Journal Pre-proof

Rotation improvement of vertical axis wind turbine by offsetting pitching angles and
changing blade numbers

Xuejing Sun, Jianyang Zhu, Zongjin Li, Guoxing Sun

PII: S0360-5442(20)32284-2
DOI: https://doi.org/10.1016/j.energy.2020.119177
Reference: EGY 119177

To appear in: Energy

Received Date: 23 June 2020

Revised Date: 22 October 2020
Accepted Date: 26 October 2020

Please cite this article as: Sun X, Zhu J, Li Z, Sun G, Rotation improvement of vertical axis wind
turbine by offsetting pitching angles and changing blade numbers, Energy, https://doi.org/10.1016/
j.energy.2020.119177.

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Credit author statement
Xuejing Sun: Modelling and simulation, Calculations, writing. Jianyang
Zhu: Conceptualization and writing. Zongjin Li: Conceptualization and revision. Guoxing
Sun: discussions, writing.

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 Rotation improvement of vertical axis wind
turbine by offsetting pitching angles and changing
blade numbers

Xuejing Sun a, Jianyang Zhu b*, Zongjin Li a, Guoxing Sun a

a
Institute of Applied Physics and Materials Engineering, University of Macau, Avenida da
Universidade, Taipa, Macau S.A.R China

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b
Institute of Robotics and Intelligent Systems, Wuhan University of Science and Technology,
Wuhan, 430081, P.R.China

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*Corresponding author – zhujy@wust.edu.cn

ABSTRACT
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To improve the power extraction performance and self-starting characteristics of the vertical axis wind
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turbine (VAWT), the effect of offsetting pitching angles and blade numbers on the performance of a
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wind-induced rotation VAWT has been systematically investigated. Different from the conventional numerical
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and experimental approach, the rotation velocity of the turbine is driven by the aerodynamic torque of the blade
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in the present study. The flow around the turbine was simulated using Fluent 6.3 code, and the governing

equation of the turbine’s rotation was coupled to the code through UDF. The optimized pitching angle β was

found to be of -4°, at which the maximum 5.89 and 5.14 times average power increasing were achieved for the

turbine with 5-blades and 3-blades, respectively. Moreover, shorter self-starting time was also observed for the

turbine with β=-4° at larger wind velocity (> 9m/s). In addition, the influence of the blade number on the

performance of the turbine depended on the wind velocity. The analysis of the flow field of the turbine showed

that the offsetting pitching angle and blade number could suppress or delay the vortex separation, and therefore

improve the overall performance of the turbine.

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Keywords: Offsetting pitching angle; Blade number; Self-starting characteristic; Mean Power coefficient;

Vortex separation

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1 introduction

To harvest wind energy in the urbane environment, straight bladed Darrieus vertical axis

wind turbine (S-VAWT) is gaining more and more attention in recent years owing to its

advantages of independent of the wind direction, friendly to the environment, low cost, and high

adaptability to unsteady turbulence flow. However, the low efficiency and poor self-starting are

the significant challenges for further development of the S-VAWT. Thus, improving the

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performance of the S-VAWT is of great importance for energy harvest and sustainable

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development [1-5].
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Different from horizontal axis wind turbine (HAWT), the angle of attack of the S-VAWT’s
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blade changes frequently during the revolution of the turbine, which makes the flow around the
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S-VAWT becomes very complex, particularly, when the turbine is working at low wind speed;
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vortex separation could be initiated when the blade is at a larger angle of attack. This phenomenon
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is known as dynamic stall, reducing the lift force of the blade sharply, and deteriorating the overall
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performance of the turbine. Therefore, to improve the performance of the S-VAWT, it is essential

to delay or eliminate the flow separation around the blade. Changing the pitching angle is a

common and feasible method to address this problem.

Rezaeiha et al [6] performed a numerical study to investigate the effect of fixed offset

pitching angles on the performance of a three-blade S-VAWT. The range of the angles was set

from -7° to +3° with an interval of 1°. The turbine with fixed offset pitching angle -2° was found

to be the optimum, and compared with the turbine with non-offset pitching angle, the power

coefficient was increased by 6.6%. Guo et al. [7] employed the pitching control method to

improve the power extraction performance of a S-VAWT. The analysis indicated that an optimal

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fixed offset pitching angle of the turbine was of -1°, at which a 4.5% more power coefficient was

achieved compared to the turbine without offset pitching angle. Armstrong et al. [8] conducted an

experimental study to investigate the aerodynamic performance of a 3-blades S-VAWT with

different fixed offset pitching angle in a wind tunnel with a wind speed of 10m/s. It was concluded

that the fixed offset pitching angle could influence the performance of the turbine significantly; if

the turbine has a fixed offset pitching angle at a value of -6°, the vortex separation of the blade

could be delayed, consequently, leading to the power coefficient of the turbine increasing. Zhang

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et al. [9] analyzed the effect of the blade pitching angle on the aerodynamic performance of a

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3-blades S-VAWT by using both experimental and numerical methods. The aerodynamic force and
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power coefficient of the turbine with different fixed offset and variable pitching angles were
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compared. It was reported that a power coefficient of 19.3% increasing was achieved by the

optimized pitching angle. Li et al. [10] proposed an optimized blade pitching control method
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based on a genetic algorithm and computational fluid dynamics simulation for an S-VAWT. It
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demonstrated that the optimized blade pitches could increase the average power coefficients of the
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turbine under eight different tip speed ratios by the range of 0.177 to 0.317, respectively.

Based on the aforementioned studies, it is concluded that the performance of the S-VAWT

can be improved significantly by changing the pitching angle of the blade. However, most of the

aforementioned studies on the pitching angle of the S-VAWT were set with a fixed angular

velocity of the turbine, which is not accurate as the rotation of the turbine induced by the wind

varies with the wind speed. Besides, to date, most conclusions of the effects of the blade pitching

angle on the S-VAWT were based on the 3-blades S-VAWT only. Therefore, in this work, the

performance of the wind-induced rotation of S-VAWT with different fixed pitching angles and

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blade numbers are analyzed. To this end, a numerical coupling model was established and

validated to simulate the interaction between the wind flow and the induced turbine rotation, and

both the starting stage and steady rotation stage of the turbine have been analyzed.

