of the Darrieus blade in hybrid VAWT is highly distinguished at θ = 180 ° 360 °. The presence of the Savonius reduces the upwind Darrieus blade torque in hybrid VAWT due to the blockage effect. The torque coefficient in γ = 0 ° decreases from 0.10 to −0.013 in 30 °. Its gradient ( 0.10 + 0.013 30 = 0.0038) is larger than the other three cases, which are 0.0022, 0.0016, and 0.0013 for γ = 90 ° , 60 °, and 30 °, respectively. In addition, the difference between the maximum and the minimum torque coefficients in the second-half rotation of γ = 0 ° is the highest among the four cases. The rapid variation of the blade forces occurring in the second-half rotation may cause fatigue loads for the Darrieus blade.

FIG. 11.

Effect of the attachment angle on the torque generation of Darrieus and Savonius blades, λ hybrid = λ D = 4 , R D R S = 5. (a) Darrieus blade in a hybrid turbine, (b) Savonius blade in a hybrid turbine.

FIG. 11.

Effect of the attachment angle on the torque generation of Darrieus and Savonius blades, λ hybrid = λ D = 4 , R D R S = 5. (a) Darrieus blade in a hybrid turbine, (b) Savonius blade in a hybrid turbine.

Close modal

To study the effects of the attachment angle on the blade forces and correlate the vortex dynamics to the blade forces, the pressure distributions along the blade surface and the vorticity fields at different phase angles are depicted in Figs. 12 and 15, respectively. The difference between maximum and minimum pressure coefficients on the Darrieus blade is denoted by Δ C p. Its variation with attachment angles is shown in Fig. 13. Through the analysis of pressure distribution and vorticity field at θ = 180 ° , 210 °, and 330 ° (denoted by phases a, b, and f), blade 1 is not significantly affected by the perturbed region due to the presence of the Savonius part, so the pressure distributions and pressure differences slightly vary with the attachment angles. It indicates that the attachment angle shows little effect on the torque when the Darrieus blade is outside of the Savonius' wake region. This also indicates the benefits of the large radius ratio to the power performance of the hybrid VAWT. In Fig. 11(a), it is observed that the torque coefficient of the Darrieus blade in the hybrid VAWT increases with the phase angle ranging from θ = 180 ° to θ = 240 °. The pressure fields of the single Darrieus and hybrid VAWT at θ = 210 ° are shown in Fig. 14. The low-pressure region at the outer side of the Darrieus blade in hybrid VAWT is larger than that in single Darrieus. The increasing size of the low-pressure region at the outer side of the Darrieus blade is attributed to the vortex flow behind the Savonius rotor. The torque is observed to increase from θ = 180 ° to θ = 240 ° in Fig. 11(a), which can be attributed to the rotation of the Darrieus blade from the high-pressure region to the low-pressure region (Savonius' wake region).

FIG. 12.

Effect of the attachment angle on the pressure distribution along the Darrieus blade, λ hybrid = λ D = 4 , R D R S = 5, (a) θ = 180 °, (b) θ = 210 °, (c) θ = 240 °, (d) θ = 270 °, (e) θ = 300 °, and (f) θ = 330 °.

FIG. 12.

Effect of the attachment angle on the pressure distribution along the Darrieus blade, λ hybrid = λ D = 4 , R D R S = 5, (a) θ = 180 °, (b) θ = 210 °, (c) θ = 240 °, (d) θ = 270 °, (e) θ = 300 °, and (f) θ = 330 °.

Close modal
FIG. 13.

Variation of pressure difference with varied attachment angles at different phases, Δ C p = C p max C p min.

FIG. 13.

Variation of pressure difference with varied attachment angles at different phases, Δ C p = C p max C p min.

Close modal
FIG. 14.

Pressure fields of the single Darrieus and hybrid VAWT at θ = 210 °: (a) single Darrieus, (b) γ = 0 °, (c) γ = 30 °, (d) γ = 60 °, and (e) γ = 90 °.

FIG. 14.

