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Reynolds Number Effect on the Optimization of a Wind
Turbine Blade for Maximum Aerodynamic Efficiency
Mingwei Ge, Ph.D.1; De Tian, Ph.D.2; and Ying Deng3
Abstract: Because of the increase in wind rotor size, the Reynolds number of an airfoil profile can reach a very high value. The effect of the
Reynolds number on the aerodynamic performance of airfoils is investigated, and its influence on the optimal design of a wind rotor aiming to
maximize the power coefficient is discussed. Six airfoils are involved—four DU and two NACA6—as well as five Reynolds numbers varying
from 106to 107, which cover most commercial wind turbines. At a higher Reynolds number, all of the airfoils exhibit better performance,
such as a higher lift coefficient, a lower drag coefficient, and a larger lift-to-drag ratio at a given angle of attack. The largest lift-to-drag ratio
and the corresponding lift coefficient and angle of attack also change with the Reynolds number, which in turn affects both the performance
and the optimal shape of a blade. The results show that a practical blade operating at a higher Reynolds number requires a duller shape with a
greater twist angle, and has a better power coefficient than those operating at lower Reynolds numbers. DOI: 10.1061/(ASCE)EY.1943-7897
.0000254.© 2014 American Society of Civil Engineers.
Author keywords: Aerodynamic design; Power coefficient; Reynolds number effect; Wind turbine blade.
Introduction
Aerodynamic design of a wind rotor is extensively studied in the
field of wind energy (Fuglsang and Madsen 1999;Benini and
Toffole 2002;Wang et al. 2009). Through excellent design using
sophisticated aerodynamic principles, modern wind turbines are
able to capture wind energy efficiently. Currently, the application of
large-scale wind turbines has become an irreversible trend in wind
power technology. Many types of 5-MW–7-MW wind turbines have
recently been successfully designed and brought into commercial
operation around the world. These include the Repower 5-MW
wind turbine, the Siemens 6-MW wind turbine, and the Vestas
7-MW wind turbine (V164) among others. (Jonkman et al. 2009;
Marsh 2012). With increasing blade sizes, the Reynolds number
R¼ρWc=υextends to a wider range, where W= relative velocity
to a local airfoil profile; c= chord length of the airfoil; ρ= density of
the air flow; and υ= dynamic viscosity. Fig. 1shows the distribution
of local Reynolds numbers along the blade span under the rated con-
dition for wind turbines HD28 (200 KW), HD50 (750 KW), UP77
(1.5 MW), UP126 (3 MW), UP136 (6 MW), and UP200 (12 MW),
all of which were designed by the United Power Company. Here air
flow is assumed to be at standard atmosphere with the properties
ρ¼1.225 kg=m3and υ¼1.793 ×10−5kg=ðm·sÞ. The local
Reynolds numbers of the airfoil profiles along the span are found
to be as high as 14 million for the 12-MW wind turbine; however,
they are approximately only 2 million for the 200-kW equipment. In
addition, for the same wind turbine operating at different altitudes,
the Reynolds number is very different because of changes in air
density and dynamic viscosity. For example, the Reynolds number
of a blade for a 1.5-MW wind turbine operating in southeastern
China is approximately 4 million, whereas for the same wind turbine
operating in Tibet in southwestern China it is only approximately
2–3 million.
Variation in the Reynolds number brings a new issue to the aero-
dynamic design of a wind rotor. Substantial change in the optimal
shape of a wind turbine blade is induced by a change in Reynolds
number (Ceyhan 2012), but related research is still lacking. In the
aviation industry, scale effects on aircraft and high-Rwind tunnel
experiments are studied extensively (Haines 1994;Rechzeh and
Hansen 2006). For better performance, the airfoils of the aircraft
are generally designed under the flight Reynolds number, so it
is very important to determine the Reynolds number effect when
optimizing a wind rotor.
In modern aerodynamic design of a blade, boosting the power
coefficient is always of interest (Johansen et al. 2009). Hence, the
power coefficient of a wind turbine is maximized through both
theoretical analysis (ideal blade without constraints) and numerical
optimization (practical blade) to give an insight into the Reynolds
number effect on the blade’s aerodynamic design. The blade
element momentum (BEM) theory (Burton et al. 2011;Lanzafame
and Messina 2007), which is widely used in wind rotor design in
scientific research and industry, is adopted in this work.
Reynolds Number Effect on Airfoil Performance
As a dimensionless number that gives a measure of the ratio of
inertial to viscous forces, the Reynolds number can significantly
influence the flow around an airfoil. Hence, the performance char-
acteristics of an airfoil, such as the lift coefficient Cl, drag coeffi-
cient Cd, and lift-to-drag ratio Cl=Cd, change considerably under
different Reynolds numbers.
Many specific airfoils for wind rotors with a high lift coefficient,
leading-edge roughness insensitivity, and good stall performance
1Lecturer, State Key Laboratory of Alternate Electrical Power System
with Renewable Energy Sources, North China Electric Power Univ.,
Beijing 102206, P.R. China (corresponding author). E-mail: gmwncepu@
163.com
2Professor, State Key Laboratory of Alternate Electrical Power System
with Renewable Energy Sources, North China Electric Power Univ.,
Beijing 102206, P.R. China.
3Professor, School of Renewable Energy, North China Electric Power
Univ., Beijing 102206, P.R. China.
Note. This manuscript was submitted on May 3, 2014; approved on
October 14, 2014; published online on December 4, 2014. Discussion per-
iod open until May 4, 2015; separate discussions must be submitted for
individual papers. This paper is part of the Journal of Energy Engineering,
© ASCE, ISSN 0733-9402/04014056(12)/$25.00.
© ASCE 04014056-1 J. Energy Eng.
