aerodynamic models for darrieus type

Renewable and Sustainable Energy Reviews
12 (2008) 1087–1109
Aerodynamic models for Darrieus-type
straight-bladed vertical axis wind turbines
Mazharul IslamÃ, David S.-K. Ting, Amir Fartaj
Department of Mechanical, Automotive and Materials Engineering, University of Windsor,
Windsor, Ont., Canada N9B 3P4
Received 20 October 2006; accepted 31 October 2006
Abstract
Since ancient past humans have attempted to harness the wind energy through diversiﬁed means
and vertical axis wind turbines (VAWTs) were one of the major equipment to achieve that. In this
modern time, there is resurgence of interests regarding VAWTs as numerous universities and
research institutions have carried out extensive research activities and developed numerous designs
based on several aerodynamic computational models. These models are crucial for deducing
optimum design parameters and also for predicting the performance before fabricating the VAWT.
In this review, the authors have attempted to compile the main aerodynamic models that have been
used for performance prediction and design of straight-bladed Darrieus-type VAWT. It has been
found out that at present the most widely used models are the double-multiple streamtube model,
Vortex model and the Cascade model. Each of these three models has its strengths and weaknesses
which are discussed in this paper.
r 2006 Elsevier Ltd. All rights reserved.
Keywords: Renewable energy; Vertical axis wind turbine; VAWT; Wind; Straight-bladed; Darrieus
Contents
1. Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1088
2. Historical background . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1090
3. Modern VAWT types . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1091
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www.elsevier.com/locate/rser
1364-0321/$ - see front matter r 2006 Elsevier Ltd. All rights reserved.
doi:10.1016/j.rser.2006.10.023
ÃCorresponding author. Tel.: +1 519 253 3000x2635; fax: +1 519 973 7007.
E-mail address: islam1f@uwindsor.ca (M. Islam).

3.1. Savonius wind turbine. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1091
3.2. Darrieus wind turbines . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1092
3.3. H-Rotors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1094
4. General mathematical expressions for aerodynamic analysis of straight-bladed Darrieus-type
VAWTs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1094
4.1. Variation of local angle of attack . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1094
4.2. Variation of local relative flow velocity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1096
4.3. Variation of tangential and normal forces . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1097
4.4. Calculation of total torque. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1097
4.5. Power output . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1098
5. Computational models for Darrieus-type straight-bladed VAWT . . . . . . . . . . . . . . . . 1098
5.1. Momentum model. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1098
5.1.1. Single streamtube model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1098
5.1.2. Multiple streamtube model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1100
5.1.3. Double-multiple streamtube model. . . . . . . . . . . . . . . . . . . . . . . . . . . . 1102
5.2. Vortex model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1103
5.3. Cascade model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1105
6. Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1108
Acknowledgments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1108
References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1108
1. Introduction
At present, there are two categories of modern wind turbines, namely horizontal axis
wind turbines (HAWTs) and vertical axis wind turbines (VAWTs), which are used mainly
for electricity generation and pumping water. The main advantage of VAWT is its single
moving part (the rotor) where no yaw mechanisms are required, thus simplifying the design
configurations significantly. Blades of straight-bladed VAWT may be of uniform section
and untwisted, making them relatively easy to fabricate or extrude, unlike the blades of
HAWT, which should be twisted and tapered for optimum performance. Furthermore,
almost all of the components of VAWT requiring maintenance are located at the ground
level, facilitating the maintenance work appreciably.
From the past experiences, it is evident that wind turbines can compete with
conventional sources in niche markets, and lower costs make them affordable options in
increasingly large markets. Environmentally benign VAWTs can be utilized for a range of
applications, including (i) electricity generation; (ii) pumping water; (iii) purifying and/or
desalinating water by reverse osmosis; (iv) heating and cooling using vapour compression
heat pumps; (v) mixing and aerating water bodies; and (vi) heating water by fluid
turbulence. In general, VAWT can sensibly be used in any area with sufficient wind, either
as a stand-alone system to supply individual households with electricity and heat, or for
the operation of freestanding technical installations. If a network connection is available,
the energy can be fed in, thereby contributing to a reduction in electricity costs. In order to
maximize the security of the energy supply, different types of VAWT can be supplemented
by a photovoltaic system or a diesel generator in a quick and uncomplicated fashion.
Through the combination of several VAWTs with other renewable energy sources and a
backup system, local electrical networks can be created for the energy supply of small
settlements and remote locations.
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M. Islam et al. / Renewable and Sustainable Energy Reviews 12 (2008) 1087–11091088