2 Physical model and parameters definition

According to the literature, three and five are the typical blade numbers used for the S-VAWT.

In this study, the effect of the blade pitching angle on the wind-induced rotation was also

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evaluated with two different blade numbers, as shown in Fig.1.

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Fig.1 Two typically S-VAWTs (a) Wind turbine with 3-blades. One extra blade is used to show force and other
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symbols. (b) Wind turbine with 5-blades.
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Based on the development of bowl-shaped floating S-VAWT in the University of Macau [11],
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NACA0018 is employed to represent the section profile of the blade. The chord length is fixed at
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c=0.15m. This study focused on two dimensions analysis with the span of the blade of 1.0 m. To
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reduce the initial rotation moment, the hollow section blade was adopted with sufficient wall

thickness to satisfy the strength requirement. The pitching center and mass center of the blade

coincide at the same point, which was set at c/3 from the leading edge of the NACA0018 airfoil.

The mass of each blade is 0.17kg and the rotation radius of the turbine was fixed at 0.5m. Hence,

the initial rotation moment of the S-VAWT with 3-blades and 5-blades are 0.1278 kg·m2 and

0.2129 kg·m2 respectively, and the parameters related to the blade in this work are summarized in

Table 1.

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 Table 1. Parameters of the blade

As compared with the variable pitching method, the fixed offset pitching method is a more

simple and effective way to be applied for real S-VAWT designing. Therefore, in this work, seven

fixed offset pitching angles (β, from -8°to 4°with an interval of 2°) were employed to explore the

effect of the blade pitching angle on the performance of the S-VAWT. The positive or negative β is

determined by the right-handed spiral rule.

Continuing our previous work [12-13], the rotation of the turbine is induced by the wind in

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this study. According to the Newton's second law, the governing equation of the wind-induced

rotation of S-VAWT is defined as: -p

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Iθ&& + Cload θ& = Qwind (1)
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where, I is the initial rotation moment, Cload is the external loading coefficient, which represents

the resistance resulted from the electricity generator and the bearing friction. Once the turbine was
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developed, Cload is constant, and it can be determined by the experiment, however, in this work,
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for both the S-VAWT with 3-blades and 5-blades, Cload is fixed at the value of 0.05 kg·m2/s; θ is
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the azimuth angle; θ&& is the angular acceleration; θ& = ω is the turbine rotation velocity; Qwind is

the aerodynamic torque, which can be calculated by:

Qwind = Ft R (2)

here, Ft is the tangential force, as shown in Fig.1, and Ft is described as:

 Ft   sin α − cos α   FL 
F  =    (3)
 n  cos α sin α   FD 

where, Fn is the normal force, FL is the lift force, FD is the drag force, α is the angle of the attack.

It is well know that the lift force and drag force of the blade are determined by the angle of the

attack α. Based on the relationship illustrated in Fig.1, the angle of the attack depends on

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tangential velocity ωR and free stream wind speed U∞:

 cos θ 
α = tan −1  +β (4)
 λ + sin θ 

herein, λ=ωR/U∞ is the tip speed ratio. Note that for the wind-induced rotation of S-VAWT, λ and

ω are not constant, they are varying with the azimuth angle.

At last, two critical parameters to evaluate the performance of the turbine are introduced: the

power coefficient and mean power coefficient:

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Qwind ω
CP = (5)
0.5 ρU ∞3 c

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T
C P = ∫ C P dt (6)
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herein, T=2π/ω is the rotation cycle.
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Based on equation (1)-(5), it can be found that the power coefficient of the turbine is dependent on

angle of the attack, therefore, fixed offset pitching angles are adjusted to produce a better angle of
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attack and maximize the mean power coefficient of the turbine.

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3 Methodology
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3.1 Method for solving wind-induced rotation S-VAWT

Based on the free-stream wind speed U∞=3-12m/s and the chord length of the blade c=0.15m,

the Reynolds number of the turbine (Re=U∞c/ν, ν is the fluid kinematic viscosity) in this work is at

3.08×104 to 1.23×105, and the match number is less than 0.04. Therefore, the fluid around the

turbine is assumed as incompressible and turbulent, the governing equations for fluid flow around

the S-VAWT are Unsteady Reynolds Average Navier-Stokes (URANS) equation that is given as:

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 ∂ui
=0
∂xi
(7)
∂ui ∂ 1 ∂p ∂  ∂ui ∂u j  ∂
+ (ui u j ) = − + ν ( + )  + (−ui' u 'j )
∂t x j ρ ∂xi x j  ∂x j ∂xi  x j

where u represents the velocity vector, the superscript ′ and ¯ represent the fluctuation and mean

values, i and j are the subscript, ν is the fluid kinematic viscosity, p is the pressure, −ui' u 'j is the

Reynolds stress. To enclose equation (7), one equation Spalart-Allmaras (S-A) turbulence model

which has been widely used to simulate the S-VAWT [14-16], is employed to solve Reynolds

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stress in equation(7). The SIMPLEC solution scheme is used for pressure-velocity coupling, the

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first order implicit scheme is employed for the temporal term discretization, and the second-order
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upwind scheme is used for pressure and momentum discretization. Fluent 6.3 with double
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precision computation mode is used as the fluid field solver.
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In order to solve equation (1), second-order finite difference method [17] is employed, the
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angular acceleration and velocity of the turbine at t are given as:

θ t + ∆t + θ t − ∆t − 2θ t
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θ&&t =
∆2 t (8)
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θ t + ∆t − θ t − ∆t
θ =
&t
2∆t

By substituting equation (8) into equation (1), we can get the time progressive form of azimuth

angle of the turbine:

4I −2I + Cload ∆t t −∆t 2I t

Qwind
θ t +∆t = θt + θ + ∆ 2t (9)
2I + Cload ∆t 2I + Cload ∆t 2I + C load ∆t I

For the wind-induced rotation VAWT, equation (7) and equation (9) have to be solved

simultaneously. To this end, a coupling method is developed, and the calculation flowchart is

summarized in Fig.2. Once the calculation is initialized, the aerodynamic torque Qwind over the

rotation center of the turbine is integrated by solving equation (7). Then the azimuth angle of the

turbine under the obtained torque at this time is calculated by equation (9), which is embedded in

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Fluent 6.3 by using the User Defined Function (UDF) with C language. In the next time step, the

azimuth angle of the turbine is updated, which is realized by a dynamic mesh technique, and a

new boundary of the wind flow field is formed. By repeating these procedures in the calculation,

the wind induced rotation VAWT starting from stationary and stopping at stable periodic rotation

is realized.