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have been developed to adapt the operational state of a wind turbine
(Timmer and Van Rooij 2003;Fuglsang and Bak 2004;Tangler and
Somers 1995). In the present paper, the following airfoils are stud-
ied: DU00-W2-401, DU00-W-350, DU97-W-300, DU97-W-250,
NACA63421, and NACA64618. Airfoils beginning with “DU”
are developed by Delft University of Technology; those beginning
with “NACA”are developed by NASA. Airfoil characteristics, in-
cluding lift coefficient Cland drag coefficient Cd, are determined
from numerical calculations by the software RFOIL.RFOIL is an
extension of XFOIL, an interactive program first developed by the
Massachusetts Institute of Technology (Drela 1989) for the design
and analysis of subsonic isolated airfoils. In RFOIL, regarding the
effect of rotation on airfoil characteristics, the boundary layer equa-
tions from XFOIL are modified for better poststall prediction (van
Rooij 1996;Snel et al. 1994), and improved prediction for maxi-
mum lift coefficient is achieved. Fig. 2(a) compares the XFOIL/
RFOIL code and measurements by Timmer and Van Rooij
(2003). Although the numerical results cannot match the measure-
ments perfectly, the main part of the polar curve is fairly well
predicted. To match the wind tunnel data better, a 9% increase
in predicted drag coefficient is suggested by Timmer (2009).
Corrected prediction results for NACA64618 at R¼3×106
and the measurement from the Langley low-turbulence pressure
tunnel (Timmer 2009) are given in Figs. 2(b and c). A very good
match between the predicted drag data with correction and the ex-
perimental results can be observed, which means that fairly reliable
results in engineering can be obtained by RFOIL with correction.
For its rapid evaluation and good reliability, the software is also
used in the optimization of wind turbine airfoils by some research-
ers (Grasso 2013;Bizzarrini et al. 2011). The present study, focuses
only on normal airfoils with smooth surfaces and all of them are set
to a standard form with a unit chord length in RFOIL’s calculations.
Adopting the correction method proposed by Timmer (2009), all the
predicted drag coefficients are multiplied by a factor of 1.09.
As shown in Fig. 1, the Reynolds numbers of local airfoils
across the range of 200-kW–6-MW wind turbines mainly vary
between 1×106and 1×107, giving an approximate range of the
Reynolds numbers for most commercial wind turbines. Hence, five
Reynolds numbers—2×106,4×106,6×106,8×106, and 1×
107are selected here with an equal difference between 1×106and
1×107to cover most commercial wind turbine blades. Lift and
drag coefficients from RFOIL prediction for the selected airfoils
are plotted against the attack angle for the proposed Reynolds num-
bers in Figs. 3(a–f). For all of the airfoils, as the Reynolds number
increases the lift coefficient increases and the drag coefficient de-
creases. Moreover, the stall angle of attack is significantly delayed
at higher Reynolds numbers for all airfoils. For thicker airfoils,
such as DU00-W2-401 and DU00-W-350, the change in lift coef-
ficient is more sensitive than that for thinner airfoils, such as DU97-
W-300, DU97-W-250, NACA63421, and NACA64618.
Figs. 4(a–f) show the lift-to-drag ratio results for the six airfoils at
different angles of attack. At a constant Reynolds number, the ratio
first increases with the attack angle to reach a maximum value; it
then decreases to a rather low value for all airfoils at 15°. In general,
the lift-to-drag ratio significantly increases as the Reynolds number
increases at the same operating attack angle. Taking the DU00-W2-
401 airfoil as an example, the lift-to-drag ratio at R¼1×107is
approximately twice that at R¼2×106under attack angles 4–7°.
To account for the drag effect on the turbine optimal power coeffi-
cient, the following relationship is proposed by Wilson et al. (1976):
CPmax
¼16
27 λN2=3
1.48 þðN2=3−0.04Þλþ0.0025λ2−Cd
Cl
1.92Nλ
1þ2λN
ð1Þ
r
R
20 40 60 80 100
0.0E+00
2.0E+06
4.0E+06
6.0E+06
8.0E+06
1.0E+07
1.2E+07
1.4E+07
6 MW
3 MW
1.5 MW
12 MW
750 KW
200 KW
Fig. 1. Distribution of Reynolds numbers with length of blade for
HD28 (200 KW), HD50 (750 KW), UP77 (1.5 MW), UP126 (3 MW),
UP136 (6 MW), and UP200 (12 MW) under the rated condition
Cd
00.01 0.02 0.03
–1
–0.5
0
0.5
1
1.5
XFOIL
RFOIL
Wind tunnel
R = 2×106
α
010 20
–0.5
0
0.5
1
wind tunnel
RFOIL
Cl
Cd
Cl
00.02 0.04 0.06
–0.5
0
0.5
1
wind tunnel
RFOIL
(a) (b) (c)
Cl
Fig. 2. Comparison of the prediction result and the measurement: (a) DU91-W2-250 polar curve (from Timmer and Van Rooij 2003);
(b) NACA64618 lift coefficient; (c) NACA64618 polar curve (predicted drag coefficient multiplied by 1.09)
© ASCE 04014056-2 J. Energy Eng.
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where N= number of blades and λ= tip-speed ratio of the wind
turbine. The sensitivity of the optimal power coefficient with the
change in Cd=Clcan be obtained as follows:
ΔCPmax ¼−
16
27 λ1.92Nλ
1þ2λNΔCd
Clð2Þ
At a constant tip-speed ratio and fixed blade number, the change
in CPmax is inversely proportional to the change in Cd=Cl. Hence, it
is predictable that at a higher Reynolds number the optimal power
coefficient is better because of the increase in lift-to-drag ratio.
Figs. 5(a and b) show the angles of attack and lift coefficients
corresponding to the maximum lift-to-drag ratio for the six airfoils
under the selected Reynolds numbers. As shown, at the point of
maximum lift-to-drag ratio, both angle of attack and lift coefficient
increase with the Reynolds number for thicker airfoils (DU00-
W2-401 and DU00-W-350). Taking DU00-W-350 as an example,
the highest Cl=Cd¼90.4occurs at α¼6.3° with Cl¼1.0581
under R¼2×106; the maximum Cl=Cd¼138.6corresponds to
α¼8.2° and Cl¼1.3795 at R¼1×107. Unlike those for
thicker airfoils, both angle of attack and corresponding lift coeffi-
cient at the highest lift-to-drag ratio decrease with Reynolds num-
ber for thinner airfoils (DU97-W-300, DU97-W-250, NACA63421,
and NACA64618). For example, the highest Cl=Cdfor the
NACA64618 airfoil occurs at α¼5.3°, corresponding to Cl¼
1.1284 at R¼2×106; the maximum Cl=Cdcorresponds to α¼
3.3° with Cl¼0.8948 at R¼1×107. For thinner airfoils, in spite
of the performance improvement at the higher Reynolds number,
both angle of attack and lift coefficient corresponding to the best
lift-to-drag ratio decrease with the Reynolds number.
