Energy 161 (2018) 753e775
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Energy
journal homepage: www.elsevier.com/locate/energy
Effect of solidity on the performance of variable-pitch vertical axis
wind turbine
A. Sagharichi a, M. Zamani b, *, A. Ghasemi c
a
Department of Mechanical Engineering, Ferdowsi University of Mashhad, Iran
Department of Mechanical Engineering, Payame Noor University(PNU), P.O. Box 19395-3697, Tehran, Iran
c
Banxin Corporation and Worcester Polytechnic Institute, MA, USA
b
a r t i c l e i n f o
a b s t r a c t
Article history:
Received 19 March 2018
Received in revised form
17 July 2018
Accepted 23 July 2018
Available online 27 July 2018
The Darrieus vertical axis wind turbine (VAWT) has been the subject of a large number of recent studies
to improve self-starting capability and aerodynamic performance. This study presents a computational
fluid dynamics simulation of the vertical axis wind turbine with fixed and variable pitch at different
solidities. In order to introduce the best efficiency value of solidity for a variable-pitch vertical axis wind
turbine, a computational modeling of two-dimensional transient flow around a VAWT at solidities between 0.2 and 0.8 and turbine with two, three and four blades is conducted. The numerical simulation
models the flow field around a turbine rotor by solving the strong nonlinearity of URANS and SST k-u
turbulence model, utilizing the semi-empirical numerical model. To simulate the turbine in a variable
pitch method, the user-defined function (UDF) code and the moving mesh method are used. The results
show that variable pitch blades with high solidities are preferred when the initial self-starting torque is
required. Moreover, the variable pitch blades are capable to produce more torque at high solidities.
Therefore, the vortex interaction between upwind vortices and blades in downwind stream is decreased.
The results verify that the power reduction of fixed pitch VAWT at high solidities could be solved by
utilizing the variable pitch technique.
© 2018 Elsevier Ltd. All rights reserved.
Keywords:
Vertical axis wind turbine
Variable-pitch
Solidity
2D numerical simulation
UDF
1. Introduction
Vertical axis wind turbine (VAWT) technologies have been
studied recently to improve the performance of small VAWTs used
in urban areas (home and office). Generally, a VAWT consists of two,
three or more blades with the main rotor shaft which blades rotate
vertically around it. Such type of turbines can take the wind from
every direction and it is not required to adjust blades to the wind
direction. It is a desirable factor for using this turbine in urban
consumption, where the wind direction constantly changes [1].
Also, the generators used in this type of turbine are located near the
ground which reduces costs [2]. Furthermore, VAWT turbines
produce less aerodynamic noise than horizontal axis wind turbines,
since it works at low speeds, and installation of such turbines is
relatively easy [3].
According to principles of aerodynamics, small VAWTs can be
classified into drag-based and lift-based turbines [4,5]. The
* Corresponding author.
E-mail address: m.zamani@um.ac.ir (M. Zamani).
https://doi.org/10.1016/j.energy.2018.07.160
0360-5442/© 2018 Elsevier Ltd. All rights reserved.
Darrieus wind turbine is a lift-based turbine which consists of a
number of curved airfoils. This type of wind turbine was designed
by Georges Jean Marie Darrieus in 1931 [6]. The Darrieus rotor
blades could be straight, helical and eggbeater types and can be
fixed pitch or controlled by variable pitch system.
Darrieus VAWTs with fixed-pitch blades have some disadvantages such as poor self-starting, low efficiency and low effective life
due to large fluctuations [7]. In contrast, in the variable pitch turbines, it is possible to adjust the angle of attack by changing the
blade pitch angle and angle of attack. By controlling the angle of
attack, the variable pitch turbines can significantly increase the
starting torque which leads to self-start of a turbine. Additionally, a
variable pitch Darrieus VAWT can generate high efficiency, and
reduce torque ripples [8].
Fig. 1 shows the schematics of a mechanism for variable-pitch
VAWT proposed by Sagharichi et al. [8]. In this design, the pitch
angle (the angle between the blade chord line and tangent to the
path) is depended on eccentricity between the rotation axis and the
cam.
Paraschivoiu et al. [9] analyzed a 7 kW prototype H-Darrieus
wind turbine with blade pitch control system. In this study,
754
A. Sagharichi et al. / Energy 161 (2018) 753e775
Fig. 1. A schematic of a proposed variable pitch mechanism.
CARDAAV code was developed, based on the “Double-Multiple
Streamtube” model to determine the performances of the straightbladed VAWT. The code was coupled using a genetic algorithm
optimizer, and azimuthal variation of the blades' pitch angle was
modeled with an analytical function with the variable coefficients
in the optimization process. In the variable pitch angle method, the
blade angle of attack is changing relating to the local flow conditions. It is predicted an increase of almost 30% in the annual energy
production will be obtained using the polynomial optimal pitch
control [9]. Bianchini et al. [10] optimized the pitch system of a
small Darrieus VAWT. The pitch modeling in an advanced BEM code
was utilized to investigate different blade pitch control strategies.
The results showed with an appropriate pitch angle, the output
power of turbine could significantly increase. Wang et al. [11]
numerically investigated a novel Darrieus vertical axis wind turbine in which blades could be deformed automatically into the
desired geometry and was concluded that the deformable blades
can improve the performance of turbine specifically at low solidities. Ouro et al. [12] studied the effect of camber of airfoil on the
flow over and aerodynamic performance of a symmetric NACA
0012 airfoil and an asymmetric NACA 4412 airfoil under the same
flow and kinematic conditions. The results revealed that a
cambered airfoil shape provides pitching airfoils have a higher liftto-drag ratio and lead to having a short delay in the shedding of the
dynamic stall vortex.
Chao Li et al. [13] used computational fluid dynamics to analyze
the unsteady two-dimensional flow simulation. The simulations
show that the optimized pitch curves can increase the average
power coefficient for different tip speed ratios (TSRs).