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Abbreviations and Acronyms
A projected frontal area of turbine
AR aspect ratio ¼ H/C
C blade chord
Cd blade drag coefficient
Cdor reference zero-lift-drag coefficient
CD turbine overall drag coefficient ¼ FD=rAV2
1
CDD rotor drag coefficient ¼ FD=rAV2
1
Cl blade lift coefficient
Cn normal force coefficient
CP turbine overall power coefficient ¼ Po=rAV3
1
CQ turbine overall torque coefficient ¼ Q=rAV2
1R
Ct tangential force coefficient
d minimum distance from the vortex filament
D blade drag force
~e unit vector
F force on blade airfoil
FD turbine overall drag force
Fn normal force (in radial direction)
Ft tangential force
Fta average tangential force
Ft non-dimensional tangential force ¼ Ct (W/VN)2
H height of turbine
HAWT horizontal axis wind turbine
ki exponent in the induced velocity relation
L blade lift force
_m mass flow rate
N number of blade
p static pressure
Po overall power
PN atmospheric pressure
Q overall torque
~r unit vector
R turbine radius
Re local Reynolds number ¼ WC/u
t blade spacing ¼ (2pR/N)
V centre line velocity along freestream velocity direction
Va induced velocity
Vad induced velocity in the downstream side
Vau induced velocity in the upstream side
Vc chordal velocity component
Vcd chordal velocity component in the downstream side
Vcu chordal velocity component in the upstream side
Ve wake velocity in upstream side
Vn normal velocity component
Vnd normal velocity component in the downstream side
M. Islam et al. / Renewable and Sustainable Energy Reviews 12 (2008) 1087–1109 1089

In this modern time, there is resurgence of interests regarding VAWTs as several
universities and research institutions have carried out extensive research activities and
developed numerous designs based on several aerodynamic computational models. These
models are crucial for optimum design parameters and also for predicting the performance
before fabricating the models or prototypes. In this paper, the authors attempt to explore
the main aerodynamic models that have been used for performance and design analysis of
straight-bladed Darrieus-type VAWT through literature survey that are organized and
briefly described in the subsequent headings.
2. Historical background
Wind energy systems have been used for centuries as a source of energy for mankind.
According to historic sources, the Babylonian emperor Hammurabi used windmills for an
ambitious irrigation project as early as the 17th century BCE [1]. Later on, Persian
inventors developed a wind-power machine, a more advanced windmill than that
developed by the Babylonians [2]. Arab geographers traveling in Afghanistan in the 7th
century wrote descriptions of windmills, which resembled our modern revolving doors [3].
Vertical windmills of this category were still used in some parts of Iran and Afghanistan in
the late 1980s, and it was estimated that they generated about 75 hp and can grind a ton of
wheat every 24 h [4].
The earliest European windmills appeared in France and England in the 12th century
and quickly spread throughout Europe. These early wood structures, called post mills,
were rotated by hand around a central post to bring the sails into the wind. By the late part
of the 13th century the typical ‘European windmill’ had been developed and this became
the norm until further developments were introduced during the 18th century. At the end
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Vnu normal velocity component in the upstream side
~Vp induced velocity at a point P on the filament
Vw wake velocity in downstream side
VG velocity contributed by circulation
VN wind velocity
VAWT vertical axis wind turbine
W relative flow velocity
Wd relative flow velocity in the downstream side
Wu relative flow velocity in the upstream side
a blade angle of attack
ad blade angle of attack in the downstream side
au blade angle of attack in the upstream side
g blade pitch angle
G circulation per unit length
y azimuth angle
l tip speed ratio ¼ Ro/VN
n kinematic viscosity
r fluid density
s solidity ¼ NC/R
o angular velocity of turbine in rad/s

of the 19th century there were more than 30,000 windmills in Europe, used primarily for
the milling of grain and water pumping [1].
3. Modern VAWT types
There have been many designs of vertical axis windmills over the centuries and currently
the vertical axis wind turbines can be broadly divided into three basic types, namely (1)
Savonius type, (2) Darrieus type, and (3) H-Rotor type. Brief descriptions of these VAWT
types are given below.
3.1. Savonius wind turbine
The Savonius-type VAWT, as shown in Fig. 1, was invented by a Finnish engineer
S.J. Savonius in 1929 [5]. It is basically a drag force driven wind turbine with two cups or
half drums fixed to a central shaft in opposing directions. Each cup/drum catches the wind
and so turns the shaft, bringing the opposing cup/drum into the flow of the wind. This cup/
drum then repeats the process, causing the shaft to rotate further, thus completing a full
rotation. This process continues all the time the wind blows and the turning of the shaft is
used to drive a pump or a small generator. Though typical values of maximum power
coefficient for other types of wind turbines vary between 30% to 45%, those for the
Savonius turbines are typically not more than 25% according to most investigators [6].
This type of turbine is suitable for low-power applications and they are commonly used for
wind speed instruments.
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Fig. 1. Savonius-type VAWT.