Fig.2 The calculation flowchart of the wind induced rotation turbine.

3.2 Computation domain and grid generation

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A quadrilateral-type topology was employed for the computational domain with one outer

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stationary and three or five inner rotation sub-domains, as shown in Fig.3. According to the
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literature [18-19], the domain is of 10D upstream, 20D downstream, and 10D in up and low
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boundary, the numerical results are independent on the domain sizes. Therefore, on the left side,
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the velocity inlet boundary condition was set for a straight line located at 10D away from the
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rotation center of the turbine; on the right side, the pressure outlet boundary condition was set for
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a straight line located at 20D away from the rotation center of the turbine; the upper and lower
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boundaries, both located 10D away from the rotation center of the turbine, were set as the

symmetry boundaries. The blade surfaces were defined as no-slip wall boundaries. The one outer

domain was stationary, the three or five inner domains and its surrounding blades were governed

by the turbine’s passive rotation, whereas the meshes around the blades were not changed during

the rotation.

To capture the detailed flow distribution around the blade, fifteen structured mesh boundary

layers were used around the blade surface at both the leading and trailing edges. Based on the

NASA y+ calculator (http://www.pointwise.com/yplus/), the first mesh cell distance from the

blade was set 0.00015c, so that the y+ was controlled to be less than 1.0.

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 Fig. 3 The computational domain and mesh structure.

4 Validation
The grid sensitivity study was carried out at fixed offset pitching angles β=-6° and U∞=6m/s

for a VAWT with either 3-blades or 5-blades. Three different grid modes were employed for each

type of VAWT as shown in Table 2. The induced rotation velocity versus flow time and mean

power coefficient for the two types of VAWT with different grid modes are illustrated in Fig. 4. As

shown in Fig.4 (a) that both the induced rotation velocity and mean power coefficient decreased

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with the increase of grid cells. However, the differences of the induced rotation velocity and mean

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power coefficient between the grid mode2 and grid mode3 are significantly smaller than those
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between grid mode1 and grid mode3. Therefore, the solution is converged at grid mode2 of grid
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resolution for the VAWT with 3-blades, and it is used for the following numerical study. For the

turbine with 5-blades, as it is presented in Fig.4 (b), the mean power coefficient is almost identical
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for the three covered grid modes; however, the differences of the curves of the induced rotation
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velocity between grid mode5 and grid mode6 are significantly smaller than those between grid
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mode4 and grid mode5, therefore, grid mode5 is used for the following numerical study on the

VAWT with 5-blades.

Table 2 Details of the grid discrete form for the VAWT

Fig.4 The induced rotation velocity versus flow time and mean power coefficient for the two types VAWT with
different grid modes (a) VAWT with 3-blades (b)VAWT with 5-blades.

Secondly, simulations were performed with iterate time step sizes of 7.5×10-5s, 1.5×10-4s,

and 3.0×10-4s for the turbine having either 3-blades or 5-blades to investigate the influence of

iterative time step size on the sensitivity of the computation. As shown in Fig.5 that the most

significant difference in the mean power coefficient is less than 1.98% and 1.21% for the turbine

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with 3-blades and 5-blades respectively. The induced rotation velocity versus flow time also

shows little variation between the iterate time step sizes of 7.5×10-5s and 1.5×10-4s. Therefore, the

moderate iteration time step of 1.5×10-4s is used for the following calculating.

Fig.5 The induced rotation velocity versus flow time and mean power coefficient for the two types VAWT with
different iterate time step sizes(a) VAWT with 3-blades (b)VAWT with 5-blades.

Finally, the validation is performed for the developed numerical strategy on its performance in

simulation of the wind-induced VAWT rotation. The wind-induced VAWT rotation with 3-blades

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was simulated in our previous study [20]. The results were compared to the literature data by

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Rainbird [21] and Asr et al. [22]. The tip speed ratio versus flow time is shown in Fig.6, where T*
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is the time for the turbine from stationary to the steady passive rotation, the present numerical
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results with different turbulence models are also presented for comparison. It can be seen that the
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numerical results reported by Asr et al. over-predicted the experiment data, while the numerical
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result with k-epsilon turbulence under-predicted the data; on the other hand, the numerical result
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with Spalart-Allmaras (S-A) turbulence model agreed with the experimental data very well,
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therefore, (S-A) turbulence model is employed for this simulation.

Fig.6 The tip speed ratio versus flow time during the startup: comparison of the present simulation data to
numerical data calculated by Asr et al. [21] and experimental data reported by Rainbird et al. [22].

5 Results and discussion

The performance of the wind-induced rotation VAWT is systematically reported in this

section. The main interest of this work is the influence of blade numbers (3-blades and 5-blades)

and fixed offset pitching angles (range from -8° to 4° with an interval of 2°) on the performance of

the turbine. In order to investigate the performance of the turbine under different free stream wind,

four different wind speeds (U∞=3, 6, 9 and 12m/s) were considered.

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5.1 Power extraction performance

5.1.1 Mean power coefficient generating

In this section, the mean power coefficient of the VAWT with 3-blades or 5-blades were

investigated as the function of free-stream wind velocity and offsetting pitching angle. Compared

with the zero offsetting pitching angle turbine, regardless of the adopted wind velocity, the mean

power coefficient for both the turbines with 3-blades or 5-blades was enhanced when a negative

offsetting pitching angle is applied (Fig.7). The maximum 5.89 times increase was achieved at

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U∞=6m/s, β=-4° for the turbine with 5-blades. Similarly, the maximum 5.14 times increase was

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achieved at U∞=9m/s, β=-4° for the turbine with 3-blades. The increase in the mean power
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coefficient is significant in applying fixed offsetting pitching angles.
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Moreover, Figure 7 also shows that the blade numbers can influence the mean power

coefficient significantly. For the turbine at low free-stream wind velocity (U∞=3 and 6m/s), the
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mean power coefficient of the turbine with 5-blades was larger than the turbine with 3-blades.
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However, when the free-stream wind velocity is higher (U∞=9 and 12m/s), the results are opposite.
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It is interesting to note that even for the turbine with 5-blades, no matter the wind velocity is, the

optimized fixed offset pitching angle was fixed at a value of β=-4°, for the turbine has 3-blades,

the optimized fixed offsetting pitching angle was varying with the wind velocity. The optimized

fixed offsetting pitching angle initially increases with the wind velocity increase, then stabilized at

a value of β=-4°. This implies that when the free-stream wind velocity is low, the turbine with

5-blades is more suitable.