The results show that airfoil performance is significantly
affected by the Reynolds number. At higher Reynolds numbers,
all airfoils clearly exhibit better performance.
Reynolds Number Effect on the Ideal Blade Design
Because of variation in the airfoil databases caused by Reynolds
number changes, the aerodynamic shape of the wind turbine blade
should be appropriately adjusted. In this section, the influence of
α
Cl
–10 –5 0 5 10 15
0
0.5
1
Cd
0.02
0.04
0.06
Cl
Cd
α
Cl
–10 –5 0 5 10 15
–0.5
0
0.5
1
1.5
Cd
0.01
0.02
0.03
0.04
0.05
α
Cl
–10 –5 0 5 10 15
–1
–0.5
0
0.5
1
1.5
Cd
0.01
0.02
0.03
0.04
Cd
Cl
α
Cl
–10 –5 0 5 10 15
–0.5
0
0.5
1
1.5
Cd
0.01
0.02
0.03
0.04
Cd
Cl
α
Cl
–10 –5 0 5 10 15
–0.5
0
0.5
1
1.5
Cd
0.01
0.02
0.03
0.04
Cl
Cd
Cd
–10 –5 0 5 10 15
0.005
0.01
0.015
0.02
α
Cl
–0.5
0
0.5
1
1.5
R = 2×106
R = 4×106
R = 6×106
R = 8×106
R = 1×107
Cd
Cl
(a) (b)
(c) (d)
(e) (f)
Fig. 3. Lift and drag coefficients for different airfoils: (a) DU00-W2-401; (b) DU00-W-350; (c) DU97-W-300; (d) DU97-W-250; (e) NACA63421;
(f) NACA 64618
© ASCE 04014056-3 J. Energy Eng.
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α
Cl/Cd
–10 –5 0 5 10 15
–20
0
20
40
60
80
100
R = 2×106
R = 4×106
R = 6×106
R = 8×106
R = 1×107
α
Cl/Cd
–10 –5 0 5 10 15
–50
0
50
100
α
Cl/Cd
–10 –5 0 5 10 15
–100
–50
0
50
100
150
α
Cl/Cd
–10 –5 0 5 10 15
–50
0
50
100
150
α
Cl/Cd
–10 –5 0 5 10 15
–50
0
50
100
150
α
Cl/Cd
–10 –5 0 5 10 15
–50
0
50
100
150
(a) (b)
(c) (d)
(e) (f)
Fig. 4. Lift-to-drag ratio for different airfoils: (a) DU00-W2-401; (b) DU00-W-350; (c) DU97-W-300; (d) DU97-W-250; (e) NACA63421;
(f) NACA64618
R
α
246810
0
2
4
6
8
10
DU00-W2–401
DU00-W–350
DU97-W–300
DU97-W–250
NACA63421
NACA64618
(×106)
R
Cl
246810
0.6
0.8
1
1.2
1.4
1.6
×106
(a) (b)
Fig. 5. Angle of attack and lift coefficient corresponding to maximum lift-to-drag ratio at different Reynolds numbers: (a) angle of attack; (b) lift
coefficient
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the airfoil databases under different Reynolds numbers on the
aerodynamic design of a wind turbine blade is investigated. Ideal
blades—that is, those with only a specific airfoil profile along the
entire span—are designed at a given tip-speed ratio. To make the
discussion more comprehensive, all blade positions (i.e., distance
from the root) are given by the local r=R, and all chord lengths are
designed in the form of c=R. This covers blades with different
lengths and different Reynolds numbers. For the so-called ideal
blade, no restrictions based on structure design, manufacture, trans-
portation, and so on, are applied in the design procedure; the chord
and twist angle distributions are determined only according to aero-
dynamic principles.
Aerodynamic Principles for Ideal Blade Optimization
As a basic theory of wind turbine aerodynamics, the blade element
momentum (BEM) theory is widely used in blade design and
assessment. To achieve the optimal blade shape, a mathematical
model based on BEM has been established by Burton et al.
(2011) that maximizes the optimal power coefficient without drag.
Unfortunately, with drag, the algebra of the analysis is more com-
plex. To cover the influence of drag, BEM results based on the drag
factor are derived in this section. For simplification, some minor
terms in the derivation are omitted to avoid the need to solve com-
plex polynomial equations for both the axial induction factor and
the tangential induction factor. Following Burton et al. (2011), both
root losses and tip losses are not considered.
In momentum theory, the flow is simplified to be a flow tube
that, because of the axial induction effect, expands after passing
through a wind rotor [Fig. 6(a)]; in BEM, the blade is assumed
to be a series of airfoils with no interactions between them. The
downstream airflow rotates with angular velocity ωin the opposite
direction of the wind rotor because of the tangential induction
effect.
Fig. 6(b) shows the blade element model. By applying momen-
tum and angular momentum conservation equations, the following
equations can be obtained from the equality of the axial force and
torque acting on the blade sector:
W2
U2
∞
Nc
RðClcos ϕþCDsin ϕÞ¼8π½að1−aÞ−ðbλμÞ2μð3Þ
W2
U2
∞
Nc
RðClsin ϕ−CDcos ϕÞ¼8πλμ2bð1−aÞð4Þ
For the inflow angle, there is
tan ϕ¼1−a
λμð1þbÞð5Þ
Dividing Eq. (3) by Eq. (4),
Clcos ϕþCDsin ϕ
Clsin ϕ−CDcos ϕ¼að1−aÞ−ðbλμÞ2
λμbð1−aÞð6Þ
Substituting Eq. (5) into Eq. (6) and simplifying yields
ð1−aÞ½að1−aÞ−λ2μ2b
−ελμ½að1−aÞð1þbÞþbð1−aÞ2þλ2μ2b2ð1þbÞ ¼ 0ð7Þ
where ε¼Cd=Clwith an order of 10−2.