Rezaeiha et al. [14] used unsteady Reynolds-averaged NavierStokes (URANS) calculations and transition SST turbulence model
to investigate the performance of a Darrieus wind turbine for
different pitch angles in the range of 7 to þ3 . The results
showed that a 6.6% increase in power coefficient can be achieved
using a pitch angle of 2 at a tip speed ratio of four.
Hang Lei et al. [15] carried out CFD time-dependent aerodynamics and performance of an offshore floating vertical axis
wind turbine in different pitching amplitudes and pitching periods.
The results showed that pitch motion can increase the variation
range of aerodynamic force coefficients.
Jain and Abhishek [16] developed an aerodynamic model based
on double multiple stream-tube theory for a small-scale VAWT
turbine. Furthermore, unsteady aerodynamic effects were modeled
using Wagner's function. The authors showed by decreasing the
amplitude of pitch change from 30 to 8 , the peak power coefficient greater than 0.3 can be achieved and maintained among tip
speed ratios of 1e2.5.
Sagharichi et al. [8] and Abdalrahman et al. [17] investigated the
effects of fixed and variable pitch blades on the performance of a
Darrieus turbine using a CFD simulation. A 2D unsteady flow
simulation with moving mesh technique using SST k-u turbulence
model was performed. The results showed that a blade pitch control system can be utilized for an H-type VAWT as a means of
improving its power generation performance. The authors also
estimated that the variable pitch angle blade increases the starting
torque and efficiency while reduces the torque fluctuation imposed
on the blades.
The effects of solidity on the performance of a fixed-pitch wind
turbine have been extensively studied [18e24]. For instance, to
study the effect of solidity on the performance of wind turbines, a
Fig. 2. Angle of attack versus tip speed ratios in one revolution.
A. Sagharichi et al. / Energy 161 (2018) 753e775
numerical analysis using multiple stream tube method was carried
out by Roh et al. [25] who conducted their research at various rotor
solidities in the range of 0.5e0.8. It is stated that by increasing the
solidity, the trend of the power curve of the straight-type Darrieus
VAWT varies from the eggbeater-type of Darrieus VAWTs. Jiang
et al. [26] numerically investigated a smallescale Darrieus wind
turbine and examined the effects of different blade numbers and
TSR on the performance of the turbine and showed that higher
755
solidity rotor leads to smaller corresponding TSR for the maximum
power coefficient. A three-dimensional CFD model using RNG k ε
turbulence model presented by Howell et al. [27]. The effects of
rotor solidity, blade roughness and tip vortices on the performance
of a VAWT have been studied. For example, Joo et al. [28] performed
a 3D unsteady numerical simulation to analyze the aerodynamic
characteristics of an H-Darrieus vertical axis wind turbine with two
straight blades. The authors illustrated that by decreasing the solidity, the effects of blockage and vortex interaction can be reduced
and the self-starting will be improved by decreasing the negative
torque region at the low TSRs. Also, they demonstrated that the
double multiple streamtube models (DMST) cannot accurately
predict the performance of H-Darrieus VAWT at high solidities.
More recently, Kirke and Paillard [29] used two different models
Fig. 5. A schematic of the computational domain and the boundary conditions.
Fig. 3. Schematic of variable pitch mechanism based on a four-bar linkage.
Table 2
The grid resolution details along the NACA 0021 airfoil.
Grid
Coarse
Medium
Fine
yþ
Number of Nodes
Airfoil
Boundary Layers
Total
230
532
1089
20
50
60
203762
513628
909796
0.8876
0.83
0.815
Fig. 4. Blade placement in variable and fixed pitch wind turbine.
Table 1
Angular velocity of rotor corresponding to
TSR.
l
u
0.5
1
1.5
2
2.5
3
8.737864
17.47573
26.21359
34.95146
43.68932
52.42718
Fig. 6. Torque coefficient versus dimensionless azimuth angle at optimum TSR.
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A. Sagharichi et al. / Energy 161 (2018) 753e775
Fig. 7. Grid resolution around rotor and blade.
(DMS and 2-D RANS) to predict vertical axial turbine performance
in both fixed and variable cases. They stated that the variable pitch
turbine can overcome two major drawbacks, inability to self-start
and over-speed, of fixed-pitch vertical axis wind turbines.
Overall, it could be stated that vertical axis wind turbines and
among them primarily Darrieus turbines are not fully developed
yet. The studies cited above did not focus on the effects of variable
pitch system at different solidities in order to improve the performance of the turbine. Therefore, the present work seeks to compare
the performance of a straight-bladed VAWT (SB-VAWT) with both
variable-pitch control system and fixed-pitch blades at variety of
Table 3
Specifications of the Darrieus wind turbine tested in Ref. [42].
Features
Value
Radius of rotor (R) [mm]
Number of blades (N)
Blade profile
Blade chord (C)[mm]
Azimuth angles (q) [degree]
Tip Speed Ratio (l or TSR)
1030
3
NACA 0021
85.8
0 to 360
0.5, 1, 1.5, 2, 2.5, 3, 3.5
Fig. 8. Validation of model with experimental results from Ref. [42].
Table 4
Blade chords versus solidities.
Solidity ()
Blade Chord (mm)
0.20
0.40
0.50
0.60
0.80
34.3
68.7
85.8
103.0
137.3
Fig. 9. Effect of solidities on fixed pitch VAWT power curve performance.
A. Sagharichi et al. / Energy 161 (2018) 753e775
757
2. CFD methodology
2.1. Solver settings
The unsteady flow field around a VAWT is numerically simulated using ANSYS Fluent 15 software package. The discretization of
the governing equations was carried out using a finite volume
scheme. In addition, to discretize the advection and viscous terms
of the momentum equations, a second order upwind and secondorder central difference schemes have been utilized, respectively.