Fig.7 The power extraction performance of the VAWT with different fixed offset pitching angles and free stream
velocity.

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5.1.2 Effect of fixed offsetting pitching angle

In order to explore the relationship between the mean power coefficient and the negative

fixed offsetting pitching angle, the variation of the instantaneous passive rotation velocity and

aerodynamic torque of the turbine with the azimuth position are examined in Fig.8, where the case

of the turbine with maximum increased mean power coefficient (U∞=6m/s, β=-4° for the turbine

with 5-blades, U∞=9m/s, β=-4° for the turbine with 3-blades) and the zero offsetting pitching

angle turbine are compared.

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For the turbine with 3-blades, the passive rotation velocity of the turbine with β of -4° has

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larger value than the turbine with β of 0° (Fig.8). There is a phase difference for the aerodynamic
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torque of the turbine with different offsetting pitching angle. Compared to the turbine with β=0°,
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the torque of the turbine with a negative offsetting pitching angle is lagged to achieve minimum or

maximum amplitude. Moreover, the turbine with β=-4° has a larger positive torque amplitude.
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This can explain the negative offsetting pitching angle having a higher mean power coefficient.
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For the turbine with 5-blades, as shown in Fig.8 (b), a similar phenomenon with the turbine with
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3-blades can be observed: the passive rotation velocity of the turbine with β=-4° had larger value

than the turbine with β=0°, and the torque of the turbine with negative offsetting pitching angle

was delayed to arrive the minimum or maximum amplitude. However, the turbine with β=-4° had

positive aerodynamic torque during the whole rotation cycle, and it also had larger mean torque,

which results in a higher mean power coefficient for the turbine with a negative offsetting pitching

angle.

Fig.8 The variation of the instantaneous passive rotation velocity and aerodynamic torque of the turbine with the
azimuth position.

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 According to equations (2)-(5), the aerodynamic torque of the turbine is determined by the

lift force of the blade, which is subjected to the angle of the attack. Fig.9 shows the instantaneous

angle of the attack for a single blade with the azimuth position. Obviously, for the turbine with

3-blades, the effect of β on the α depends on the U∞. When the U∞ is 6m/s, the turbine with β=-4°

had a slightly larger positive and smaller negative amplitude of the α than the turbine with β=0°.

For two turbines under investigation, both peak α values closed to 90°, which indicates that both

of turbines were under the deep dynamic stall. On the other hand, when the U∞ is 9m/s the turbine

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with β=-4° has a smaller amplitude of the α than the turbine with β=0°, the peak α of the turbine

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with β=0° was close to 90°, while the peak α of the turbine with β=-4° closed to 25° indicating
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that the turbine with β=-4° was working out of dynamic stall angle (near 15° for NACA0018
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airfoil) for most of the rotation cycle, which is the reason for the turbine to have better power

extraction performance. For the turbine with 5-blades, a similar effects were observed when the
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turbine was at,U∞=6m/s, the amplitude of the α of the turbine with β=0° has lager value than the
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turbine with β=-4°. Moreover, the peak α of the turbine with β=0° closed to 90°, which indicates
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that the turbine with β=0° is under the deep dynamic stall, while the peak α of the turbine with

β=-4° was close to 25°; when the U∞ of the turbine was 9m/s. The turbine with β=-4° has a slightly

larger positive and smaller negative amplitude of the α than the turbine with β=0°, and the positive

amplitude of the α of the turbine with β=-4° is closer to the stall angle,, which is the reason for the

turbine to extract more power at this condition.

Fig.9 The variation of the instantaneous angle of the attack for a single blade with the azimuth position.

Fig.10 and Fig.11 show the contours of vorticity magnitude and static pressure of the turbines.

Due to the symmetry of the turbine rotation, for the turbine with 3-blades, only the turbine at θ

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=0°, 30°, 60° and 90° were considered, and for the turbine with 5-blades, only the turbine at θ =0°,

18°, 36° and 54° were investigated. It is clear in these figures that the fixed offsetting pitching

angle could influence the flow around the turbine significantly. Obviously, separated vortex was

observed around the turbine with β=0°, which resulted in a smaller pressure difference between

the pressure and suction surface of the blade (as shown in the first line of Fig.10(b) and Fig.11(b)),

and led to a smaller aerodynamic force of the blade. However, for the turbine with β=-4°, almost

for the whole rotation cycle, the generated vortex was attached to the blade surface, which

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generated a larger pressure difference between the pressure and suction surface of the blade (as

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shown in the second line of Fig.10(b) and Fig.11(b)), and resulted in a larger aerodynamic force of
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the blade. From these analyses, it is suggested that the fixed offsetting pitching angle could
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suppress the vortex separation from the blade surface, which is the reason for the turbine with a

negative fixed offsetting pitching angle having better power extraction performance.
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Fig.10 The contours of vorticity magnitude and static pressure of the 3-blades turbine with β=0°
and β=-4° during a rotation cycle (a) vorticity magnitude (b) static pressure.
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Fig.11 The contours of vorticity magnitude and static pressure of the 5-blades turbine with β=0°
and β=-4° during a rotation cycle (a) vorticity magnitude (b) static pressure.

5.1.3 Effect of blade numbers

As mentioned in section 5.1.1, the mean power coefficient of the turbine is different with the

changing of blade number and free stream wind velocity. When the wind velocity is low, no matter

what fixed offsetting pitching angle is, the turbine with 5-blades has a larger mean power

coefficient than the turbine with 3-blades. When the wind velocity is high, regardless of the

offsetting pitching angle, the turbine with 3-blades has a larger mean power coefficient than the

turbine with 5-blades. Therefore, in this section, the cases of the turbine with β=-4° at U∞=6 and

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12m/s were investigated in detail to explore how the blade numbers influence the power extraction

of the turbine.