3.2. Darrieus wind turbines
The modern Darrieus VAWT was invented by a French engineer George Jeans Mary
Darrieus. He submitted his patent in 1931 [7] in the USA which included both the
‘‘Eggbeater (or Curved Bladed)’’ and ‘‘Straight-bladed’’ VAWTs. Sketches of these two
variations of Darrieus concepts are shown in Figs. 2 and 3, respectively. The Darrieus-type
VAWTs are basically lift force driven wind turbines. The turbine consists of two or more
aerofoil-shaped blades which are attached to a rotating vertical shaft. The wind blowing
over the aerofoil contours of the blade creates aerodynamic lift and actually pulls the
blades along. The troposkien shape eggbeater-type Darrieus VAWT, which minimizes the
bending stress in the blades, were commercially deployed in California in the past.
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Fig. 2. Curved-blade (or ‘‘Egg-beater’’ type) Darrieus VAWT.

In the small-scale wind turbine market, the simple straight-bladed Darrieus VAWT,
often called giromill or cyclo-turbine, is more attractive for its simple blade design. This
conﬁguration fall into two categories: ﬁxed pitch and variable pitch. It has been found out
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Fig. 3. Straight-bladed Darrieus VAWT.

from the previous research activities that fixed pitch VAWTs provide inadequate starting
torque [6]. Contemporary variable pitch blade configuration has potential to overcome the
starting torque problem but it is overly complicated, rendering it impractical for smaller
capacity applications. Majority of the previously conducted research activities on VAWT
focused on straight bladed VAWTs equipped with symmetric airfoils (like NACA 4-digit
series of 0012, 0015, 0018) which were unable to self-start. This inability to self-start is due
to several factors (like technical, inadequate research work & funding), and the most
dominant ones are due to aerodynamic factors. According to Internet survey, there are
handfuls of commercial straight-bladed VAWTs products, but no reliable information
could be obtained from an independent source regarding the performance of these
products and the claims made by the manufacturers are yet to be authentically verified.
At present, development of large-scale straight-bladed VAWT is limited to the research
stage only, although large eggbeater Darrieus wind turbine had reached the market
commercially in the past before disappearing away later. However, in the small-scale wind
turbine market, the simple straight-bladed Darrieus seems to be more cost effective than
the eggbeater Darrieus as few companies had marketed this type of wind turbine before,
i.e. the Pinson/Asi cycloturbine which utilized an end tail for changing pitch. This
particular giromill model was stated in Drees’ [8] research paper of having 3.6 m diameter
and 2.4 m height. With 3 blades at chord length of 29 cm, the rotor has solidity of 0.24.
Another pitch changing research prototype was built by Grylls et al. [9]. It has a diameter
of 2.4 m and a height of 1.6 m. Using 3 blades with a chord length of 14.5 cm only, the
rotor has a solidity of 0.18. Wind tunnel results for this prototype indicated the rotor was
able to self-start at wind speed of 3.5 m/s, provided the pitch angle change is larger than
plus or minus 41.
3.3. H-Rotors
H-Rotors, as shown in Fig. 4, were developed in the UK through the research carried
out during the 1970–1980s when it was established that the elaborated mechanisms used to
feather the straight-bladed Darrieus VAWT blades were unnecessary. It was found out
that the drag/stall effect created by a blade leaving the wind flow would limit the speed that
the opposing blade (in the wind flow) could propel the whole blade configuration forward.
The H-Rotor was therefore self-regulating in all wind speeds reaching its optimal
rotational speed shortly after its cut-in wind speed.
4. General mathematical expressions for aerodynamic analysis of straight-bladed Darrieus-
type VAWTs
Though the straight-bladed Darrieus-type VAWT is the simplest type of wind turbine,
its aerodynamic analysis is quite complex. Before comparative analysis of the main
aerodynamic models, the general mathematical expressions, which are common to most of
the aerodynamic models, are described in this section.
4.1. Variation of local angle of attack
The flow velocities in the upstream and downstream sides of the Darrieus-type VAWTs
are not constant as seen in Fig. 5. From this figure one can observe that the flow is
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considered to occur in the axial direction. The chordal velocity component Vc and the
normal velocity component Vn are, respectively, obtained from the following expressions:
Vc ¼ Ro þ Va cos y, (1)
Vn ¼ Va sin y, (2)
where Va is the axial ﬂow velocity (i.e. induced velocity) through the rotor, o is the
rotational velocity, R is the radius of the turbine, and y is the azimuth angle. Referring to
Fig. 5, the angle of attack (a) can be expressed as
a ¼ tanÀ1 Vn
Vc