Pressure fields of the single Darrieus and hybrid VAWT at θ = 210 °: (a) single Darrieus, (b) γ = 0 °, (c) γ = 30 °, (d) γ = 60 °, and (e) γ = 90 °.

Close modal

At θ 240 ° (phase c), blade 1 rotates into the shed vortex from the advanced Savonius blade. For γ = 0 ° and 90 °, the leading edge of blade 1 is approaching the shed vortex from the advanced Savonius blade. In Figs. 12(c) and 13, the pressure differences of γ = 0 ° and 90 ° at θ = 240 ° are larger than Δ C p of single Darrieus, leading to a higher torque generation compared to the single Darrieus. The pressure differences at γ = 30 ° and 60 ° are lower than the single Darrieus, but their torque coefficients are still higher than the single Darrieus. It is worth noting that forces and moments are output in their total and constituent components in OpenFOAM.29 In this work, the sum of pressure and viscous contributions to moments is equal to the blade torque. Because blade 1 interacts with the shed vortex of Savonius directly at γ = 0 ° and 90 ° as shown in Fig. 15(c), where the pressure contribution dominates the torque generation. While blade 1 at γ = 30 ° and 60 ° cannot directly interact with shed vortices, in which case the viscous contribution dominates the torque generation.

FIG. 15.

Effect of γ on vorticity fields at six phase angles, λhybrid=4,λS=0.8,RDRS=5: (a) θ=180°, (b) θ=210°, (c) θ=240°, (d) θ=270°, (e) θ=300°, and (f) θ=330° (Left to right: Single Darrieus, γ=0°,γ=30°,γ=60°,γ=90°).

FIG. 15.

Effect of γ on vorticity fields at six phase angles, λhybrid=4,λS=0.8,RDRS=5: (a) θ=180°, (b) θ=210°, (c) θ=240°, (d) θ=270°, (e) θ=300°, and (f) θ=330° (Left to right: Single Darrieus, γ=0°,γ=30°,γ=60°,γ=90°).

Close modal

After θ = 240 ° (after phase c), blade 1 goes through the large separated region behind the Savonius, where the Darrieus blade has a strong interaction with the Savonius' wake and a significant velocity deficit. Compared to the single Darrieus case, less power can be converted by the rotor.10 At θ = 270 ° (phase d), the pressure coefficients and the adverse pressure gradient at the leading edge decrease as the attachment angle increases. In Fig. 13, the pressure difference of the hybrid VAWT is lower compared to the single Darrieus. At this phase, the Darrieus blade torque drops to a certain amount and experiences a subsequent increase due to the flow perturbances of Savonius. It is observed that blade 1 interacts with the shed vortices from the two Savonius blades at γ = 30 ° 90 °. This interaction between blade 1 and wake makes the flow around the blade more perturbed so that the flow near the blade internal side differs much from the external side, in which case the pressure contribution dominates the torque generation. This leads to a torque decrease with the pressure difference decrease. The torque generation at γ = 30 ° 90 ° is more than γ = 0 °, but the pressure differences of the former three cases are lower than that of the latter case because blade 1 is situated in between the shed vortices from the advancing and advanced Savonius blades in the case of γ = 0 ° where viscous contribution dominates the torque generation. So, the attachment angle of 30 ° 90 ° can be used as a flow control guide to reduce power losses from downstream blades of the hybrid VAWT.

At θ = 300 ° (phase e), blade 1 is rotating out of the Savonius wake disturbances. The shed vortex from the advancing Savonius blade appears on the internal side of blade 1, and the pressure difference between the internal and the external sides of blade 1 increases compared to θ = 270 ° (phase d). The torque coefficient of blade 1 at phase e yields a higher magnitude than that at phase d. In Fig. 15(e), for cases where γ = 30 ° and 60 °, blade 1 is notably influenced to a greater extent by the nearby shed vortices originating from the advancing Savonius blade in comparison to cases where γ = 0 ° and 90 °. The pressure differences and torque coefficients at γ = 30 ° and 60 ° are higher, and the Darrieus blades have higher adverse pressure gradients compared to those at γ = 0 ° and 90 °.