Under the optimal condition, the order of bis approximately
10−2to 10−3; the order of ais approximately 10−1. Compared with
the other terms in Eq. (7), then, εbis a small quantity that can be
ignored and Eq. (7) can be simplified to
að1−aÞ−λ2μ2b−ελμa¼0ð8Þ
Hence
b¼að1−ελμ −aÞ
λ2μ2ð9Þ
The values of aand bthat provide maximum possible efficiency
can be determined by differentiating Eq. (4) by either factor and
setting the result equal to zero
d
da bð1−aÞ¼0ð10Þ
Substituting Eq. (9) into Eq. (10) yields
3a2−2ð2−ελμÞaþð1−ελμÞ¼0ð11Þ
Solving Eq. (11) gives the solution
a¼2
3−ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
1−ελμ þðελμÞ2
p3−ελμ
3ð12Þ
From Eqs. (12) and (9), it can be found that both flow induction
factors, aand b, can be affected by the drag coefficient of the airfoil
through ε.
An equation similar to the one without drag can be obtained
from Eq. (4)
Nc
2πRλCl¼4að1−ελμ −aÞð1−aÞ
½1−a−εð1þbÞλμffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
ð1−aÞ2þ½λμð1þbÞ2
pð13Þ
At a given tip-speed ratio, a simple equation can be used to cal-
culate the chord length distribution. In the ideal blade design, the
airfoil should operate at the point of the largest lift-to-drag ratio to
U
p
d
p
d
U
d
p
w
U
p
W
(1 )Ua
(1 )rb
(a)
(b)
Fig. 6. (a) Stream tube over a wind turbine; (b) model of a blade
element
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reduce drag losses (Burton et al. 2011). Hence, in Eq. (13) both Cl
and Cdare selected to correspond to the greatest lift-to-drag ratio;
ε¼Cd=Clis the reciprocal of the greatest lift-to-drag ratio; λis
the given tip-speed ratio; μis the normalized local radius r=R;
and Nis the number of blades. The axial induction factor acan
be solved by Eq. (12), and the tangential induction factor can be
solved by Eq. (9). For the twist angle, there is
β¼arctan1−a
λμð1þbÞ−αð14Þ
where angle of attack is also selected to corresponding to the
greatest lift-to-drag ratio.
Because the effect of drag seems to be of little significance in
calculating aor b,εis omitted in some frameworks of wind rotor
optimization. However, the role of drag in evaluating CPis very
important. The power coefficient can be calculated as
CP¼QΩ
ρU3
∞πR2=2¼λ2Z1
0
8ð1−aÞbμ3dμð15Þ
Under the optimal condition, the optimal power coefficient can
be obtained by substituting Eq. (9) into Eq. (15)
CPmax ¼λ2Z1
0
8ð1−aÞað1−ελμ −aÞ
λ2μ2μ3dμð16Þ
From Eqs. (12)–(16), it can be concluded that at a certain tip-
to-speed ratio, the optimal power coefficient correlates only with ε.
Fig. 7shows the relationships of CPmax with εbased on Eq. (16).
For engineering applications, the results can be fitted by an empiri-
cal linear formula
CPmax ¼0.593 −0.565λε ð17Þ
where the first term = Betz limit and the second term = loss induced
by drag.
As shown in Fig. 7,CPmax is clearly reduced by drag through ε
with a slope approximately proportional to λ. As validation, the
simple empirical linear formula obtained in this paper is well doc-
umented by Eq. (1), as proposed by Wilson et al. (1976) from
BEM theory, with the effect of the tip correction induced by blade
N. Because the tip and root losses are omitted in the present process
of deduction, Eq. (1) can be simplified to Eq. (18) by setting
N→∞, which agrees well with Eq. (17)
CPmax ¼16
27 ð1−0.96λεÞ¼0.593 −0.569λε ð18Þ
Optimal Design of Ideal Blades under Different
Reynolds Numbers
Different from a practical blade, the ideal blade covers different
lengths and Reynolds numbers. Moreover, the Reynolds number
distribution along span r=Rcan be very different between blades
with different lengths or different rotational speeds. In the present
discussion, the change in blade shape at the same local r=Rcaused
by the change in Reynolds number is the main concern. Therefore,
the variation in Reynolds number from root to tip is not considered
in the ideal blade design so as to reveal only the influence of the
airfoil database under different Reynolds numbers. The airfoil data-
bases of DU00-W-350 and NACA64618 under different Reynolds
numbers are selected to design the ideal blade under a constant
tip-speed ratio.
From Eq. (1), it is clear that the optimal blade should operate at
the greatest lift-to-drag ratio to obtain the maximum power coef-
ficient. As the Reynolds number changes, both the lift coefficient
Cland the drag coefficient Cdcorresponding to the greatest lift-to-
drag ratio clearly change. Hence, the optimal chord length should
be adjusted accordingly to provide suitable shaft torque to drive the
wind rotor. Fig. 8shows the distribution of the required chord
length from Eq. (13). As the tip-speed ratio increases, the chord
length distribution decreases, indicating that a high designed
tip-speed ratio requires a slender blade whereas a low designed tip-
speed ratio requires a duller blade. For all ideal blades, the maxi-
mum chord length appears at locations between 0.05 Rand 0.1 R,
with values from 0.11 Rto 0.46 R. And then, the maximum chord
length will exceed 6.6 m for a 60-m blade. But generally, the maxi-
mum chord length for a practical blade is generally much smaller
than for the ideal blade, considering structure, transportation, and
other limitations. As shown in Fig. 8, in the dimensionless aspect,
optimal design of the ideal blade is significantly affected by a
change in Reynolds number. At a constant tip-speed ratio, for the
design using the NACA64618 database, the chord length increases
with Reynolds number. In contrast, for the design using the
DU00-W-350 database, the chord length decreases with Reynolds
number. Therefore, at the same tip-speed ratio, the ideal blade de-
signed using thin airfoils requires a duller geometry in chord
length; a design using thicker airfoils requires a slender blade.
Similarly, as the Reynolds number changes, the twist angle
needs to be adjusted to ensure that the blade operating under
the optimal condition is under the greatest lift-to-drag ratio point.
The distribution of the required twist angle can be computed by
Eq. (14). For λ¼4, the twist angle distribution exhibits a variation
similar to that in chord length, as shown in Fig. 9. The optimal twist
angle increases with the Reynolds number for the design using
NACA64618, but decreases for the ideal blade using DU00-
W-350. Under other tip-speed ratios, the twist angle changes sim-
ilarly to when λ¼4.
To elucidate the sensitivity of the Reynolds number effect, a
factor of the chord length change rate caused by the change in
Reynolds number is defined as
fc¼cR=cR0−1ð19Þ
where cR0= chord length of the optimal blade at the reference
Reynolds number, R0, and cR= chord length of the optimal design
at a given Reynolds number, R.