Fig.12 illustrates the variation of the instantaneous passive rotation velocity and aerodynamic

torque with the azimuth position for the turbine with 3-blades or 5-blades at β=-4°, U∞=6 and

12m/s. Compared with the turbine with 3-blades, the turbine with 5-blades had a larger passive

rotation velocity in the whole rotation cycle, and it generated smoother and larger mean

aerodynamic torque (Fig. 12a). Moreover, in the whole rotation cycle, the turbine with 5-blades

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had positive aerodynamic torque, while negative aerodynamic torque was observed for the turbine

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with 3-blades. These results make the turbine with 5-blades to have a better power extraction
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performance when the wind velocity is low (U∞=6m/s). On the other hand, when the wind velocity
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is high(U∞=12m/s), compared to the turbine with 5-blades, the turbine with 3-blades had a larger

passive rotation velocity for the whole rotation cycle, although almost identical mean aerodynamic
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torque was generated for the two considered turbine, which resulted in the turbine with 3-blades to
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have a larger mean power coefficient.

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Fig.13 plots the instantaneous angle of the attack for a single blade with the azimuth position

for the above discussed turbine. The amplitude of the α of the turbine with 3-blades had lager

value than the turbine with 5-blades when the wind velocity was at U∞=6m/s, and the peak α of

the turbine 3-blades was close to the value of 90°, which indicates that the turbine with 3-blades is

under the deep dynamic stall. This is the reason for the sharply changing aerodynamic torque

generating for the turbine with 3-blades. On the other hand, when the wind velocity was at

U∞=12m/s; similarly, the variation trend of α with azimuth position was observed for the turbine

with 3-blades or 5-blades, however, the peak α of the turbine with 5-blades was slightly larger

Page 16 of 23
than the turbine with 3-blades, which is the reason for the two turbines to generate almost identical

mean aerodynamic torque as shown in Fig.12 (b).

Fig.12The variation of the instantaneous passive rotation velocity and aerodynamic torque of the turbine with the
azimuth position.

Fig.13 The variation of the instantaneous angle of the attack for a single blade with the azimuth position.

Fig.14 and Fig.15 show the contours of the vorticity magnitude of the turbines. Again, for the

turbine with 3-blades, only the turbine at θ =0°, 30°, 60° and 90° were considered, and for the

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turbine with 5-blades, only the turbine at θ =0°, 18°, 36° and 54° were examined due to the

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symmetry. It is concluded from Fig.14 that when the wind velocity is low, except the fixed
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pitching angle, the blade numbers also can suppress the vortex separation from the blade surface.
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However, when the wind velocity is larger, the more the blade numbers of the turbine had, the

more the complicated vortex around the turbine was. For the turbine with 5-blades, the separated
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vortex of the adjacent blade could interact with each other, which is the reason for the turbine with
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5-blades to deteriorate power extraction performance.

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Fig.14 The contours of vorticity magnitude the turbine with different blade numbers during a rotation cycle at
U∞=6m/s (a) turbine with 3-blades (b) turbine with 5-blades

Fig.15 The contours of vorticity magnitude the turbine with different blade numbers during a rotation cycle at
U∞=12m/s (a) turbine with 3-blades (b) turbine with 5-blades .

5.2 Self-starting performance

5.2.1 Self-starting time

To date, the capacity of the self-starting of the VAWT is not well defined [23-24]. In this

work, the same as our previous study [20], the turbine is considered to have self-starting capacity

when the turbine can reach steady passive rotation without external activation. The self-starting

Page 17 of 23
performance of the turbine is evaluated by the time of the turbine from initial static to the steady

passive cycle rotation (self-starting time). Based on this definition, the covered turbine in this

work all can be self-starting; however, the self-starting time is different for the turbine with

different fixed offsetting pitching angles and blade numbers.

As stated in section 5.1, applying the negative fixed offsetting pitching angle could enhance

the power extraction performance of the turbine, and the optimized β was -4° for the turbine with

5-blades and the turbine with 3-blades when the wind velocity is high. Therefore, in this section,

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the self-starting performance is investigated for the turbine with β=0° and β=-4° at U∞=3 to 12m/s.

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Fig.16 shows the variation of the instantaneous passive rotation velocity with the flow time of
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the covered turbine in this section. It is concluded from this figure that when the wind velocity
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was at 3 and 6m/s, for all the covered turbines, the self-starting time almost had identical value,

which indicates that the fixed offsetting pitching angle and blade number influence the
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self-starting performance of the turbine slightly. However, when the wind velocity was at 9 and
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12m/s, both the blade number and fixed offsetting pitching angle could influence the self-starting
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time significantly. The turbine with 5-blades had less self-starting time than that of the turbine

with 3-blades, and the turbine with a negative fixed offsetting pitching angle had less self-starting

time than the turbine with zero fixed offsetting pitching angle. This conclusion indicates that when

the wind velocity is low, the turbine with 3-blades and 5-blades almost have identical self-starting

performance, on the other hand when the wind velocity is high, the turbine with 5-blade and

applied negative fixed offsetting pitching angle has a better self-starting performance.

Fig.16 The variation of the instantaneous passive rotation velocity with flow time.

Page 18 of 23
5.2.2 Effect of fixed offsetting pitching angle

To explore the effect of fixed offsetting pitching angle on the self-starting performance of the

turbine in detail, the cases of the 3-blades turbine with β=0° and -4° at U∞=3 and 12m/s are

selected for instance.

Fig.17 illustrates the instantaneous aerodynamic torque with the flow time of the covered

turbine in this section. It is evident in Fig.17 (a) that when the wind velocity was at 3m/s, during

the self-starting process (when the rotation velocity does not reach steady cycle, t< 3.0s), the

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turbine with β=0° and -4° almost had identical aerodynamic torque. This is the reason for the two

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turbines had almost an identical self-starting time, as shown in Fig.16 (a). On the other hand, as
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shown in Fig.17 (b) that during the self-starting process (t<3s), the turbine with β=-4° had larger
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aerodynamic torque than the turbine with β=0°, which resulted in the turbine with β=-4° to have a

larger rotation acceleration. This also can explain the turbine had smaller self-starting times as
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shown in Fig.16 (d).