To capture the flow properties around the rotational domain, a
sliding mesh technique has been used and a rotational term is
added to the equations. In each time-step, the solver performs
mesh manipulation for the sliding interface boundary conditions.
Because of transient CFD problems, the pressure implicit splitting of
operators (PISO) algorithm was used for coupling of the velocity
Fig. 10. Effect of solidities on variable pitch VAWT power curve performance.
Fig. 12. Blade number effects on power coefficient of fixed pitch wind turbine.
Fig. 11. Maximum power coefficient versus solidity.
solidities in order to find the best efficiency value of solidity which
increases output power. It was stated that increasing solidity is a
good way to increase the output torque of a turbine especially in
low TSRs [11] but it has some drawbacks such as increasing weight,
cost, and inertia. However, increasing the solidity is one of the
present methods for reducing the dead band region.
In this paper, we intend to prove that instead of increasing solidity for the purpose of improving self-starting and output power
at low TSRs, the variable pitch method can be used to produce
higher starting torque at lower TSRs and solidities, so that the
drawbacks due to increasing of solidities could be eliminated. For
this purpose, the power coefficient of the rotor at different solidities
was exactly calculated and the results were compared at different
TSRs.
Fig. 13. Blade number effects on power coefficient of variable pitch wind turbine.
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A. Sagharichi et al. / Energy 161 (2018) 753e775
and the pressure equations [7,30]. PISO algorithm performs two
additional corrections: a neighbor correction and a skewness
correction. After one or more additional PISO loops, the corrected
velocities satisfy the continuity and momentum equations precisely. This iterative process is called a momentum correction or
“neighbor correction”. The PISO algorithm also neglects the velocity
correction in the first step, but then performs one at a later stage,
which leads to additional corrections for the pressure. After the
initial solution of the pressure-correction equation, the pressurecorrection gradient is recalculated and used to update the mass
flux corrections. This process, which is referred to as “skewness
correction”, significantly reduces convergence difficulties associated with highly distorted meshes [31e33].
It should be noted that angular time-step has a significant
impact on the accuracy of the solution and it was proven that the
time-step corresponding to the rotation of 1 does not provide a
precise solution, especially for small tip speed ratios and complex
geometries [34]. Also, a simulation with a smaller time-step did not
show significant changes [8]. Therefore, the time step was selected
based on the rotation of the turbine for Dq ¼ 0.5 in the unsteady
simulation. Simulations showed that the solution converged after 6
to 8 full rotations and the results of final rotation were selected for
further studies.
2.2. Key performance parameters
The relationship between the angle of attack a, the azimuth
angle q, and the tip speed l is:
a ¼ tan1
sin q
l þ cos q
(1)
Where tip speed ratio (l or TSR) are defined as:
l¼
R:u
U
(2)
The variation of the angle of attack during one revolution at
different tip speed ratios has been plotted in Fig. 2. It is illustrated
that by increasing the tip speed ratio, the angle of attack decreases.
The solidity s of vertical axis wind turbine is the vital parameter
to define the rotor geometry which can be calculated by:
s¼
Nc
R
(3)
Where N is the number of blades, c is the chord length, u is angular
velocity, U is free stream velocity and R is the radius of the turbine.
As shown in the previous equation, the solidity is depended on the
number of blades, the chord of the blades, and R, the radius of the
rotor. In order to study the solidity effects in the present study, the
number of blades and the chord length are taken as a variable
parameter while the rotor radius is assumed to be constant.
By combining different parameters which affect the performance of a darrieus wind turbine, the equation of torque and power
coefficients can be defined by:
Cm ¼
T
0:5rARU 2
P
CP ¼
0:5rAU 3
(4)
Fig. 14. Instantaneous torque coefficient of wind turbine with a) 2, b) 3 and c) 4 blades
at TSR ¼ 3.
(5)
Where Cp and Cm are the power coefficient and torque coefficient,
respectively.
A. Sagharichi et al. / Energy 161 (2018) 753e775
Fixed Pitch
Variable Pitch
Fixed Pitch
Variable Pitch
0.2
0.15
0.15
0.1
0.1
Torque Coefficient
Torque Coefficient
0.2
759
0.05
0
-0.05
0.05
0
-0.05
-0.1
-0.1
-0.15
-0.15
0.2
0.4
0.6
Time (S)
0.8
0.2
1
a)
0.6
Time (S)
0.8
1
b)
Fixed Pitch
Variable Pitch
0.04
0.4
Fixed Pitch
Variable Pitch
0.2
Torque Coefficient
Torque Coefficient
0.15
0.02
0
-0.02
0.1
0.05
0
-0.05
-0.1
-0.04
-0.15
0.2
0.4
0.6
Time (S)
0.8
1
0.2
c)
Fixed Pitch
Variable Pitch
0.2
0.25
0.15
Torque Coefficient
0.2
Torque Coefficient
0.8
d)
Fixed Pitch
Variable Pitch
0.3
0.4
0.6
Time (S)
0.15
0.1
0.05
0
0.1
0.05
0
-0.05
-0.05
-0.1
-0.1
-0.15
0.2
0.4
0.6
Time (S)
e)
0.8
1
-0.2
0.2
0.4
Time (S)
0.6
0.8
f)
Fig. 15. Total torque coefficient for turbine with a) 2-bladed b) 4-bladed c) s ¼ 0.2, d) s ¼ 0.4, e) s ¼ 0.5, f) s ¼ 0.6, g)s ¼ 0.8 at TSR ¼ 1.
760
A. Sagharichi et al. / Energy 161 (2018) 753e775
Fig. 16. Instantaneous blade torque coefficient for a) Fixed pitch 2 bladed b)Variable pitch 2 bladed c) Fixed pitch 3 bladed d)Variable pitch 3 bladed, e)Fixed pitch 4 bladed f)
Variable pitch 4 bladed at TSR ¼ 1.