. (3)
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Fig. 4. H-Rotor-type VAWT.

Substituting the values of Vn and Vc and non-dimensionalizing,
a ¼ tanÀ1 sin y
ðRo=V1Þ=ðVa=V1Þ þ cos y

, (4)
where V1 is the freestream wind velocity. If we consider blade pitching then,
a ¼ tanÀ1 sin y
ðRo=V1Þ=ðVa=V1Þ þ cos y

À g, (5)
where g is the blade pitch angle.
4.2. Variation of local relative flow velocity
The relative flow velocity (W) can be obtained as (Fig. 5),
W ¼
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
V2
c þ V2
n
q
. (6)
Inserting the values of Vc and Vn (Eqs. (1) and (2)) in Eq. (6), and non-dimensionalizing,
one can find velocity ratio as,
W
V1
¼
W
Va
:
Va
V1
¼
Va
V1
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
Ro
V1

Vau
V1

þ cos y
2
þ sin2
y
s
. (7)
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Fig. 5. Flow velocities of straight-bladed Darrieus-type VAWT.

4.3. Variation of tangential and normal forces
The directions of the lift and drag forces and their normal and tangential components
are shown in Fig. 6. The tangential force coefficient (Ct) is basically the difference between
the tangential components of lift and drag forces. Similarly, the normal force coefficient
(Cn) is the difference between the normal components of lift and drag forces. The
expressions of Ct and Cn can be written as
Ct ¼ C1 sin a À Cd cos a, (8)
Cn ¼ C1 cos a þ Cd sin a. (9)
The net tangential and normal forces can be defined as
Ft ¼ Ct
1
2rCHW2
, (10)
Fn ¼ Cn
1
2rCHW2
, (11)
where r is the air density, C is the blade chord and H is the height of the turbine.
4.4. Calculation of total torque
Since, the tangential and normal forces represented by Eqs. (10) and (11) are for any
azimuthal position, so, they are considered as a function of azimuth angle y. Average
tangential force (Fta) on one blade can be expressed as
Fta ¼
1
2p
Z 2p
0
FtðyÞ dy. (12)
The total torque (Q) for the number of blades (N) is obtained as
Q ¼ NFtaR. (13)
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Fig. 6. Force diagram of a blade airfoil.