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Fig.17 The variation of the instantaneous aerodynamic torque with flow time for the turbine with 3-blade.

Fig.18 and Fig.19 plot the contours of vorticity magnitude and static pressure of the turbines

as conducted in this section at t=3s. For the covered turbines, almost identical vorticity and static

pressure were generated when the wind velocity was at 3m/s, which indicates that the self-starting

performance is influenced slightly by the fixed offsetting pitching angle at this condition. However,

when the wind velocity was at 12m/s, the fixed offsetting pitching angle can suppress the vortex

shedding from the blade surface, which caused a larger pressure difference around the blade of the

turbine with applying negative offsetting pitching angle, and induced a larger acceleration. This is

the reason for the turbine with β=-4° has better self-starting performance at 12m/s.

Page 19 of 23
 Fig.18 The contours of vorticity magnitude and static pressure of the 3-blades turbine with β=0° and β=-4° at
U∞=3m/s and t=3s (a) vorticity magnitude (b) static pressure.

Fig.19 The contours of vorticity magnitude and static pressure of the 3-blades turbine with β=0° and β=-4° at
U∞=12m/s and t=3s (a) vorticity magnitude (b) static pressure.

5.2.3 Effect of blade numbers

To explore the effect of the blade number on the self-starting performance of the turbine in

detail, the cases of the turbine with β=-4° at U∞=3 and 9m/s are selected for instance.

Fig.20 illustrates the instantaneous aerodynamic torque with the flow time of the covered

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turbine in this section. It is found in Fig.20 (a) that when the wind velocity was at 3m/s, during the

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self-starting process (t<10.0s), the turbine with 3-blades and 5-blades almost had identical
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aerodynamic torque. This is the reason for the two turbines to have almost identical self-starting
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times. On the other hand, it can be seen from Fig.20 (b) that during the self-starting process
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(t<6.0s, especially during 1s<t<3s), the turbine with 5-blades had larger aerodynamic torque than
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the turbine with 3-blades, which led to the turbine with 5-blades to have a larger rotation
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acceleration. This s explains a smaller self-starting time of the turbine , as shown in Fig.16 (d).
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Fig.20 The variation of the instantaneous aerodynamic torque with flow time for the turbine with different blade
numbers.

Fig.21 and Fig.22 plot the contours of vorticity magnitude and static pressure of the turbines,

as discussed in this section at t=4.0s. A similar vortex structure was observed around the turbine

with 3-blades or 5-blades (Fig.21).Iincreasing the blade number of the turbine could delay the

vortex shedding from the blade surface, as shown in Fig.22, which induced a longer time of the

larger pressure difference between the pressure and suction surface of the blade, and led to a larger

acceleration of the turbine with more blades.

Fig.21 The contours of vorticity magnitude and static pressure of the turbine with different blade number at β=-4°,

Page 20 of 23
 U∞=3m/s and t=4s (a) vorticity magnitude (b) static pressure.

Fig.22 The contours of vorticity magnitude and static pressure of the turbine with different blade number at β=-4°,
U∞=9m/s and t=4s (a) vorticity magnitude (b) static pressure.

6 Conclusions
The advantages of applying for fixed offsetting pitching angle and changing blade number in

VAWT were demonstrated in this work. A numerical method based on unsteady Navier-Stokes

equations and Newton’s second law was established and validated to accurately simulate flow

around the turbine. Seven different offsetting pitching angles and two different blade numbers of

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the VAWT were investigated. The main conclusions of this work are summarized as:

1) -p
The power extraction performance: maximum 5.89 and 5.14 times of increase in average
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power coefficients were achieved for the turbine with 5-blades and 3-blades, respectively, at
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an offsetting pitching angle of β=-4° comparing with a turbine with zero offsetting pitching

angle. The effect of the blade number on the power extraction was subject to the free-stream
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wind velocity. For the negative offsetting pitching angle, the turbine with 5-blades had larger
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mean power coefficient than the turbine with 3-blades at low wind velocity(U∞<6m/s), while
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at high wind velocity(U∞>9m/s), the turbine with 3-blades had larger mean power coefficient.

2) The self-starting characteristics: At low wind velocity(U∞<6m/s), the self-starting time of the

turbine was almost identical, which indicated that the offsetting pitching angle and blade

number had little effect on the self-starting characteristics. While at high wind

velocity(U∞>9m/s), the turbine with more blade and an applying β=-4° had a smaller

self-starting period.

3) The flow structure around the turbine: For a turbine with appropriate offsetting pitching angle

(β=-4°) and blade number (5-blades), the vortex around the blade surface could be

Page 21 of 23
 suppressed or delayed, which results in a longer period of a larger pressure difference

between the pressure and suction surface of the blade, leading to higher mean power

coefficient and acceleration.

4) All in all, the turbine with 5-blades and applying offsetting pitching angle (β=-4°) is

beneficial to the recently developed prototype of VAWT with liquid lifting force in

Macau[17].

It should be pointed out that comparing with changing blade numbers, variance of offsetting

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pitching angle can influence the performance of the turbine more significantly. In the future works,

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the performance of the turbine with 5-blades applying variable changing offsetting pitching angle
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will be analyzed.
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Acknowledgement
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This work was funded by The Science and Technology Development Fund, Macau
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SAR (File no. 0083/2018/A2); Multi-Year Research Grant (File no.

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MYRG2019-00135-IAPME), Research & Development Grant for Chair Professor

(File no. CPG2020-00002-IAPME), and Start-up Research Grant (File no.

SRG2017-00093-IAPME) from University of Macau.

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Slotted Airfoil Blades[J]. Energy Conversion and Management: X, 2019: 100026.
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without winglets: CFD simulations[J]. Energy Sources, Part A: Recovery, Utilization, and Environmental
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[3] Pallotta A, Pietrogiacomi D, Romano G P. HYBRI–A combined Savonius-Darrieus wind turbine:
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[4] Zhu J, Tian C. Effect of Rotation Friction Ratio on the Power Extraction Performance of a Passive
Rotation VAWT[J]. International Journal of Rotating Machinery, 2019, 2019.