2.3. Turbulence model
In computational fluid dynamics, the k ε and k u turbulence
models are common two-equation turbulence models. The k ε
model has been shown to be useful for free-shear layer flows with
relatively small pressure gradients [34]. The k u model is one of
the most commonly used turbulence models. The model attempts
to predict turbulence by two partial differential equations for two
variables, the turbulence kinetic energy (k) and the specific rate of
dissipation (u) [35]. In free shear layers away from surfaces, the
standard k ε model is utilized while the k u model is used in the
sub-layer of the boundary layer. For retaining the robust and accurate formulation of the k u model in the near wall region,
furthermore, to take the advantage of the free-stream independence of the k ε model in the outer part of the boundary layer, the
SST k u model has been used in this study which was proposed by
Menter [36]. The SST k u turbulence model is a two-equation
eddy-viscosity model which has become very popular. The use of
a k u formulation in the inner parts of the boundary layer makes
the model directly usable all the way down to the wall through the
viscous sub-layer; hence the SST k u model can be used as a LowReynolds turbulence model without any extra damping functions.
A. Sagharichi et al. / Energy 161 (2018) 753e775
mt ¼
a* a1 k
max½a1 u; F2 761
(9)
Where constant a1 is equal to 0.31 and coefficient F2 is calculated by
the following relation:
("
F2 ¼ tanh
!#2 )
pffiffiffi
2 k 500w
max * ;
b ud ud2
(10)
d is the distance to the nearest surface. Because the low-Reynolds
effects only modify the near wall boundary layer, low-Reynolds
corrections of Wilcox [38] are applied only to the k u part of
the SST model for a*, a1 and b* coefficients.
2.4. Variable pitch method
Fig. 17. Single blade torque coefficient of wind turbine at s ¼ 0.8 and TSR ¼ 2.5.
Then the SST formulation switches to a k ε behavior in the freestream layers. In this paper, for achieving the turbulence effect of
numerical simulation, SST k u turbulence model which has the
capability of capturing proper behavior in the near-wall layers and
separated flow regions is used [21,37]. The combined model of the
k u model and the k ε model (SST k u model) expressed as
[38]:
vðrkÞ=vt þ v ruj k
"
#
h
v
vk
ðm þ sk mt Þ
b* ruk 1
vxj ¼
vxj
vxj
i
00 00
þ a1 Mt2 ð1 F1 Þ þ Pk þ ð1 F1 Þp d
(6)
"
v
vu
ðm þ su mt Þ
vðruÞ=vt þ v ruj u vxj ¼
vxj
vxj
þ 2ð1 F1 Þ
q ¼ u:t
rsu2 vk vu
bru2
u vxj vxj
*
r
u
00 00
ð1 F1 Þp d þ a Pk
mt
k
(7)
Here Pk is the production of turbulence. Turbulent Mach numpffiffiffiffiffiffiffiffiffiffiffiffiffi
and
pressure
dilatation,
ber,
Mt ¼ 2k=c2
*
p d ¼ g2 Pk Mt2 þ g3 rb kuMt2 corrects the compressibility effects
of the turbulent compressible flow. The coefficients of transport
equations, sk , s6 , a and b are calculated by the blending function,
(
"
!
#)
pffiffi
4rs62 k
k 500w
F1 ¼ tanh min max * ; ud2 ; CD d2 , using the following
00
00
b ud
k6
equation
4 ¼ F1 41 þ ð1 F1 Þ42
b ¼ a:sinðqÞ
(11)
Where a is blade pitch amplitude. Also, Azimuth angle q can be
written as a function of u:
#
þ ð1 F1 Þb a1 Mt2 ru2
In this study, a variable pitch wind turbine has been numerically
investigated and the results have been compared with a fixed pitch
case. Recently, Chen et al. [39] showed that a sinusoidal variable
pitch angle can significantly improve the performance of a VAWT.
In addition, in 2014, Kozak [40] showed that a sinusoidal pitch can
be used to optimize the angle of attack at different azimuth angles
(q). The pitching movement of the blades reduces the variation of
the angle of attack in upstream and downstream phases. Therefore,
it reduces the destructive effects of the dynamic stall at various
azimuthal angles [40]. The blade could change its pitch angle by
using a variable pitch mechanism which would be convenient for
the self-starting of wind turbines especially at low tip speed ratios
[41]. A schematic of variable pitch mechanism based on a four-bar
linkage with four blades has been shown in Fig. 3. Fig. 4 also indicates the difference between the position of the blades in a variable and a fixed-pitch wind turbine at different azimuth angles. It
is shown that the blade has been controlled to keep the angle of
attack below of the dynamic stall angle. In UDF, the blade pitch
angle b can be defined as:
(8)
Where 4 represents each of sk , s6 , a and b, and subscripts 1 and 2
correspond to k u and k ε turbulence models, respectively.
According to the SST model, the eddy viscosity mt is:
(12)
Where t is time of motion and u is the rotating speed of rotor zone
which is computed by TSR and velocity input U, as shown in
Equation (2). By combining two above equations, the blade pitch
function can be expressed as:
b ¼ a:sinðu:tÞ
(13)
The frequency of pitching oscillation (up) can be obtained by:
up ¼ a:u:cosðqÞ
(14)
In the present study, a sinusoidal pitch angle with an amplitude
of a ¼ 10 and as a function of the azimuthal angle has been used to
adjust the movement of blades during a full rotation for different
solidities. In previous work [8] a sinusoidal variable pitch mechanism with amplitude a ¼ 3 , a ¼ 10 , a ¼ 20 , a ¼ 36 has been
analyzed and the results proved that a sinusoidal pitch with a ¼ 10
has the best performance in self-starting while the power coefficient of the variable-pitch blade with a ¼ 20 and a ¼ 36 decreases
at high TSRs. Therefore, due to the better performance of turbine for
the amplitude of 10 , this amplitude has been chosen.