4.5. Power output
The total power (P) can be obtained as
P ¼ Q Á o. (14)
5. Computational models for Darrieus-type straight-bladed VAWT
In the past, several mathematical models, based on several theories, were prescribed
for the performance prediction and design of Darrieus-type VAWTs by different
researchers. The key components of all the computational models can be broadly
described as:
calculations of local relative velocities and angle of attacks at different tip speed ratios
and azimuthal (orbital) positions;
calculation of ratio of induced to freestream velocity (Va=V1) considering the blade/
blade-wake interaction;
mathematical expressions based on different approaches (Momentum, Vortex or
Cascade principles) to calculate normal and tangential forces;
‘Pre-Stall airfoil characteristics’ (Cl, Cd Cm) for the attached regime at different
Reynolds numbers;
‘Post-Stall Model’ for Stall Development and Fully Stalled regimes;
‘Finite Aspect Ratio consideration’;
‘Dynamic Stall Model’ to account for the unsteady effects;
‘Flow Curvature Model’ to consider the circular blade motion.
According to literature survey, the most studied and best validated models can be
broadly classified into three categories—(1) Momentum model, (2) Vortex model and (3)
Cascade model. It should be noted that not all the models consider all the key components
described above. Descriptions of these three main categories of VAWT computational
models are presented below.
5.1. Momentum model
Different momentum models (also specified as Blade Element/Momentum or BEM
model) are basically based on calculation of flow velocity through turbine by equating the
streamwise aerodynamic force on the blades with the rate of change of momentum of air,
which is equal to the overall change in velocity times the mass flow rate. The force is also
equal to the average pressure difference across the rotor. Bernoulli’s equation is applied in
each streamtube. The main drawback of these models is that they become invalid for large
tip speed ratios and also for high rotor solidities because the momentum equations in these
particular cases are inadequate [10]. Over the years, several approaches were attempted to
utilize this concept, which are discussed briefly in the following headings.
5.1.1. Single streamtube model
In 1974 Templin proposed the single streamtube model which is the first and most simple
prediction method for the calculation of the performance characteristics of a Darrieus-type
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VAWTs [11]. In this model the entire turbine is assumed to be enclosed within a single
streamtube as shown in Fig. 7. This model first incorporated the concept of the windmill
actuator disc theory into the analytical prediction model of a Darrieus-type VAWT. In this
theory the induced velocity (rotor axial flow velocity) is assumed to be constant
throughout the disc and is obtained by equating the streamwise drag with the change in
axial momentum.
In the assumption, the actuator disc is considered as the surface of the imaginary body
of revolution. It is assumed that the flow velocity is constant throughout the upstream and
downstream side of the swept volume. This theory takes into account the effect of airfoil
stalling on the performance characteristics. The effects of geometric variables such as blade
solidity and rotor height–diameter ratio have been included in the analysis. The effect of
zero-lift-drag coefficient on the performance characteristics has also been included. Wind
shear effect cannot be incorporated into the model.
Now, according to Gluert Actuator Disk theory, the expression of the uniform velocity
through the rotor is
Va ¼
V1 þ Vw
2
, (15)
where Vw is the wake velocity. All the calculations in this model are performed for a single
blade whose chord equals the sum of the chords of the actual rotor’s blades. The
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Fig. 7. Schematic of single streamtube model.

streamwise drag force (FD) due to the rate of change of momentum is
FD ¼ _m Á V1 À Vwð Þ, (16)
where _m (¼ ArVa) is the mass flow rate. The rotor drag coefficient (CDD) is defined as
CDD ¼
FD
1=2rAV2
a
. (17)
From Eqs. (16) and (17), we can find that
CDD ¼ 4
V1 À Va
Va

(18)
and
Va
V1
¼
1
1 þ CDD=4

. (19)
The overall torque and power coefficient of the VAWT can be determined from Eqs.
(13) and (14) by utilizing the expression of Va=V1 derived in Eq. (19) above.
This model can predict the overall performance of a lightly loaded wind turbine but
according to the inquest, it always predicts higher power than the experimental results. It
does not predicts the wind velocity variations across the rotor. These variations gradually
increase with the increase of the blade solidity and tip speed ratio. In 1980, Noll and Ham
presented an analytical method for the performance prediction of a cyclically pitched
straight-bladed vertical-axis wind turbine using the single streamtube model [12]. They
added the effect of strut drag, turbulent wake state and dynamic stall to their analytical
method.
5.1.2. Multiple streamtube model
In 1974, Wilson and Lissaman [13] introduced the Multiple streamtube model which was
an improvement to single streamtube model. In this model the swept volume of the turbine
is divided into a series of adjacent, aerodynamically independent parallel streamtubes as
shown in Fig. 8. The blade element and momentum theories are then employed for each
streamtube. In their model they considered the flow as inviscid and incompressible for the
calculation of the induced velocity through the streamtube. As a result, there appears only
the lift force in the calculation of the induced velocity. Wilson and Lissaman [13]
considered the theoretical lift for their calculation, which is given by
Cl ¼ 2p sin a. (20)
In this model, the induced velocity ratio can be found from the following expression:
Va
V1
¼ 1 À
k
2
:
Nc
R
:
R$
V1
: sin y