Overall, the toque generations of the four attachment angles at θ = 240 ° and 210 ° are higher compared to the single Darrieus blade, shown in Fig. 11. Therefore, the shed vortex from the advanced Savonius blade increases the torque generation. This dependency of the torque increase on the shed vortex from the advanced Savonius blade is observed in various attachment angle cases.

2. Effect of Savonius' rotation speed

As the inner Savonius perceives a lower incoming flow during the rotation of the hybrid VAWT, the equivalent tip speed ratio for Savonius is different from the original tip speed ratio. To study the effect of Savonius' rotation on the hybrid VAWT, Savonius and Darrieus are assumed to be mounted in separate rotation axes. The relation between the original tip speed ratio and the power coefficient of the hybrid VAWT is shown in Table VI. It shows a slight power increase for the case with a lower λS. Therefore, slowing down the inner Savonius after start-up is beneficial for the power performance of the hybrid VAWT. Reducing the size of the inner Savonius leads to the same effect, but it is difficult to achieve during turbine operation. Apart from the mean power of the hybrid VAWT, the torque variation during rotation is also analyzed.

TABLE VI.

Power coefficient of hybrid VAWT with different rotation speeds of Savonius, λ hybrid = 4.0 , γ = 0 °.

λ S = 0.2 λ S = 0.4 λ S = 0.6 λ S = 0.8
c p hybrid  0.400 07  0.389 82  0.384 62  0.369 38 
λ S = 0.2 λ S = 0.4 λ S = 0.6 λ S = 0.8
c p hybrid  0.400 07  0.389 82  0.384 62  0.369 38 

The torque variations of blade 1 in hybrid VAWT and single Darrieus are compared in Fig. 16. The force fluctuation in the second-half revolution is investigated. The torque coefficient at λ S = 0.8 has the most rapid change compared to the other three cases. This is due to the large shedding frequency of Savonius's wake. So, there is a higher possibility of the occurrence of fatigue. From θ = 180 ° to θ = 240 °, the same trends of torque increase as the varied γ cases are observed due to the blade rotation from high-pressure region to low-pressure region (Savonius' wake region).

FIG. 16.

Effect of Savonius' tip speed ratio on torque variation of the top Darrieus blade in hybrid VAWT and single Darrieus, λ hybrid = λ D = 4 ( γ = 0 ° , λ S = Ω S R S U ).

FIG. 16.

Effect of Savonius' tip speed ratio on torque variation of the top Darrieus blade in hybrid VAWT and single Darrieus, λ hybrid = λ D = 4 ( γ = 0 ° , λ S = Ω S R S U ).

Close modal

To analyze flow physics in the second-half revolution of hybrid VAWT with varied λS, six angular positions from 180 ° to 330 ° are denoted by phases a to f. The pressure distribution along the blade surface, the pressure difference between the maximum and minimum values, and the vorticity field around the rotor at the six phases are shown in Figs. 17–19, respectively. The torque coefficients of blade 1 in the single Darrieus rotor at θ = 180 ° and 330 ° (phases a and f) are similar to those in the hybrid VAWT when varying the rotation speed of the inner Savonius because blade 1 only interacts with the wake shedding from the Darrieus blades, with no disturbances introduced from the Savonius wake. At θ = 210 ° (phase b), blade 1 rotates toward the shed vortex shed from the advanced Savonius blade tip. As the size of the shed vortex blob grows with the decreasing rotation speed, the pressure difference between the internal and external sides of blade 1 and its torque coefficient at λ S = 0.2 is higher compared to that at λ S = 0.4−0.6. At θ = 240 ° (phase c) λ S = 0.6 and 0.8, the shed vortex from the advanced Savonius blade appears at the internal side of blade 1, resulting in a stronger adverse pressure gradient and larger pressure difference at the leading edge. The torque generation is also higher than λ S = 0.2 and 0.4.

FIG. 17.