In Equation (14), u is the rotating speed of rotor zone which can
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A. Sagharichi et al. / Energy 161 (2018) 753e775
Fig. 18. a) Average torque coefficient of single blade, b) Percentage difference for torque coefficient of fixed and variable pitch, c) Average torque coefficient of a fixed blade, d)
Average torque coefficient of a variable pitch blade in upwind and downwind of turbine.
be computed by TSR, R (radius of the turbine) and velocity input U
(refer to Equation (2)). Since the velocity inlet and radius of the
turbine are constant, the value of rotor rotational speeds varies
corresponding to TSR which are shown in Table 1. Ultimately, from
Equation (14) the frequency of pitching up can be computed using
azimuth angle q, pitch amplitude a and rotor rotational speed u.
Therefore, different frequency related to different azimuth angles
will be achieved.
2.5. Computational domain and boundary conditions
The computational domain and boundary conditions are shown
in Fig. 5. In addition, the no-slip wall boundary condition was
selected around the surface of blades. The interface boundary
conditions are used around the circumference of rotational
domains to ensure the continuity in the flow field and the solver
performs mesh manipulation between two adjacent domains in
each time steps.
The dimensions of the computational domain have been chosen
to minimize the blockage effects around the turbine [7]. The width
of the domain is set 14R (where R is the radius of the turbine) and
the distance of the inlet and the outlet boundary condition from the
rotor axis is 7R and 14R, respectively. The computational domain
has been divided into three regions: the outer stationary domain,
the inner rotational domain and the blade domains which are
surrounded by the rotor domain. The free-stream velocity is set at
9 m/s in this study. Also, the pressure gradient at the downstream
end of the domain is assumed to be zero. For variable pitch simulation, the four dynamic sub-domains (consist of a rotor and three
blade domains) can be rotated independently which is added as a
A. Sagharichi et al. / Energy 161 (2018) 753e775
763
Fig. 19. a) Average torque coefficient of single blade, b) Percentage difference for torque coefficient of fixed and variable pitch blade, c) Average torque coefficient of a fixed blade, d)
Average torque coefficient of a variable pitch blade in upwind and downwind of turbine.
user-defined function (UDF) to ANSYS/FLUENT software. Therefore,
the rotor is rotated merely around z-axis while the three blades can
be rotated not only around the z-axis but also around the axis
passing through 25% of their chord length. For such motions, a code
has been programmed in C and added into the solver as a UDF. The
UDF provides the blades fluctuation to obtain the desired angle of
attack.
2.6. Grid generation
A grid refinement study has been carried out to ensure the
independency of the numerical results from grid mesh resolution.
Three unstructured grid resolutions consist of coarse, medium and
fine resolutions with 203762, 513628 and 909796 nodes are performed. The grid resolution details along the NACA 0021 airfoil and
boundary layers are described in Table 2. The numerical accuracy of
the SST k u model was assured when the maximum value of yþ is
less than one. Therefore, the first grid spacing on the blade surfaces
is set so that yþ is approximately equal to one in all cases. Also, grid
lines close to the surface are perpendicular to the blade surfaces [7].
Therefore, the boundary layer grid generation was used around the
blade surfaces.
Fig. 6 shows the torque coefficient versus dimensionless azimuth angle at optimum TSR for the three grid resolutions. As it is
presented, the differences between torque coefficient of medium
and fine grids are significantly smaller than those between coarse
and fine grids. Therefore, the results are converged at the medium
size of grid resolution and the domain with 513628 nodes is used
for following numerical simulation of VAWT in this paper. Fig. 7
shows the grid resolution generated around the rotor and blade
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A. Sagharichi et al. / Energy 161 (2018) 753e775
Fig. 20. a) Average torque coefficient of a single blade, b) Percentage difference for torque coefficient of fixed and variable pitch, c) Average torque coefficient of a fixed blade, d)
Average torque coefficient of a variable pitch blade in upwind and downwind of turbine.
for the medium mesh. Due to the crucial importance of the grids
quality and quantity around the blades on the simulation results,
grid with a higher density has been used around the blades.
2.7. Validation
In order to validate the numerical simulation, the power coefficients are compared to the experimental result which is presented by Castelli et al. [42]. In this study, both experimental and 2D numerical tests are investigated for prediction the performance
of a small scale high solidity H-type Darrieus VAWT which had
three NACA0021 blades. The chord length of the blade is 85.8 mm,
and the diameter and the height of the blade are 1030 mm and
1456.4 mm, respectively. Table 3 shows the dimensions and parameters of the CFD model and Darrieus wind turbine tested by
Castelli et al.
The comparison between the numerical and experimental results has been shown in Fig. 8. This numerical simulation is done
using different turbulence models such as RNG k ε, SST Transition,
SST k u, and the curves of CP versus l are plotted in Fig. 8. As is
evident from this figure, the SST k u turbulence model could
predict flow properties relatively better compared to two another
turbulence models. SST k u model has an appropriate behavior in
adverse pressure gradients and correctly estimates the onset of
flow separation under adverse pressure gradients [8]. On the other
hand, the RNG k-e and SST-Transition model have the weak ability
in predicting the behavior of flow and excessively overestimated or
underestimated the experimental power coefficient. Therefore,
based on this figure, SST k u model has been chosen.