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[5] Du L, Ingram G, Dominy R G. Experimental study of the effects of turbine solidity, blade profile, pitch
angle, surface roughness, and aspect ratio on the H‐Darrieus wind turbine self‐starting and overall
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[6] Rezaeiha A, Kalkman I, Blocken B. Effect of pitch angle on power performance and aerodynamics of a
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[7] Guo Y, Li X, Sun L, et al. Aerodynamic analysis of a step adjustment method for blade pitch of a
VAWT[J]. Journal of Wind Engineering and Industrial Aerodynamics, 2019, 188: 90-101.
[8] Armstrong S, Fiedler A, Tullis S. Flow separation on a high Reynolds number, high solidity vertical axis
wind turbine with straight and canted blades and canted blades with fences[J]. Renewable Energy, 2012, 41:
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[9] Zhang L, Liang Y, Liu X, et al. Effect of blade pitch angle on aerodynamic performance of
straight-bladed vertical axis wind turbine[J]. Journal of Central South University, 2014, 21(4): 1417-1427.
[10] Li C, Xiao Y, Xu Y, et al. Optimization of blade pitch in H-rotor vertical axis wind turbines through

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computational fluid dynamics simulations[J]. Applied energy, 2018, 212: 1107-1125.
[11] Li Z, Diao S, Hanif A, Pei H, and Sun G. Vertical axis aerogenerator. Chinese patent, CN105715456B,

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2018.
[12] Zhu J, Jiang L, Zhao H. Effect of wind fluctuating on self-starting aerodynamics characteristics of
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VAWT[J]. Journal of Central South University, 2016, 23(8): 2075-2082.
[13] Zhu J, Tian C. Effect of Rotation Friction Ratio on the Power Extraction Performance of a Passive
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Rotation VAWT[J]. International Journal of Rotating Machinery, 2019, (2019):1-10.
[14] Liu K, Yu M, Zhu W. Enhancing wind energy harvesting performance of vertical axis wind turbines
lP

with a new hybrid design: A fluid-structure interaction study[J]. Renewable Energy, 2019, 140: 912-927.
[15] Liu Q, Miao W, Li C, et al. Effects of trailing-edge movable flap on aerodynamic performance and
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noise characteristics of VAWT[J]. Energy, 2019: 116271.

[16] Shukla V, Kaviti A K. Performance evaluation of profile modifications on straight-bladed vertical axis
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wind turbine by energy and Spalart Allmaras models[J]. Energy, 2017, 126: 766-795.
[17] Logan G L. A first course in the finite element method. Publishing house of electronics industry,
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2003,pp:443-445.
[18] Almohammadi KM, Ingham DB, Ma L, et al. Computational fluid dynamics (CFD) mesh independency
techniques for a straight blade vertical axis wind turbine. Energy 2013;58:483–93
[19] Rezaeiha A, Montazeri H, Blocken B. Towards accurate CFD simulations of vertical axis wind turbines
at different tip speed ratios and solidities: Guidelines for azimuthal increment, domain size and convergence.
Energy Convers Manage 2018;156:301–16.
[20] Sun X, Zhu J, Hanif A, et al. Effects of blade shape and its corresponding moment of inertia on self-
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wind turbine[J]. Sustainable Energy Technologies and Assessments, 2020,38:100648.
[21] Rainbird J. The Aerodynamic Development of a Vertical Axis Wind Turbine (MEng Thesis). England:
Durham University; 2007.
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wind turbines comprising NACA 4-digit series blade airfoils[J]. Energy, 2016, 112: 528-537.
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Gas Turbines Power 2015;138(6). V03BT46A008.
[24] Dominy R, Lunt P, Bickerdyke A, et al. Self-starting capability of a Darrieus turbine. Proc Inst Mech
Eng Part A: J Power Energy 2007;221(1):111–20.

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List of Figures
Fig.1 Two typically S-VAWTs (a) Wind turbine with 3-blades. One extra blade is
used to show force and other symbols. (b) Wind turbine with 5-blades.
Fig.2 The calculation flowchart of the wind induced rotation turbine.
Fig. 3 The computational domain and mesh structure.
Fig.4 The induced rotation velocity versus flow time and mean power coefficient for
the two types VAWT with different grid modes (a) VAWT with 3-blades (b)VAWT
with 5-blades.
Fig.5 The induced rotation velocity versus flow time and mean power coefficient for
the two types VAWT with different iterate time step sizes(a) VAWT with 3-blades
(b)VAWT with 5-blades.
Fig.6 The tip speed ratio versus flow time during the startup: comparison of the

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present simulation data to numerical data calculated by Asr et al. [21] and

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experimental data reported by Rainbird et al. [22].
Fig.7 The power extraction performance of the VAWT with different fixed offset
pitching angles and free stream velocity.
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Fig.8 The variation of the instantaneous passive rotation velocity and aerodynamic
torque of the turbine with the azimuth position.
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Fig.9 The variation of the instantaneous angle of the attack for a single blade with the
azimuth position.
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Fig.10 The contours of vorticity magnitude and static pressure of the 3-blades turbine
with β=0° and β=-4° during a rotation cycle (a) vorticity magnitude (b) static pressure.
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Fig.11 The contours of vorticity magnitude and static pressure of the 5-blades turbine
with β=0° and β=-4° during a rotation cycle (a) vorticity magnitude (b) static pressure.
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(b) β=-4°, U∞=12m/s

Fig.12The variation of the instantaneous passive rotation velocity and aerodynamic
torque of the turbine with the azimuth position.
Fig.13 The variation of the instantaneous angle of the attack for a single blade with
the azimuth position.
Fig.14 The contours of vorticity magnitude the turbine with different blade numbers
during a rotation cycle at U∞=6m/s (a) turbine with 3-blades (b) turbine with
5-blades.
Fig.15 The contours of vorticity magnitude the turbine with different blade numbers
during a rotation cycle at U∞=12m/s (a) turbine with 3-blades (b) turbine with
5-blades.
Fig.16 The variation of the instantaneous passive rotation velocity with flow time.
Fig.17 The variation of the instantaneous aerodynamic torque with flow time for the
turbine with 3-blade.
Fig.18 The contours of vorticity magnitude and static pressure of the 3-blades turbine

Page 1 of 17
with β=0° and β=-4° at U∞=3m/s and t=3s (a) vorticity magnitude (b) static pressure.
Fig.19 The contours of vorticity magnitude and static pressure of the 3-blades turbine
with β=0° and β=-4° at U∞=12m/s and t=3s (a) vorticity magnitude (b) static
pressure.
Fig.20 The variation of the instantaneous aerodynamic torque with flow time for the
turbine with different blade numbers.
Fig.21 The contours of vorticity magnitude and static pressure of the turbine with
different blade number at β=-4°, U∞=3m/s and t=4s (a) vorticity magnitude (b) static
pressure.
Fig.22 The contours of vorticity magnitude and static pressure of the turbine with
different blade number at β=-4°, U∞=9m/s and t=4s (a) vorticity magnitude (b) static

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pressure.