Effect of λS on the pressure distribution along the Darrieus blade, λ hybrid = λ D = 4 , R D R S = 5: (a) θ = 180 °, (b) θ = 210 °, (c) θ = 240 °, (d) θ = 270 °, (e) θ = 300 °, and (f) θ = 330 °.

FIG. 17.

Effect of λS on the pressure distribution along the Darrieus blade, λ hybrid = λ D = 4 , R D R S = 5: (a) θ = 180 °, (b) θ = 210 °, (c) θ = 240 °, (d) θ = 270 °, (e) θ = 300 °, and (f) θ = 330 °.

Close modal
FIG. 18.

Variation of pressure difference with varied Savonius' tip speed ratios at different phases, Δ C p = C p max C p min.

FIG. 18.

Variation of pressure difference with varied Savonius' tip speed ratios at different phases, Δ C p = C p max C p min.

Close modal
FIG. 19.

Effect of λS on vorticity fields at six phase angles, λhybrid=4,γ=0°,RDRS=5: (a) θ=180°, (b) θ=210°, (c) θ=240°, (d) θ=270°, (e) θ=300°, and (f) θ=330° (Left to right: Single Darrieus, λS=0.2,λS=0.4,λS=0.6,λS=0.8).

FIG. 19.

Effect of λS on vorticity fields at six phase angles, λhybrid=4,γ=0°,RDRS=5: (a) θ=180°, (b) θ=210°, (c) θ=240°, (d) θ=270°, (e) θ=300°, and (f) θ=330° (Left to right: Single Darrieus, λS=0.2,λS=0.4,λS=0.6,λS=0.8).

Close modal

At θ = 270 ° (phase d), blade 1 is behind the Savonius and experiences a large velocity deficit. The pressure difference between the internal and external sides of the leading edge reduces, leading to the reduction of torque generation. λ S = 0.8 has more power losses than the other cases due to Savonius' high induction to the flow field. So, reducing the rotation speed of the inner Savonius would decrease the power losses downstream. Compared to the single Darrieus, the pressure differences of blade 1 in hybrid VAWT at four Savonius' tip speed ratios are lower, shown in Figs. 17 and 18. It is observed that the pressure difference decreases on the order of single Darrieus, λ S = 0.4 , λ S = 0.8 , λ S = 0.2 , λ S = 0.6, but the blade torque decreased on the order of single Darrieus, λ S = 0.4 , λ S = 0.2 , λ S = 0.6 , λ S = 0.8. The outlier λ S = 0.8 may be because the pressure contribution cannot dominate the blade torque generation. At θ = 300 ° (phase e), blade 1 rotates outward of the shed vortex from the advancing Savonius blade. At λ S = 0.4, the near vortex has a bigger impact on blade 1 compared to the other Savonius' tip speed ratios, leading to the highest Δ C p and cq at θ = 300 °.

The effects of the attachment angle and Savonius' rotation speed on the blade–vortex interaction and performances of a hybrid VAWT have been thoroughly investigated in the present paper. The studies were conducted employing a computational fluid dynamic approach. Four attachment angles for the hybrid VAWT and four rotation speeds for the inner Savonius have been evaluated systematically. Several conclusions can be drawn from the paper:

  1. The two-bladed Savonius without the gap distance can self-start at most phase angles except 50 ° 69 °.

  2. At γ = 60 ° and 90 °, the power coefficient of the hybrid VAWT is larger compared to 0 ° and 30 °.

  3. In the downwind part, the torque variation is dependent on the blade–vortex interaction. The torque coefficient of the Darrieus blade in hybrid VAWT is higher than that in the single Darrieus, while the blade interacts with the shed vortex from the advanced Savonius blade. This dependency is observed in varied attachment angles and varied tip speed ratio cases.

  4. The effect of Savonius' rotation on the performance of the hybrid VAWT is studied. The results indicate that slowing down the Savonius in hybrid VAWT leads to less induction to the flow field and more available energy downstream. Lower λS is beneficial to the power performance of the Darrieus part in the hybrid VAWT regardless of the complex mechanics of the rotation axis.