The numerical result related to SST k u model matches well
A. Sagharichi et al. / Energy 161 (2018) 753e775
765
Fig. 21. a) Average torque coefficient of a single blade, b) Percentage difference for torque coefficient of fixed and variable pitch, c) Average torque coefficient of a fixed blade, d)
Average torque coefficient of a variable pitch blade in upwind and downwind of turbine.
with the experimental data and it well predicts the power coefficients trend at wide ranges of TSRs. The slight difference between numerical and experimental data could be a result of
simulating turbine in two dimensions while the experiment was
performed using a three-dimensional model. Also, the rotor hub
and blade connections have not been included in present study. As
a result of the absence of these features in 2D simulation, for TSRs
higher than 2, the present numerical power coefficients are slightly
higher than the experimental results. This difference was observed
in different research such as Mohamed et al. [43], Nobile et al. [44],
Sun et al. [45].
3. Numerical results and discussion
In this section, the numerical results of the 2D simulation are
presented. In order to study the solidity effects, the number of
blades and the chord length are taken as a variable parameter while
the rotor radius is assumed to be constant.
Seven different values for solidity (0.2, 0.33, 0.4, 0.5, 0.6, 0.66
and 0.8) have been considered to study the effects of solidity on the
performance of a three-bladed VAWT with fixed and variable pitch
blades. The blade chord lengths corresponding to these solidities
are presented in Table 4. Also, the flow field around the wind turbine with two, three and four blades has been analyzed at several
TSRs, different angular velocities with a constant wind speed of
9 m/s. Figs. 9 and 10 show the effect of solidity on the power coefficient of the fixed and variable pitch angle of a three-bladed
VAWT, respectively. It is shown that the maximum power coefficient increases as solidity increases. According to Mohammed's
study [46], each Darrieus VAWT has an individual best efficiency
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A. Sagharichi et al. / Energy 161 (2018) 753e775
Fig. 22. a) Average torque coefficient of a single blade, b) Percentage difference for torque coefficient of fixed and variable pitch blade, c) Average torque coefficient of a fixed blade,
d) Average torque coefficient of a variable pitch blade in upwind and downwind of turbine.
value of solidity. It is seen in Fig. 9 that the optimal TSR is around 2.5
for the fixed-pitch angle turbine while the optimum TSR for the
variable pitch angle wind turbine is around 2 as shown in Fig. 10. So,
in variable pitch turbine, the optimum TSR occurs at lower TSRs. In
other words, in variable pitch turbine, it is possible to extract energy at lower TSR. Moreover, the ideal value of the solidity for the
fixed and variable pitch wind turbines is 0.6 and 0.8, respectively.
This leads to an increased power/cost ratio for a variable pitch angle
VAWT compared to a fixed-pitch VAWT. Also in both fixed and
variable pitch cases, a higher TSR should be used to cover the
smaller swept area at low solidities. It clarifies that for higher solidities, the rotor can produce more torque at medium and low TSRs
[19]. It is also observed that, at low solidities, the CP l curve is
quite flat near the optimum power coefficient which confirms the
results by G. Bedon et al. [22]. Finally, it can be illustrated that both
the operational zone of a turbine and design TSR (for MPPT condition) are reduced by increasing of solidity.
The variation of the maximum power coefficient versus solidity,
for the fixed and the variable pitch angle wind turbine, are illustrated in Fig. 11. It is shown that at lower solidity (s ¼ 0.2), the
power coefficient of the fixed pitch angle wind turbine is more than
the variable one. However, in variable pitch angle wind turbine, the
increasing of the solidity improves the optimum power coefficient.
The highest power coefficient in the fixed pitch angle wind turbine
occurs at s ¼ 0.6 and 0.8, respectively. From the aforesaid results, it
can be stated that the difference between the solidity which
maximum power coefficient occurs is related to the reduction of
wind speed which passes through the interior of the fixed pitch
VAWT. This is the main reason which affects the performance of
VAWT and deteriorates the power production in higher solidity
A. Sagharichi et al. / Energy 161 (2018) 753e775
767
Fig. 23. a) Average torque coefficient of a single blade, b) Percentage difference for torque coefficient of fixed and variable pitch blade, c) Average torque coefficient of a fixed blade,
d) Average torque coefficient of a variable pitch blade in upwind and downwind of turbine.
[20].
However, in the variable pitch angle wind turbine, by increasing
the solidity, the power coefficient increases. On the other hand, at
high TSR and solidity, the blockage occurs which the pressure in
front of the turbine increases and the air flow cannot pass through
it [8]. The advantage of variable pitch angle technique is the ability
to decrease the angle of attack at azimuthal angles where it exceeds
the stall angle. Therefore, in variable pitch wind turbine with high
solidity, the blades can be rotated while the angle of attack is lower
than the fixed pitch turbine. It leads to reduce the vortex shedding
strength which prevents the interaction between upwind vortexes
and blades in the downstream region. As a result, the performance
and the power production of variable pitch angle wind turbine will
be increased compared to the fixed pitch angle ones at the higher
solidities.
Figs. 12 and 13 illustrate the variation of power coefficient with
different number of the blades for the fixed and the variable pitch
wind turbine, respectively. It should be noted that the solidity will
be changed by changing the number of the blades due to Equation
(3). Fig. 12 shows that the maximum power coefficient occurs at the
four-blade for the fixed pitch angle wind turbine. Employing more
blades in the fixed pitch angle wind turbine reduces the operational
zone which limits the maximum power production. Fig. 13 shows
that the maximum power coefficient is obtained at three-bladed in
variable pitch angle wind turbine. Also, the higher power coefficient is obtained using variable pitch angle wind turbines
compared to the fixed pitch angle ones. It is also found that the tip
speed ratio related to the maximum power coefficient decreases
with increasing the number of blades. Therefore, there is an optimum in number of blade in variable pitch turbines which
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A. Sagharichi et al. / Energy 161 (2018) 753e775
Fig. 24. a) Average torque coefficient of a single blade, b) Percentage difference for torque coefficient of fixed and variable pitch blade, c) Average torque coefficient of a fixed blade,
d) Average torque coefficient of a variable pitch blade in upwind and downwind of turbine.
maximized the power coefficient. Fig. 13 also illustrates that the
maximum power coefficient has been shifted to lower TSRs by
increasing the number of the blades (increasing solidity).