, (21)
where k is a factor found through iteration. In this model, the induced velocity varies over
the frontal disc area both in the vertical and horizontal directions [14]. Atmospheric wind
shear can be included in this model. However, this model still is inadequate in its
description of flow field and it can be applied only for a fast running lightly loaded wind
turbine.
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In 1975, Strickland [15] presented another multiple streamtube model for a Darrieus-
type VAWT. In this model, induced velocity is found by equating the blade elemental
forces (including airfoil drag) and the change in the momentum along each streamtube.
The wind shear effects have also been included in the calculation of the model. This model
predicts the overall performance reasonably, especially when the rotor is lightly loaded. It
displays improvement over the single streamtube model.
The basic difference between Wilson’s and Strickland’s models is that Wilson used the
theoretical lift force only in the calculation of induced velocity while Strickland added the
effect of drag force as well for the similar calculation. Among these two models, Wilson’s
model gives fast convergence while Strickland’s model gives slow convergence due to
added complicacy.
Another theory based on the multiple streamtube model including the effects of airfoil
geometry, support struts, blade aspect ratio, turbine solidity and blade interference was
presented by Muraca et al. [16]. The effect of flow curvature is evaluated by considering the
flow over a flat plate. They derived an expression of lift distribution on the plate with the
variable angle of attack from the leading to the trailing edge points of the flat plate and
averaged the distributed lift force over the whole surface of the flat plate. According to
them, the effect of flow curvature on the performance characteristics is insignificant for a
low chord to radius ratio.
In 1977, Sharpe gave an elaborated description of the multiple streamtube model in a
report. The principal idea of his model is similar to Strickland’s model. Additionally, he
incorporated the effect of Reynolds number into the calculation [17]. In 1980, another
improved version of the multiple streamtube model was presented by Read and Sharpe [18]
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Fig. 8. Schematic of multiple streamtube model.

for VAWT. In their model the parallel streamtube concept is dispensed with and the
expansion of the streamtube is included. It is strictly applicable to low solidity lightly
loaded wind turbines with large aspect ratio (H/c). It can predict the instantaneous
aerodynamic blade forces and the induced velocities better than those by the conventional
multiple streamtube model. But the prediction of overall power coefﬁcients cannot be
made with reasonable accuracy. It usually gives lower power than that obtained
experimentally.
5.1.3. Double-multiple streamtube model
In 1981, Paraschivoiu [19] introduced double multiple streamtube theory for the
performance prediction of a Darrieus wind turbine. As shown in Fig. 9, in this model the
calculation is done separately for the upstream and downstream half cycles of the turbine.
At each level of the rotor, the upstream- and downstream-induced velocities are obtained
using the principle of two actuator discs in tandem. The concept of the two actuator discs
in tandem for a Darrieus wind turbine was originally given by Lapin [20]. For both the
upstream and downstream half cycles vertical variation of the induced velocity (like that in
the multiple stream tube model) is considered while in the horizontal direction induced
velocity is assumed to be constant (like that of a single streamtube model). For the
upstream half-cycle, the wake velocity is represented by
Ve ¼ V1i 2
Vau
V1i
À 1

¼ V1i 2uu À 1ð Þ, (22)
where V1i is the local ambient wind velocity (which is different at different heights of the
turbine bladed due to the effect of wind shear), Vau is the induced velocity and
uu( ¼ Vau=V1i) is the interference factor for the upstream half-cycle. For the downstream
half-cycle of the rotor, Ve is the input velocity. The induced velocity for the downstream
half-cycle is Vad which can be written as
Vad ¼ udVe ¼ ud 2uu À 1ð ÞV1i, (23)
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Fig. 9. Schematic of double-multiple streamtube model.

where, ud( ¼ Vad=Ve) is the interference factor for the downstream half-cycle. The
streamtube induced velocity is calculated by a double iteration, one for each part of the
rotor.
The double-multiple streamtube model with constant and variable interference factors
(induced velocity ratios), including secondary effects for a Darrieus wind rotor was
examined by Paraschivoiu et al. [21]. They found a relatively significant influence of the
secondary effects, namely, the blade geometry and profile type, the rotating tower and the
presence of struts and aerodynamically spoilers, especially at high tip speed ratios. They
considered the variation of the induced velocity as a function of azimuth angle that gives a
more accurate calculation of the aerodynamic loads. In the paper presented by
Paraschivoiu and Delclaux [22], they made improvements in the double-multiple
streamtube model. They considered the induced velocity variation as a function of the
azimuth angle for each streamtube.
The double-multiple streamtube model gives better correlation between the calculated
and experimental results, especially for the local aerodynamic blade forces with the
multiple streamtube models. However, this model gives over prediction of power for a high
solidity turbine and there appears to be a convergence problem for the same type of
turbine especially in the downstream side and at the higher tip speed ratio.
5.2. Vortex model
The Vortex models are basically potential flow models based on the calculation of the
velocity field about the turbine through the influence of vorticity in the wake of the blades.
The turbine blades are represented by bound or lifting-line vortices whose strengths are
determined using airfoil coefficient datasets and calculated relative flow velocity and angle
of attack.
A simple representation of the vortex system associated with a blade element is shown in
Fig. 10. The VAWT blade element is replace by a ‘‘bound’’ vortex filament sometimes
called ‘‘substitution’’ vortex filament or a lifting line. The strengths of the bound
vortex and each trailing tip vortex are equal as a consequence of the Helmholtz theorems
of voticity [23]. According to Fig. 10, the strengths of the shed vortex systems have
changed on several occasions. On each of these occasions, a spanwise vortex is shed whose
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Fig. 10. Vortex system for a single blade element.