For convenience, the smallest of the five Reynolds numbers
studied in the present paper, 2×106, is chosen to be the reference.
ε
CP
0 0.01 0.02 0.03 0.04 0.05
0.3
0.4
0.5
0.6
λ= 4
λ= 12
λ= 10
λ= 8
Fig. 7. Relationships of Cpmax with εfor the optimal design (triangles =
results of Eq. (16); lines = results of Eq. (17) fitted in this paper)
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Figs. 10 and 11 show the distribution of fcat different Reynolds
numbers for the ideal blades using the DU00-W-350 and
NACA64618 airfoils. It can be observed that the factor increases
with tip-speed ratio and monotonically grows with the span of
the blade, reaching its maximum value at μ¼1.AtR¼1×107,
the factor achieves approximately 23.5% for the design using the
DU00-W-350 airfoil and approximately 26% for the design using
the NACA64618 airfoil. For further analysis, factor fccan be
divided into two parts:
fc¼fc
cl þfc
cd ð20Þ
The first part relates to the lift of the airfoil; the second part, to
the drag. For a specific airfoil, fc
cl is a constant value at a constant
Reynolds number. For the thinner NACA64618 airfoil, fc
cl is
positive and increases with the Reynolds number. However, for
the thicker DU00-W-350 airfoil, fc
cl is negative and decreases with
the Reynolds number according to the variation of Clat the optimal
condition. As shown in Figs. 10 and 11,fc
cd is negative near the
root, which means that the chord length in the optimal design can
be reduced because of the drag at the higher Reynolds number. In
the postmedian of the blade, fc
cd becomes positive and increases
in the spanwise direction; hence, at these locations the drag factor
positively contributes to the chord length at a higher Reynolds
number. Overall, the Reynolds number effect on the chord length
is mainly dominated by fc
cl (fc
cd accounts for less than 5% of fc).
Similarly, the change in twist angle can be divided into two
parts. The first part is the change of inflow angle, which correlates
only with ε; the second part is the change in optimal operating
angle of attack due to the variation in Reynolds number:
Δβ ¼Δϕ −Δα ð21Þ
r/R
c/R
0.2 0.4 0.6 0.8 1
0.1
0.2
0.3
0.4
DU00-W–350 R = 2×106
DU00-W–350 R = 4×106
DU00-W–350 R = 6×106
DU00-W–350 R = 8×106
DU00-W–350 R = 1×107
NACA64618 R = 2×106
NACA64618 R = 4×106
NACA64618 R = 6×106
NACA64618 R = 8×106
NACA64618 R = 1×107
r/R
c/R
0.2 0.4 0.6 0.8 1
0.1
0.2
0.3
r/R
c/R
0.2 0.4 0.6 0.8 1
0.1
0.2
r/R
c/R
0.2 0.4 0.6 0.8 1
0.05
0.1
0.15
(a) (b)
(c) (d)
Fig. 8. Distribution of chord length for optimization of the ideal blade at different Reynolds numbers: (a) λ¼4; (b) λ¼6; (c) λ¼8; (d) λ¼10
r/R
Twist
0.2 0.4 0.6 0.8 1
5
10
15
20
25
30
Fig. 9. Distribution of twist angle of the optimal blade at different
Reynolds numbers; λ¼4
r/R
f
0.2 0.4 0.6 0.8 1
–0.24
–0.22
–0.2
–0.18
R = 4×106
R = 1×107
R = 8×106
R = 6×106fcl
c
fcl
c
fcl
c
fcl
c
c
λ= 4
λ= 6
λ= 8
λ= 10
fcl
c
fcl
c
fcl
c
fcl
c
Fig. 10. Distribution of fcat different Reynolds numbers for designs
using the DU00-W-350 airfoil
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For ideal blades, using DU00-W-350 and NACA64618, respec-
tively, Δϕ caused by drag is only a minor term (less than 0.01°) in
comparison with the contribution of Δα at different Reynolds
numbers. Because the influence of drag on twist angle is very
small, a detailed analysis is not given here.
Although the drag component contributes only slightly to the
change in optimal blade shape, its contribution to the change in
power coefficient is considerable. A factor representing the rate
of change in the power coefficient caused by the change in
Reynolds number is defined as
fCP¼CPmaxjR
CPmaxjR0¼2×106
−1ð22Þ
As factor fc, the reference Reynolds number R0here is also set
to 2×106. Fig. 12 shows factor fCPat different Reynolds numbers
for the ideal blades in the present study. As shown, the rate of
change in CPmax increases with tip-speed ratio at a constant
Reynolds number. It can also be observed that, at a constant tip-
speed ratio, factor fCPincreases with Reynolds number. The rate
of change in CPmax induced by the change in Reynolds number for
the design using the thicker airfoil is much greater than that for the
design using the thinner airfoil. For the design using DU00-W-350,
factor fCPcan achieve approximately 4.2% at R¼1×107under
the tip-speed ratio λ¼10. For the design using NACA64618, an
approximate 1.1% CPmax increase can be obtained, which is still
relevant.
These results indicate that the optimal shape of the ideal blade is
significantly affected by a change in Reynolds number. Because of
the variation in airfoil performance induced by Reynolds number
changes, the optimal blade design changes accordingly. The change
in chord length is mainly caused by the change in lift coefficient.
The change in twist angle is mainly caused by the change in
optimal angle of attack. Because of the increase in the Reynolds
number, the boosting of CPmax is quite significant.
Aerodynamic Design of Practical Blades at Different
Reynolds Numbers
In a practical blade design, many restrictions are applied to the
aerodynamic shape for the sake of manufacture, transportation,
structure design, and so on. Therefore, one cannot simply design
for an optimal power coefficient in each section; rather, designing
a practical blade requires comprehensive consideration under spe-
cific limited conditions. A genetic algorithm is introduced to opti-
mize practical design. For a given blade, the power coefficient is
evaluated using a more accurate BEM theory in which tip losses are
considered and a suitable modification is made for wind turbine
application.