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Page 2 of 17
 of
(a) (b)

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Fig.1 Two typically S-VAWTs (a) Wind turbine with 3-blades. One extra blade is used
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to show force and other symbols. (b) Wind turbine with 5-blades.
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Fig.2 The calculation flowchart of the wind induced rotation turbine.

Page 3 of 17
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Fig. 3 The computational domain and mesh structure.
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Fig.4 The induced rotation velocity versus flow time and mean power coefficient for
the two types VAWT with different grid modes (a) VAWT with 3-blades (b)VAWT
with 5-blades.

Page 4 of 17
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Fig.5 The induced rotation velocity versus flow time and mean power coefficient for
the two types VAWT with different iterate time step sizes(a) VAWT with 3-blades
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(b)VAWT with 5-blades.

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Fig.6 The tip speed ratio versus flow time during the startup: comparison of the
present simulation data to numerical data calculated by Asr et al. [21] and
experimental data reported by Rainbird et al. [22].

Page 5 of 17
 (a)U∞=3m/s (b)U∞=6m/s

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(c)U∞=9m/s (d)U∞=12m/s

Fig.7 The power extraction performance of the VAWT with different fixed offset
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pitching angles and free stream velocity.

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Page 6 of 17
 (a) VAWT with 3-blades

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(b) VAWT with 5-blades

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Fig.8 The variation of the instantaneous passive rotation velocity and aerodynamic
torque of the turbine with the azimuth position.
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Page 7 of 17
 (a) VAWT with 3-blades U∞=6 m/s (b) VAWT with 3-blades at U∞=9 m/s

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(c) VAWT with 5-blades at U∞=6 m/s (d) VAWT with 5-blades at U∞=9 m/s
Fig.9 The variation of the instantaneous angle of the attack for a single blade with the
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azimuth position.
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Page 8 of 17
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(a)

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(b)
Fig.10 The contours of vorticity magnitude and static pressure of the 3-blades turbine
with β=0° and β=-4° during a rotation cycle (a) vorticity magnitude (b) static pressure.

Page 9 of 17
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(a)

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(b)
Fig.11 The contours of vorticity magnitude and static pressure of the 5-blades turbine
with β=0° and β=-4° during a rotation cycle (a) vorticity magnitude (b) static pressure.

Page 10 of 17
 (a) β=-4°, U∞=6m/s

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(b) β=-4°, U∞=12m/s

Fig.12 The variation of the instantaneous passive rotation velocity and aerodynamic
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torque of the turbine with the azimuth position.

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(a) β=-4°, U∞=6m/s (b) β=-4°, U∞=12m/s

Fig.13 The variation of the instantaneous angle of the attack for a single blade with
the azimuth position.

Page 11 of 17
 (a)

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(b)
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Fig.14 The contours of vorticity magnitude the turbine with different blade numbers
during a rotation cycle at U∞=6m/s (a) turbine with 3-blades (b) turbine with
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5-blades.
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(a)

(b)
Fig.15 The contours of vorticity magnitude the turbine with different blade numbers
during a rotation cycle at U∞=12m/s (a) turbine with 3-blades (b) turbine with
5-blades.

Page 12 of 17
 (c)U∞=9m/s (d)U∞=12m/s

Fig.16 The variation of the instantaneous passive rotation velocity with flow time.

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(a) U∞=3m/s (b) U∞=12m/s

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Fig.17 The variation of the instantaneous aerodynamic torque with flow time for the
turbine with 3-blade.
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Page 13 of 17
 (a) (b)
Fig.18 The contours of vorticity magnitude and static pressure of the 3-blades turbine
with β=0° and β=-4° at U∞=3m/s and t=3s (a) vorticity magnitude (b) static pressure.

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(a) (b)
Fig.19 The contours of vorticity magnitude and static pressure of the 3-blades turbine
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with β=0° and β=-4° at U∞=12m/s and t=3s (a) vorticity magnitude (b) static pressure.
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Page 14 of 17
 (a) U∞=3m/s (b) U∞=9m/s

Fig.20 The variation of the instantaneous aerodynamic torque with flow time for the
turbine with different blade numbers.

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(a) (b)
Fig.21 The contours of vorticity magnitude and static pressure of the turbine with
different blade number at β=-4°, U∞=3m/s and t=4s (a) vorticity magnitude (b) static
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pressure.
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(a) (b)
Fig.22 The contours of vorticity magnitude and static pressure of the turbine with
different blade number at β=-4°, U∞=9m/s and t=4s (a) vorticity magnitude (b) static
pressure.

Page 15 of 17
List of Tables
Table 1. Parameters of the blade.
Table 2 Details of the grid discrete form for the VAWT.

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Page 16 of 17
 Table 1. Parameters of the blade

Blade shape NACA0018

Chord length c=0.15m

mass 0.17kg

Mass center Distance c/3 from the leading edge of the airfoil

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Table 2 Details of the grid discrete form for the VAWT

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nodes on each blade surface total cells

VAWT with 3-blades

grid mode1
grid mode2
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210
420
57160
90834
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grid mode3 840 145586
grid mode4 210 80838
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VAWT with 5-blades grid mode5 420 132354

grid mode6 840 216198
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Highlights
• The influence of pitching angle and blade number on the S-VAWT were investigated.
• Maximum average extraction power increase was achieved with pitching angle β of -4°.
• Self-starting time of the turbine is subject to wind-velocity and blade number.
• The turbine with 5-blade and a pitching angle of -4° showed the best performance.

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Declaration of interests

☒ The authors declare that they have no known competing financial interests or personal relationships
that could have appeared to influence the work reported in this paper.

☐The authors declare the following financial interests/personal relationships which may be considered
as potential competing interests:

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