strength is equal to the change in the bound vortex strength as dictated by Kelvin’s
theorem [23].
The fluid velocity at any point in the flow field is the sum of the undisturbed wind stream
velocity and the velocity induced by the entire vortex filaments in the flow filed. The
velocity induced at a point in the flow field by a single vortex filament can be obtained from
the Biot–Savart law, which relates the induced velocity to the filament strength. Referring
to the case shown in Fig. 11, for a straight vortex filament of strength G and length l,
induced velocity ~Vp at a point P on the filament is given by,
~Vp ¼ ~e
G
4pd
cos y1 À cos y2ð Þ, (24)
where d is the minimum distance of the point P from the vortex filament, ~e and ~r are the
unit vectors. It should be noted that if point P should happen to lie on the vortex filament,
Eq. (24) yields indeterminate results, since ~e cannot be defined and the magnitude of ~Vp is
infinite. The velocity induced by a straight filament on itself is, in fact, equal to zero.
In order to allow closure of the Vortex model, a relationship between the bound vortex
strength and the velocity induced at a blade segment must be obtained. A relationship
between the lift L per unit span on a blade segment and the bound vortex strength GB is
given by the Kutta–Joukowski law. The lift can also be formulated in terms of the airfoil
lift coefficient CL. Equating these two expressions for the lift, yields the required
relationship between the bound vortex strength and the induced velocity at a particular
blade segment as,
GB ¼ 1
2cCLW, (25)
where, c is the blade chord. After determining the induced velocity distribution, it becomes
straight-forward to obtain performance characteristics of VAWT as described in Section 4.
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Fig. 11. Velocity induced at a point by a vortex filament.

In 1975, Larsen [24] first introduced the idea of Vortex model. The Vortex system for a
single blade element of a VAWT. He used his Vortex model for the performance prediction
of a cyclogiro windmill. His model is a two-dimensional one but if the vortex trailing from
the rotor blade tips is considered, it may be said that it is not strictly two-dimensional.
However, in his model angle of attack is assumed small, as a result, the stall effect is
neglected.
Fanucci and Walters [25] presented a two-dimensional Vortex model applicable to a
straight-bladed VAWT. In their analysis they considered the angle of attack very small
which eliminates the stall effect. Holme [26] presented a Vortex model for a fast running
vertical-axis wind turbine having a large number of straight, very narrow blades and a high
height–diameter ratio (in order to make a two-dimensional flow assumption). The analysis
is valid for a lightly loaded wind turbine only. Wilson [27] also introduced a two-
dimensional vortex analysis to predict the performance of a giromill. In his method he did
not take the stall effect into account, because the angle of attack was assumed to be small.
In 1979, Strickland et al. [28] presented an extension of the Vortex model which is a
three-dimensional one and the aerodynamic stall is incorporated into the model. They
presented the experimental results for a series of two-dimensional rotor configurations.
Their calculated values show more or less good correlation with the experimental results
for the instantaneous blade forces and the near wake flow behind the rotor. Strickland et
al. [29] made improvements on the prior Vortex model (quasi-steady Vortex model). The
latest model is termed as the dynamic Vortex model, since, in this model the dynamic
effects are included. The improvements over the prior model are that it includes the
dynamic stall effect, pitching circulation and added mass effect. They repeated the
experiment with the test model as is mentioned in Ref. [30] and found appreciable
variations with the prior results. The correlation with their calculated values by the
dynamic Vortex model and the latest experimental results of the local blade forces and
wake velocities seem to be reasonable in some cases.
In 1984, Cardona [31] incorporated the effect of flow curvature following the method
given by Migliore et al. [32] into the original aerodynamic Vortex model presented by
Strickland et al. [30]. They also added a modified form of the dynamic stall effects. They
found an improved correlation with the calculated and experimental results for the
instantaneous aerodynamic blade forces and the overall power coefficients.
The main disadvantage of Vortex model is that it takes too much computation time.
Furthermore, this model still rely on significant simplifications, like potential flow is
assumed in the wake and the effect of viscosity in the blade aerodynamics is included
through empirical force coefficients [33].
5.3. Cascade model
The periodic equidistant arrangement of several blades or vanes of turbomachinaries is
called a cascade. Hence, the cascade is the basic element of the turbomachine, and cascade
flow is the essential physical phenomenon for the operation of the machine [34]. The
Cascade model was proposed by Hirsch and Mandal [35] to apply the cascade principles,
widely used for turbomachinaries, for the analysis of VAWTs for the first time. In this
model, the blade airfoils of a turbine are assumed to be positioned in a plane surface
(termed as the cascade) with the blade interspace equal to the turbine circumferential
distance divided by the number of blades as shown in Fig. 12. The relationship between the
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wake velocity and the free stream velocity is established by using Bernoulli’s equation
while the induced velocity is related to the wake velocity through a particular semi-
empirical expression.
In this model, the aerodynamic characteristics of each element of the blade are obtained
independently, like the double-multiple streamtube theory, for the upwind and downwind
halves of the rotor considering the local Reynolds number and the local angle of attack as
shown in Fig. 13. After determination of the local relative flow velocity and the angle of
attack, the VAWT is developed into a cascade configuration that is shown in Fig. 12. The
cascade is considered in a plane normal to the turbine axis. If the blade represented by A at
an azimuth angle y is considered as the reference blade, the flow conditions on the other
two blades represented by B and C, are assumed to be equal to those of the reference
blade. This process is continued for one complete revolution of the reference blade with a
step of dy.
To find the induced velocity, a relationship between the wake velocity and the induced
velocity is introduced. For the upstream side this is expressed as
Vau
V1
¼
Ve
V1
ki
(26)
and for the downstream side, this is expressed as
Vad
Ve
¼
Vw
Ve
ki
, (27)
where Ve and Vw are the wake velocities in the upstream and downstream side. The value
of the exponent ki is found from a fit of experimental results. The induced velocity ratio for
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Fig. 12. Development of blade into a cascade configuration.