Design Cases for the Practical Blades
In this section, two types of design are studied to show the
Reynolds number effect on the practical blade. First, the design of
38-m, 60-m, and 75-m blades respectively for 1.5-MW, 3-MW, and
6-MW wind turbines is investigated in the dimensionless aspect of
the blade length. Then a 60-m blade is designed for a 3-MW wind
turbine at different altitudes. Fig. 13(a) shows the distribution of the
Reynolds number along the local r=Rof the blades for the three
wind turbines: UP77 (1.5 MW), UP126 (3 MW) and UP136
(6 MW) under the rated condition. The Reynolds number of the
airfoil profile is approximately 4 million for UP77 and approxi-
mately 7–9 million for UP136. Fig. 13(b) shows the distribution
of the Reynolds number along the local r=Rof the blade for
UP126 at three altitudes: 0, 2, and 4 km. The Reynolds number
changes with changes in air density and dynamic viscosity at differ-
ent altitudes. The Reynolds number for each r=Rin Fig. 13 is not
the practical Reynolds number used in the present design; the real
Reynolds number for each profile is determined by the blade shape
and its operational condition. Blades of similar length for the same
MW-class wind turbines are presented here only to give the
approximate range of Reynolds numbers.
BEM Theory for Practical Blade Assessment in the
Design Procedure
An advanced BEM theory proposed by Lanzafame and Messina
(2007) is used in the present aerodynamic design. Based on the
r/R
fc
0.2 0.4 0.6 0.8 1
0.11
0.12
0.13
0.14
0.15
0.16
0.17
r/R
fc
λ= 4
λ= 6
λ= 8
λ= 10
R = 4×106
R = 6×106fcl
c
fcl
c
r/R
0.2 0.4 0.6 0.8 1
0.22
0.23
0.24
0.25
0.26
0.27
r/R
fc
λ= 4
λ= 6
λ= 8
λ= 10
R = 8×106fcl
c
R = 1×107fcl
c
(a) (b)
Fig. 11. Distribution of fcat different Reynolds numbers for the design using the NACA64618 airfoil: (a) R¼4×106and R¼6×106;
(b) R¼8×106and R¼1×107
fCP
46810
–0.02
0
0.02
0.04
, DU00-W–35 0
, DU00-W–35 0
, DU00-W–35 0
, DU00-W–35 0
R
fCP
46810
0
, NACA64618
, NACA64618
, NACA64618
, NACA64618
Fig. 12. Reynolds number effect factor of power coefficient fCPat
different Reynolds numbers for the ideal blade
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Glauert propeller theory, suitable modifications are made for wind
turbines. The improvement in axial and tangential induction factors
proposed by Buhl (2005) is adopted to eliminate the numerical
instability that occurs when the Glauert correction is implemented
in conjunction with the presence of tip losses. When a>0.4, the
correlation is written as Eq. (23) after Lanzafame and Messina
(2007)
a¼18F−20 −3ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
CNð50 −36FÞþ12Fð3F−4Þ
p36F−50 ð23Þ
where F= Prandtl tip loss factor. The new expression for the tan-
gential induction factor is written as
b¼1
2ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
1þ4
λ2
r
að1−aÞ
s−1ð24Þ
The mathematical model has been validated by experimental
data collected by NREL in NASA-Ames wind tunnel tests (Buhl
2005). In BEM theory, the factors aand bfor a given blade element
can be solved by an iteration method. For further details, see
Lanzafame and Messina (2007). The power coefficient of the wind
turbine blade can then be computed using aand b
CP¼RR
0
1
2ρW2NcðClsin ϕ−CDcos ϕÞΩrdr
1
2ρU3
∞πR2
¼RR
0NcλðClsin ϕ−CDcos ϕÞ½ð1−aÞ2þλ2ð1þbÞ2dr
πR2
ð25Þ
where
ϕ¼arctan1−a
λð1þbÞð26Þ
Design Procedure
All of the blades are designed using the profiles of Cylinder,
DU00-W2-401, DU00-W-350, DU97-W-300, DU97-W-250,
NACA63421, and NACA64618 airfoils arranged spanwise from
the root to the tip. A constant distribution of relative thickness
is used to specify the arrangement of the airfoils for all of the
designs, to show only the influence of the Reynolds number.
Generally, at the root the blade sections should have thick profiles,
which are essential to carry intensive loads from the entire blade.
Approaching the tip, thinner sections should be used with reduced
load, higher linear velocity, and increasing critical aerodynamic
performance (Schubel and Crossley 2012). For many commercial
wind turbine blades, such as LM40.3P2 for 1.5-MW wind turbines,
LM45.3 P for 2-MW wind turbines, LM61.5P2 for 5–6-MW wind
turbines, Sinoma45.3 for 2-MW wind turbines, Sinoma50.5 for
3-MW wind turbines, and others, airfoils with a relative thickness
of approximately 40% are arranged at the maximum chord location.
In the present study, following the settings of these commercial
blades, airfoils with a relative thickness of 100–40% are arranged
from root to maximum chord length and those with a relative thick-
ness of 40–18% are used on the sections from maximum chord
length to tip, as shown in Fig. 14. The chord distribution is opti-
mized between the maximum chord location and the tip, whereas
the twist angle is optimized from the root to the tip.
As design variables, the chord and twist angle distributions
are both parameterized by Besair curves. Besair curves are widely
used in computer graphics and related fields to model smooth
curves. The curve is defined by a set of control points P0through
Pn, where nis the order of the Besair curve. For the control points
P0;P1;:::;Pi;:::;Pn−1;Pn, the parametric curve is given by
BðtÞ¼X
n
t¼0n
iPið1−tÞn−iti;t∈½0,1ð27Þ
r/R
R
00.20.40.60.81
0.0E+00
2.0E+06
4.0E+06
6.0E+06
8.0E+06
1.0E+07 6MW
3 MW
1.5 MW
r/R
R
0 0.2 0.4 0.6 0.8 1
0
2E+06
4E+06
6E+06
H = 0 km, ρair = 1.225
H = 2 km, ρair = 1.007
H = 4 km, ρair = 0.820
(a) (b)
Fig. 13. Distribution of Reynolds number along the local r=Rof the blade under the rated condition: (a) UP77, UP126, and UP136; (b) UP126 at
altitudes 0, 2, and 4 km
r/R
Relative thickness
00.20.40.60.81
20
40
60
80
100
Fig. 14. Distribution of relative thickness for the practical blade design
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Seven and six control points are used in the parameterization for
the chord and twist angle distributions, respectively
rcmax ¼xc1<xc2<xc3<xc4<xc5<xc6¼xc7¼rtip
cmax ¼yc1¼yc2>yc3<yc4<yc5<yc6<xc7¼ctip
rtmax ¼xt1<xt2<xt3<xt4<xt5<xt6¼rtip
βmax ¼yt1¼yt2>yt3>yt4;yt5;yt6ð28Þ
To meet practical requirements, some artificial constraints are
applied to the control points empirically: xc1¼7.6, 12, and 15
are taken for the 38-m, 60-m, and 75-m blades, respectively, and
the corresponding yc1are taken as 3.15, 4.0, and 5.5. The param-
eters ctip ¼0.02,rtmax ¼0.0, and β¼16 are used for all design
cases. The coordinates of the control points are taken as optimiza-
tion variables (a total of 16 variables):
X¼ðxc2; :::;xc5;yc3;:::;yc6;xt2;:::;xt5;yt3;:::;yt6Þð29Þ
The maximum power coefficient is set as the optimization goal.