  5. From θ = 180 ° to 240 °, the blade torque increases with the increasing phase angle because the Darrieus blade rotates from high-pressure region to low-pressure region (Savonius' wake region), leading to the torque increase in the rotation direction.

  6. When the Darrieus blade does not interact directly with shed vortices, e.g., blade 1 is situated in between vortices at θ = 270 ° , λ S = 0.8, the viscous contribution would dominate the torque generation.

  7. A rapid force variation occurs when the Darrieus blade interacts with the Savonius' wake. The Darrieus blade in hybrid VAWT with γ = 0 ° and λ S = 0.8 has the largest gradient of force drop, which might lead to fatigue for the hybrid VAWT.

From the power and start-up performance point of view, it is suggested to apply an attachment angle of 60 ° or 90 °, and low λS of 0.2 to improve the rotor performance.

Exploring blade–vortex interactions in Darrieus–Savonius combined vertical axis wind turbines offers a promising research avenue. By characterizing flow patterns and linking them with blade torque variations, we gain insights that can improve turbine performance. This study illuminates the complex vortex dynamics that shape the behavior of the hybrid VAWT, guiding design enhancements and operational strategies. These insights inform design improvements and operational strategies, advancing both theoretical understanding and practical renewable energy solutions. This study uses 2 D simulation. To extend the comprehensive study of vortex dynamics in hybrid VAWT, 3 D effects can be taken into account in future work. The studied baseline hybrid VAWT will be validated against published experimental results. It may not be the optimal case, but the correlation of vortex dynamics and blade torque can provide preliminary insights to improve its performance. A parametric study can be conducted to further extend this work. Different rotation speeds of Darrieus and Savonius parts increase the complexity of the design for the rotation axis of the hybrid VAWT. Investigating the design approach for the rotation axis of the hybrid VAWT would facilitate the control of the Savonius component's operation, enabling adjustments to enhance the start-up and power performances of the hybrid VAWT.

Jingna Pan gratefully acknowledges financial support from China Scholarship Council (Grant No. 201906450032).

The authors have no conflicts to disclose.

Jingna Pan: Conceptualization (equal); Formal analysis (equal); Investigation (equal); Methodology (equal); Validation (equal); Visualization (equal); Writing – original draft (equal); Writing – review & editing (equal). Carlos J. Ferreira: Conceptualization (equal); Project administration (equal); Supervision (equal); Writing – review & editing (equal). Alexander van Zuijlen: Conceptualization (equal); Project administration (equal); Supervision (equal); Writing – review & editing (equal).

The data that support the findings of this study are available from the corresponding author upon reasonable request.