Fig. 14 shows the total torque coefficient of a variable and a fixed
pitch wind turbine with two, three and four blades when TSR is
equal to 3. It is shown that there are N peaks in instantaneous
torque coefficient curves, and the distance between two consecutive peaks is 2Np , where N is the number of blades. Since the drag
force is the dominant force at such TSR (TSR ¼ 3), two blades of
wind turbines generate more torque at peak regions compared to
the three and four blades wind turbines. Therefore, in a higher
number of blades more drag and less lift are produced which
lowered the power and torque of the turbine.
Fig. 15 shows the total torque coefficient for the variable and the
fixed pitch angle of a turbine with different solidities and number of
the blades at TSR ¼ 1. It shows that the turbine torque coefficient is
higher in the variable pitch angle blades. Also, the maximum difference between torque coefficient is illustrated for the solidity of
0.5 (turbine with three blades). Moreover, it shows that with
increasing the solidity, lower torque fluctuation will be imposed on
the turbine structure. However, there are drawbacks in use of turbines with more than three blades. Firstly, more blades will increase cost, weight, and inertia. Also, more blades make more
vibration. Increasing the chord length will also increase weight. As
the solidity increases, the aspect ratio will decrease and losses at
the blade tip will increase. So, in this case, it seems that the most
efficient turbines in variable pitch turbine should have 3 blades or
the solidity of 0.5 [47].
Fig. 16 shows the instantaneous blade torque coefficient for
three rounds of blade rotation at tip speed ratio of 2.5. It is shown
A. Sagharichi et al. / Energy 161 (2018) 753e775
Fig. 25. MPPT curve of the fixed and the variable pitch wind turbine for different
solidities.
that the variable pitch angle method increases the maximum torque which leads to improving the power output. When the vortices
combine and elongate at the downstream of the turbine and reduce
the blade torque which affects the turbine's performance. Also, the
lift force and the energy extraction from the downstream of the
turbine will be decreased by reducing the relative velocity. These
vortices contact the blades at the downstream region which leads
to reduce the torque coefficient and increase torque fluctuations. In
azimuth angles between 180 and 360 (540e720 for the second
round and 900 to 1080 for the third round of rotation) due to the
presence of these vortices, the torque production has been
decreased. As this figure shows, in turbines with the fixed-pitch
angle, the blades interact with stronger vortices for higher blade
solidity and higher tip speed ratio.
This explains the reason for the steeper slope of reducing the
power output for turbines which operate at tip speed ratio of more
than 3 in Fig. 9. On the other hand, as shown in Fig. 16, in turbines
with variable pitch angle, the problem of reducing torque production in the downstream region due to high solidity and high TSR has
been solved. For example, a four-blade turbine which was not able
to generate torque at TSR of 2.5 in the downstream part, with aid of
variable pitch method can produce power in all over of azimuth
angles, except at a small rotation angle (between 220 and 240 ).
The similar behavior is observed for the turbine with a solidity
of 0.8. Fig. 17 shows the variation of the instantaneous blade torque
coefficient for the fixed and the variable pitch turbine at s ¼ 0.8 and
TSR of 2.5. As can be seen in the variable pitch angle blades, higher
torque is produced at the downstream of the turbine which increases the efficiency of the variable pitch angle turbine. The lower
efficiency of turbines using the fixed pitch angle blades in greater
solidities is due to hitting the separated vortices to the blades
located in the downstream region.
Figs. 18e24 show a comparison between the performance of a
fixed and a variable pitch blade for VAWT with 2,3 and 4 blades and
s ¼ 0.2,0.4,0.6 and 0.8, respectively. In each figure, part a shows
average torque coefficient of single blade in upwind and downwind
of VAWT, b shows percentage difference for torque coefficient of
fixed and variable pitch blade in upwind and downwind of the
769
turbine. Part c, and d present the produced average of torque for a
blade in terms of tip speed for the fixed and the variable pitch
techniques in upwind and downwind of VAWT, respectively. The
peak of average torque coefficient for a variable pitch angle turbine
with 2 blades is higher than the fixed pitch angle turbine while the
tip speed ratio is in the range of 0.5e3. However, the average torque
decreases in the variable pitch angle turbine for the tip speed ratio
of more than 3. The main reason for increasing the torque coefficient of the turbine with the variable pitch angle is increasing the
average torque for the blade 1 in the upstream region. On the other
hand, during the tip speed ratio of more than 3, the average blade
torque is reduced compared to the fixed pitch angle of the wind
turbine.
However, in the turbine with 3 blades, more power is generated
in the variable pitch technique compared to the fixed pitch angle
method for different tip speed ratios.
Fig. 18b shows that in the variable pitch angle turbine, more
power is produced in the upstream region when the tip speed ratio
is less than 1.75. Also, it shows that by increasing the tip speed ratio
to more than 1.75, there generated power is negligible in the
downstream region for the fixed pitch angle turbine due to the
interaction of the vortices with the blades. Moreover, the generated
power is increased more than 600% in the variable pitch angle
turbine compared to the fixed pitch angle turbine due to the lower
interaction of the vortices with the blades. Fig. 18c and d shows that
the average torque becomes zero in the fixed pitch angle turbine for
the tip speed ratio of more than 3, while the variable pitch angle
turbine generates power in this range of tip speed ratio. Therefore,
the operation range increases drastically for the variable pitch
angle turbine in high TSRs. The same trend is also observed for a
three and four-bladed turbine, and at high tip speed ratios, the
output power of turbine has been increased in the downstream
part. Therefore, a variable pitch turbine has the capability to overcome the big problem of fixed pitch VAWT turbines at high tip
speed ratios, which is the lack of power production due to
increasing solidity.