The GA-II algorithm (Deb et al. 2002) is introduced to optimize the
design:
Y¼maxðCPÞð30Þ
The simple design procedure is given here:
1. Specify the number of individuals (NI) in each generation and
the number of generations for the optimization (NGO). In this
procedure, NI ¼96 and NGO ¼80;
2. Produce the first generation of optimization variables
X1
0;X2
0;:::;XNI
0randomly;
3. Calculate the distributions of chord length and twist angle for
each individual using Eq. (27);
4. Determine the power coefficient for each individual by
Eq. (25); if the number of this generation NG < NGO, go
to (5); else go to (6);
5. Produce the optimization variables of the next generation
X1
NGþ1;X2
NGþ1; :::;XNI
NGþ1using the GA-II algorism; then
go to (3); and
6. End the optimization and choose the blade with the
greatest CP.
Results of the Practical Design
Fig. 15 shows the results of the practical design at λ¼10 for the
38-m, 60-m, and 75-m blades. From the root to the 40% length of
the blades, where the thicker airfoils are used, the chord length is
mainly dominated by artificially applied constraints. At the part of
the blade tip where the power is mainly produced with the arrange-
ment of thinner airfoils, a sharper tip can be observed in the 38-m
blade than in the other two, resulting from the change in Reynolds
number. Fig. 15(b) shows the twist angle distribution of the three
blades. To guarantee the smoothness of the blade shape, the twist
angle is mainly determined by the thinner airfoils, where most of
the power is produced. The twist angle of the optimal design
changes similarly to the chord. Fig. 16 shows the distributions
of chord length and twist angle of the 60-m blades at different
altitudes. Because of the change in Reynolds number induced by
the change in air density and dynamic viscosity, the optimal shape
also clearly varies. In the practical design, both chord length and
twist angle changes are consistent with the theoretical analysis.
r/R
c/R
0 0.2 0.4 0.6 0.8 1
0.02
0.04
0.06
0.08
38-m blade
60-m blade
75-m blade
r/R
Twist
0 0.2 0.4 0.6 0.8
0
5
10
15
(a) (b)
Fig. 15. Distribution of chord length and twist angle for the 38-m, 60-m, and 75-m blades at λ¼10: (a) chord length; (b) twist angle
r/R
c/R
0 0.2 0.4 0.6 0.8 1
0.02
0.04
0.06
0.08
H = 0 km
H = 2 km
H = 4 km
r/R
Twist
0 0.2 0.4 0.6 0.8
0
5
10
15
(a) (b)
Fig. 16. Distribution of chord length and twist angle for the 60-m blade for the 3-MW wind turbine at different altitudes at λ¼10: (a) chord length;
(b) twist angle
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Fig. 17 shows the CP–λcurves for the practical design. The
power coefficient for a blade operating at a higher Reynolds num-
ber is better than that for one operating at a lower Reynolds number.
As pointed out earlier, the Reynolds number affects the largest
lift-to-drag ratio of the airfoils and the change of the ratio in turn
affects the maximum power coefficient. Hence, at higher Reynolds
numbers, a blade exhibits a larger CPmax. Furthermore, an overall
lift-to-drag ratio reduction at the small angles of attack can be ob-
served at lower Reynolds numbers for all airfoils, as shown in
Fig. 4, which substantially lessens the power coefficients at a wide
range of tip-speed ratios.
Summary and Conclusions
The aerodynamic design of a wind turbine blade is dramatically
affected by variations in airfoil performance at different Reynolds
numbers. To provide an insight into the Reynolds number effect,
both theoretical analysis and practical design were investigated by
maximizing the power coefficient.
First, the DU00-W2-401, DU00-W-350, DU97-W-300, DU97-
W-250, NACA63421, and NACA64618 airfoils were studied
with RFOIL. To show the Reynolds number effect on airfoil
performance, five typical Reynolds numbers—2×106,4×106,
6×106,8×106, and 1×107—were selected between 1×106
and 1×107; this range covers most commercial wind turbine
blades. The results show that at higher Reynolds numbers all air-
foils exhibit better performance characteristics at a given angle of
attack, such as higher lift coefficient, lower drag coefficient, and
higher lift-to-drag ratio. Both angle of attack and lift coefficient
at the point of the highest lift-to-drag ratio decrease with the
Reynolds number for thinner airfoils (DU97-W-300, DU97-W-250,
NACA63421, NACA64618), but increase for thicker airfoils
(DU00-W2-401 and DU00-W-350).
Using the airfoils database established by RFOIL, the so-called
ideal blade with no constraints was designed only in aerodynamics
terms by a simple mathematical model. For simplification, some
minor terms in the model were omitted to avoid having to solve
complex polynomial equations for both aand b. Based on this
model, the Reynolds number effects on chord length, twist angle,
and power coefficient were analyzed in detail. It was found that the
change in optimal shape mainly comes from the variation in lift
coefficient and attack angle corresponding to the highest lift-to-
drag ratio, and that the drag terms contribute only a small amount.
However, the maximum power coefficient CPmax is highly corre-
lated with the drag terms. For the design using only DU00-W-350,
CPmax increases by approximately 4.2% at R¼1×107compared
with that at R¼2×106with the tip-speed ratio λ¼10; for the
design using NACA64618, an approximate 1.1% CPmax increase
can be obtained.