2D

Two-dimensional

3D

Three-dimensional

AMI

Arbitrary mesh interface

CFD

Computational fluid dynamics

Cp

Pressure coefficient

cp

Power coefficient

c p max

Peak power coefficient

cq

Torque coefficient

e

Gap width

k

Turbulent kinetic energy

PIMPLE

PISO-SIMPLE

PISO

Pressure Implicit with Splitting of Operators

p

Pressure

R1

Semi-major axis of the elliptic Savonius blade

R2

Semi-minor axis of the elliptic Savonius blade

RD

Darrieus rotor radius

Rhybrid

Radius of hybrid VAWT

RS

Savonius rotor radius

SIMPLE

Semi-Implicit Method for Pressure Linked Equations

SST

Shear stress transport

TSR

Tip speed ratio

URANS

Unsteady Reynolds-averaged Navier–Stokes

U

Inflow velocity

u

Sum of the mean velocity and the fluctuating velocity

u τ

Friction velocity

VAWT

Vertical axis wind turbine

w

Turbulent dissipation rate

x

Coordinate in the horizontal axis

y

Coordinate in the vertical axis

γ

Attachment angle

Δ C p

Difference of maximum and minimum pressure coefficients

Δ y 1

Distance between the cell and the nearest wall

ε

Overlap ratio of Savonius blades

θ

Phase angle

λhybrid

Tip speed ratio of hybrid VAWT

λmax

Maximum tip speed ratio of the rotor

λmin

Minimum tip speed ratio of the rotor

λS

Tip speed ratio of Savonius

μ

Dynamic viscosity

μt

Turbulent viscosity

υ

Kinematic viscosity

ρ

Flow density

Φ

Arc angle of the Savonius blade

Ω

Rotational speed

ω

Vorticity

2D

Two-dimensional

3D

Three-dimensional

AMI

Arbitrary mesh interface

CFD

Computational fluid dynamics

Cp

Pressure coefficient

cp

Power coefficient

c p max

Peak power coefficient

cq

Torque coefficient

e

Gap width

k

Turbulent kinetic energy

PIMPLE

PISO-SIMPLE

PISO

Pressure Implicit with Splitting of Operators

p

Pressure

R1

Semi-major axis of the elliptic Savonius blade

R2

Semi-minor axis of the elliptic Savonius blade

RD

Darrieus rotor radius

Rhybrid

Radius of hybrid VAWT

RS

Savonius rotor radius

SIMPLE

Semi-Implicit Method for Pressure Linked Equations

SST

Shear stress transport

TSR

Tip speed ratio

URANS

Unsteady Reynolds-averaged Navier–Stokes

U

Inflow velocity

u

Sum of the mean velocity and the fluctuating velocity

u τ

Friction velocity

VAWT

Vertical axis wind turbine

w

Turbulent dissipation rate

x

Coordinate in the horizontal axis

y

Coordinate in the vertical axis

γ

Attachment angle

Δ C p

Difference of maximum and minimum pressure coefficients

Δ y 1

Distance between the cell and the nearest wall

ε

Overlap ratio of Savonius blades

θ

Phase angle

λhybrid

Tip speed ratio of hybrid VAWT

λmax

Maximum tip speed ratio of the rotor

λmin

Minimum tip speed ratio of the rotor

λS

Tip speed ratio of Savonius

μ

Dynamic viscosity

μt

Turbulent viscosity

υ

Kinematic viscosity

ρ

Flow density

Φ

Arc angle of the Savonius blade

Ω

Rotational speed

ω

Vorticity

The mesh independence study is conducted to determine the appropriate mesh size for OpenFOAM simulations of VAWT. A single-stage two-bladed semicircular Savonius rotor without gap width23 is arranged in a computational domain of 30 R S × 50 R S. The distance between the inlet and the Savonius rotation axis is 15 R S. The inflow wind speed is 7 m/s with a corresponding rotation speed of 11.2 rad/s. Three sets of meshing with varying mesh resolutions were generated with the first row's cell height of 0.046, 0.051, and 0.056 mm and the growth factor of 1.2. In this study, the first row's cell height was calculated with the premise of the dimensionless wall distance y + = ρ u τ · Δ y 1 μ39 lower than one. The three sets of mesh are named coarse mesh, medium mesh, and fine mesh, with the total numbers of mesh being around 1.77 × 105, 1.83 × 105, and 1.90 × 105 and blade discretization of 296, 312, and 332, respectively. Pave meshing scheme is applied for all surfaces. The discretization of the inlet and outlet boundaries uses a double-sided successive ratio of 0.95 to refine the wake region. The maximum values of the calculated y+ occur at the convex side of blades with an average value below one. This indicates an effective meshing strategy.40 The power coefficients from the three mesh configurations are shown in Table VII. The values of the maximum skewness for the three mesh configurations are 0.72, 0.73, and 0.66, respectively. The medium and coarse mesh configurations differ by 0.15% and 0.57% from the fine mesh, which indicates that the medium mesh yields reasonably accurate results. In order to properly control convergence, the simulation residual was set as 1 × 10−5. Among all the simulations in this work, the interpolation scheme has a second order of accuracy. The azimuthal increment of 0.24 ° / step was chosen according to the published time step independence study for a Savonius rotor with a similar tip speed ratio, wind speed, and diameter.41,42 The simulation time is over 12 revolutions. The power coefficients of Savonius from different revolutions are shown in Fig. 20. It demonstrates that there is little change in the power coefficient between the sixth and seventh revolutions. So, the simulation results are considered as converged from the sixth revolution.