This trend is also repeated for Figs. 21e24. Also since a fixed
pitch angle turbine acts like a rigid body for higher tip speed ratios
and solidity, the power generation decreases with increasing of the
solidity and the tip speed ratio. In other words, in these conditions,
the effective angle of attack is too small which decreases the
pressure gradient on the top and bottom sides of the blades.
Therefore, the power generation would be reduced.
Fig. 25 represents the MPPT curve of the fixed and the variable
pitch angle VAWT for different solidities. In the variable pitch angel
turbine, more power is generated in the lower solidity (0.2, and
0.5), while in the fixed pitch angle turbine, more power is achieved
at the higher solidity (0.8). Also, this figure presents that, for the
lower range of rotational speeds, the variable, and the fixed pitch
angle turbines produce similar power for the solidity of 0.2 and 0.5.
In contrast, for the solidity of 0.8, the fixed pitch angle turbine
generates more power at the lower rotational speed. In the turbine
with a lower number of blades and with higher rotational speeds,
the maximum value of torques is reduced, resulting in smaller
gearbox and generator. The MPPT curve shows that a wind turbine
with lower solidity or fewer blade numbers is more optimal than
other cases. In fact, a high rotational speed of the rotor is desirable
in order to reduce the required gearbox ratio [48].
Figs. 26e28 indicate the vorticity distribution around the rotor
for a complete revolution of the fixed and the variable pitch angle
wind turbine at TSR ¼ 3 for three different solidities of 0.2, 0.5 and
0.8, respectively. It is shown that the wake region at the downstream in the variable pitch angle rotor is smaller compared to the
fixed pitch angle rotor. This is related to the weaker and the smaller
vortices that are produced by the variable pitch angle blade which
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A. Sagharichi et al. / Energy 161 (2018) 753e775
Fig. 26. Vorticity distribution at a complete rotation for fixed pitch (left) and variable pitch wind turbine (right) at s ¼ 0.2.
are dissipated faster. Figs. 29 and 30 show the pressure contour
with the streamline for both fixed and variable pitch blade at tip
speed ratio at TSR ¼ 2 and s ¼ 0.5, respectively. It shows that more
separation occurs at the rotation angle of 90e180 on fixed pitch
blade. Moreover, in the variable pitch angle method, the streamlines are attached to the NACA0021 blade surface, while a separation is detected for the fixed-pitch blade. This justifies the
decreasing torque coefficient in high solidity in Fig. 17. Furthermore,
the pressure gradient on leading edge of the turbine in the fixedpitch turbine is wider than fixed-pitch one which leads more torque production [48].
The variable pitch mechanism decreases the angle of attack and
weaker vorticity in the downstream region are formed. In other
words, in fixed-pitch wind turbines with greater solidities (as can
be seen in Fig. 29), the chance of encountering detached vortices
from the upstream to downstream part is increased. Therefore, lift
A. Sagharichi et al. / Energy 161 (2018) 753e775
771
Fig. 27. Vorticity distribution at a complete rotation for fixed pitch (left) and variable pitch wind turbine (right) at s ¼ 0.5.
force and as a consequence, the output power has been reduced. In
contrast, in a variable pitch VAWT, the flow separation, the vortices
interaction and dynamic stall and torque oscillation have been
reduced. It can be predicted that, in variable pitch wind turbine, the
vibrations and fatigue stress acting on the rotor and structure will
be reduced. The reason for the increase of turbine power in high
solidities in variable pitch cases can be explained by these figures
that by using the variable pitch angle technique, the chance of
encountering detached vortices from the upstream to downstream
at high solidities is reduced which leads to decrease the flow
separation and drag force on the blades.
4. Conclusions
The present study investigates the effect of solidity on variable
and fixed pitch blade of a VAWT. For this purpose, a VAWT with
two, three and four straight blades at five different solidities in the
range of 0.2e0.8 was investigated. The main finding of the obtained
results is:
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A. Sagharichi et al. / Energy 161 (2018) 753e775
Fig. 28. Vorticity distribution at a complete rotation for fixed pitch (left) and variable pitch wind turbine (right) at s ¼ 0.8.
In cases which are required more initial self-starting torque, the
use of rotors with high solidities and variable pitch angle blades
is preferable.
The difference between the power coefficients of the fixed and
the variable pitch turbine is larger in the higher solidities and
this is related to the reduction of wind speed which passes
through the interior of the fixed pitch VAWT.
A variable pitch turbine has the capability to overcome the lack
of power production due to increasing solidity which can be
seen in the fixed pitch VAWT turbines at high tip speed ratios.
Variable pitch angle VAWT reduces the flow separation and drag
force on the blade at medium and high solidities. Also, it will
reduce the vibrations and fatigue stress acting on the rotor and
shaft.
A. Sagharichi et al. / Energy 161 (2018) 753e775
773
Fig. 29. Pressure distribution and streamlines for a fixed-pitch blade during full rotation at TSR ¼ 2.5 and s ¼ 0.8.
At low tip speed ratios which a turbine is at high risk of dynamic
stall, the use of a variable-pitch turbine at the solidity of 0.5 is
recommended.
The drag force of the variable pitch angle is lower compared to
the fixed pitch angle. This is due to the fact that the variable
pitch blade produces fewer and smaller vortices with faster
dissipation, especially at the higher range of solidities.
The variable pitch angle VAWT has higher power production
and is more controllable compared to the fixed pitch VAWT, at
high solidities.
Generally, the variable pitch angle technique decreases the
chance of encountering detached vortices from the upstream to
downstream at the higher range of solidities that studied in this
paper. Therefore, the vortex shedding strength, flow separation and
drag force on the blade are declined.
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Fig. 30. Pressure distribution and streamlines for a variable-pitch blade during full rotation at TSR ¼ 2.5 and s ¼ 0.8.
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