the downstream side can be written in the form,
Vad
V1
¼
Vad
Ve
Ve
V1
¼
Ve
V1
Vw
Ve
ki
. (28)
The expression of the exponent ki is written in accordance with Ref. [35] as
ki ¼ ð0:425 þ 0:332sÞ, (29)
where s ¼ NC=R . The ﬁnal expression for the overall torque is found from,
Q ¼ rR3 H
R
Z 2p
0
ðW2
2 sin a2 cos a2 À W2
1 sin a1 cos a1Þ dy, (30)
where W1 and W2 are the relative velocities in the cascade inlet and outlet. Detail
description of this model can be found from Hirsh and Mandal [35].
The Cascade model can predict the overall values of both low and high solidity turbines
quite well. It takes reasonable computation time. It does not make any convergence
problem even at the high tip speed ratios and high solidities. The instantaneous blade
forces calculated by this model show improved correlation in comparison to those
calculated by the conventional Momentum model. The theory also incorporated the
effect of the local Reynolds number variation at different azimuth angles (orbital position),
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Fig. 13. Horizontal section of a straight-bladed Darrieus-type VAWT with ﬂow velocities in the upstream and
downstream sides.

zero-lift-drag coefficients, finite aspect ratios and the flow curvature effect in the
calculation process.
Subsequently, to improve the analytical capability of this model, two important effects
of the dynamic stall and flow curvature with blade pitching were taken into account by
Mandal and Burton [36]. The calculated values of the wake velocities after these
modifications become comparable with those by the complex dynamic Vortex model.
6. Conclusions
Several aerodynamic models have been analyzed in this paper which are applied for
better performance prediction and design analysis of straight-bladed Darrieus-type
VAWT. At present the most widely used models are the double-multiple streamtube
model, free-Vortex model and the Cascade model. It has been found that, each of these
three models has their strengths and weaknesses. Though among these three models, the
Vortex models are considered to be the most accurate models according to several
researchers, but they are computationally very expensive and in some cases they suffer
from convergence problem. It has also been found that the double-multiple streamtube
model is not suitable for high tip speed ratios and high-solidity VAWT. On the other hand,
the Cascade model gives smooth convergence even in high tip speed ratios and high solidity
VAWT with quite reasonable accuracy.
Acknowledgments
This work was financially supported by Natural Sciences and Engineering Research
Council of Canada (NSERC). Authors would also like to thank Umm Tahzin for drawing
the sketches for this paper.
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