TABLE VII.

Power coefficient of Savonius rotor from different mesh configurations, λ S = 0.8 , U = 7 m/s.

Mesh Fine Medium Coarse
Number of cells  1.90 × 105  1.83 × 105  1.77 × 105 
Blade discretization  332  312  296 
First row's cell height (mm)  0.056  0.051  0.046 
Number of layers  26  28  30 
Power coefficient cp  0.2614  0.2618 (+0.15%)  0.2629 (+0.57%) 
Mesh Fine Medium Coarse
Number of cells  1.90 × 105  1.83 × 105  1.77 × 105 
Blade discretization  332  312  296 
First row's cell height (mm)  0.056  0.051  0.046 
Number of layers  26  28  30 
Power coefficient cp  0.2614  0.2618 (+0.15%)  0.2629 (+0.57%) 
FIG. 20.

Convergence of power coefficient for the Savonius rotor at 0.24 ° / step, λ S = 0.8 , U = 7 m/s, medium mesh configuration.

FIG. 20.

Convergence of power coefficient for the Savonius rotor at 0.24 ° / step, λ S = 0.8 , U = 7 m/s, medium mesh configuration.

Close modal

A mesh independence study of a two-bladed Darrieus rotor with the same geometrical parameter as the Darrieus part in the hybrid VAWT and the same meshing strategy as the Savonius rotor is conducted. The rotation axis of the Darrieus rotor is 10 R D from the inlet and 24 R D from the outlet. To optimize the limits of cell size, three mesh configurations (coarse, medium, fine) are simulated with cell numbers of 1.38 × 105, 1.58 × 105, and 2.02 × 105, respectively. The values of the maximum skewness for the three mesh configurations are 0.71, 0.69, and 0.51, respectively. The simulation residual is set as 1 × 10−5. An azimuthal increment of 0.24°/step is seen as sufficient according to the convergence studies of Rezaeiha et al.43 and Edwards et al.44 The average y + over the blade surface is below one with the maximum value at the leading edge.40 The power coefficients from different mesh configurations and mesh details are shown in Table VIII. The convergence of the power coefficient from different revolutions is depicted in Fig. 21. The results show that the medium mesh configuration yields reasonably accurate results with a −2.83% difference from the fine mesh configuration. The power coefficient of the Darrieus rotor is seen as converged from the twelfth revolution. The normal force coefficient of the single Darrieus is validated against the experimental results,30 shown in Fig. 22. The results indicate a comparable agreement between the numerical and experimental results.

TABLE VIII.

Power coefficient of the Darrieus rotor from different mesh configurations, λ D = 4.0 , U = 4.01 m/s.

Mesh Fine Medium Coarse
Number of cells  2.02 × 105  1.58 × 105  1.38 × 105 
Blade discretization  92  70  62 
First row's cell height (mm)  0.018  0.022  0.018 
Number of layers  28  28  30 
Power coefficient cp  0.4343  0.4220 (−2.83%)  0.3604 (−17.0%) 
Mesh Fine Medium Coarse
Number of cells  2.02 × 105  1.58 × 105  1.38 × 105 
Blade discretization  92  70  62 
First row's cell height (mm)  0.018  0.022  0.018 
Number of layers  28  28  30 
Power coefficient cp  0.4343  0.4220 (−2.83%)  0.3604 (−17.0%) 
FIG. 21.

Convergence of power coefficient for the Darrieus rotor at 0.24 °/step, λ D = 4.0 , U = 4.01 m/s, medium mesh configuration.

FIG. 21.

Convergence of power coefficient for the Darrieus rotor at 0.24 °/step, λ D = 4.0 , U = 4.01 m/s, medium mesh configuration.

Close modal
FIG. 22.

Comparison of numerical and experimental normal force coefficients of the Darrieus blade.

FIG. 22.

Comparison of numerical and experimental normal force coefficients of the Darrieus blade.

Close modal
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