Finally, under the same tip-speed ratio, blades for different MW-
class wind turbines and the 60-m blade for the 3-MW wind turbine
operating at different altitudes were designed to reveal the Reyn-
olds number effect more clearly. The practical design agrees well
with the theoretical analysis of the ideal blade. Blades operating at a
lower Reynolds number have a sharper shape, a smaller twist angle,
and a lower power coefficient than do blades operating at a higher
Reynolds number.
Acknowledgments
The authors acknowledge the support of the National Natural
Science Foundation of China (11402088) and the the Fundamental
Research Funds for the Central Universities.
Notation
The following symbols are used in this paper:
a=axial induction factor;
b=tangential induction factor;
Cl=lift coefficient;
Cd=drag coefficient;
CP=power coefficient;
CPmax =maximum power coefficient;
c=airfoil chord length;
N=number of blades;
P=power;
Q=shaft torque;
R=radius of rotor;
R=Reynolds number;
r=blade local radius;
U∞=wind speed in the far field;
W=relative wind velocity to an airfoil;
α=angle of attack;
β=twist angle;
ε=Cd=Cl;
λ=tip-speed ratio;
μ=normalized local radius r=R;
ρ=air density;
υ=dynamic viscosity;
ϕ=inflow angle;
Ω=rotor rotational speed; and
ω=rotational angular velocity.
λ
CP
46810121416
0.2
0.3
0.4
0.5
38-m blade
60-m blade
75-m blade
CP
4 6 8 10121416
0.2
0.3
0.4
0.5
60-m blade H = 0 km
60-m blade H = 2 km
60-m blade H = 4 km
λ
(a) (b)
Fig. 17. CP-λcurves of the practical designs: (a) 38-m blade for a 1.5-MW wind turbine, 60-m blade for a 3-MW wind turbine, and 75-m blade for a
6-MW wind turbine; (b) 60-m blades for the 3-MW wind turbines operating at altitudes of 0 km, 2 km, and 4 km
© ASCE 04014056-11 J. Energy Eng.
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References
Benini, E., and Toffolo, A. (2002). “Optimal design of horizontal-axis wind
turbines using blade-element theory and evolutionary computation.”
J. Solar Eng., 124(4), 357–363.
Bizzarrini, N., Grasso, F., and Coiro, D. P. (2011). “Genetic algorithms
in wind turbine airfoil design.”EWEA, EWEC2011, Brussels, Belgium,
14–17.
Buhl, M. L. (2005). “A new empirical relationship between thrust coeffi-
cient and induction factor for the turbulent windmill state.”Technical
Rep. NREL/TP-500-36834, National Renewable Energy Laboratory,
Golden, CO.
Burton, T., Jenkins, N., Sharpe, D., and Bossanyi, E. (2011). Wind energy
handbook, Wiley, West Sussex, U.K.
Ceyhan, O. (2012). “Towards 20MW wind turbine: High Reynolds number
effects on rotor design.”50th AIAA ASM Conf., AIAA, Washington,
DC.
Deb, K., Pratap, A., Agarwal, S., and Meyarivan, T. A. M. T. (2002). “A fast
and elitist multi objective genetic algorithm: NSGA-II.”IEEE Trans.
Evol. Comput., 6(2), 182–197.
Drela, M. (1989). “XFOIL: An analysis and design system for low
Reynolds number airfoils.”Low Reynolds number aerodynamics,
Springer, Berlin, 1–12.
Fuglsang, P., and Bak, C. (2004). “Development of the RisΦwind turbine
airfoils.”Wind Energy, 7(2), 145–162.
Fuglsang, P., and Madsen, H. A. (1999). “Optimization method for wind
turbine rotors.”J. Wind Eng. Ind. Aerodyn., 80(1), 191–206.
Grasso, F. (2013). “Development of thick airfoils for wind turbines.”
J. Aircr., 50(3), 975–981.
Haines, A. B. (1994). “Scale effects on aircraft and weapon aerodynamics.”
AGARD-AG-323, A. D. Young, ed., Advisory Group for Aerospace Re-
search and Development, Neuilly-Sur-Seine, France.
Johansen, J., et al. (2009). “Design of a wind turbine rotor for maximum
aerodynamic efficiency.”Wind Energy, 12(3), 261–273.
Jonkman, J., Butterfield, S., Musial, W., and Scott, G. (2009). “Definition
of a 5-MW reference wind turbine for offshore system development.”
Technical Rep. NREL/TP-500-38060, National Renewable Energy
Laboratory, Golden, CO.
Lanzafame, R., and Messina, M. (2007). “Fluid dynamics wind turbine
design: Critical analysis, optimization and application of BEM theory.”
Renewable Energy, 32(14), 2291–2305.
Marsh, G. (2012). “Offshore reliability.”Renewable Energy Focus, 13(3),
62–65.
Rechzeh, D., and Hansen, H. (2006). “High Reynolds-number wind tunnel
testing for the design of airbus high-lift wings.”New results in
numerical and experimental fluid mechanics, Vol. 92, Springer, Berlin,
1–8.
Schubel, P. J., and Crossley, R. J. (2012). “Wind turbine blade design.”
Energies, 5(9), 3425–3449.
Snel, H., Houwink, R., and Bosschers, J. (1994). Sectional prediction of lift
coefficients on rotating wind turbine blades in stall, Netherlands Energy
Research Foundation, Petten, Netherlands.
Tangler, J. L., and Somers, D. M. (1995). “NREL airfoil families for
HAWTs.”National Renewable Energy Laboratory, Golden, CO.
Timmer, W. A. (2009). “An overview of NACA 6-digit airfoil series
characteristics with reference to airfoils for large wind turbine blades.”
AIAA 2009–268, AIAA, Washington, DC.
Timmer, W. A., and Van Rooij, R. P. J. O. M. (2003). “Summary of the
Delft University wind turbine dedicated airfoils.”J. Solar Energy
Eng., 125(4), 488–496.
van Rooij, R. (1996). “Modification of the boundary layer calculation
in RFOIL for improved airfoil stall prediction.”Rep. IW-96087 R,
TU Delft, Delft, Netherlands.
Wang, X. D., Shen, W. Z., Zhu, W. J., Sørensen, J. N., and Jin, C. (2009).
“Shape optimization of wind turbine blades.”Wind Energy, 12(8),
781–803.
Wilson, R. E., Lissaman, P. B., and Walker, S. N. (1976). Aero-
dynamic performance of wind turbines, Oregon State Univ., Corvallis,
OR.
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