AERODYNAMIC ANALYSIS OF VERTICAL AND HORIZONTAL AXIS WIND
TURBINES

By
Anwelli Soala Okpue

A THESIS
Submitted to
Michigan State University
in partial fulfillment of the requirements
for the degree of
MASTER OF SCIENCE
Mechanical Engineering
2011

ABSTRACT
AERODYNAMIC ANALYSIS OF VERTICAL AND HORIZONTAL AXIS WIND
TURBINES

By
Anwelli Soala Okpue
The power generated by horizontal axis and vertical axis wind turbines is strongly dependent on
the aerodynamic performance of the turbines. This includes the lift and drag forces generated by
the individual blades and interactions between them in a rotating configuration; as well the
aerodynamic effects of the turbine’s structure and the terrain. To maximize the efficiency of
wind turbines, experimental and computation analysis of the blade and system is normally
needed. This research is part of larger efforts at Michigan State University to develop sustainable
low speed wind energy systems and is focused on the aerodynamic modeling of vertical and
horizontal axis wind turbines. The main objective is to better understand the effects of various
parameters on the performance of these turbines. A computational model was developed as part
of this research for studying the aerodynamics of vertical axis turbines. The model is refined via
experimental data and other computational results available in literature. A parametric study was
conducted with this model. The effects of parameters like the tip speed ratio, airfoil shape/type,
Reynolds number are investigated in detail. The aerodynamics of variable pitch vertical axis
wind turbines is also studied. Our results indicate that a significant improvement in aerodynamic
efficiency

of

vertical

axis

wind

turbines

is

possible

by

variable

pitching.

ACKNOWLEDGMENTS
I would like to express my sincerest gratitude to those who made this work possible. My thanks
go out to the Department of Mechanical Engineering at Michigan State University for the
financial support provided during the course of my study and research efforts. I am very grateful
to my Graduate Advisor, Dr. Farhad Jaberi for his wisdom, enthusiasm, and patience. Also for
the countless other advice I have received from him in developing me professionally and
personally. I will also like to thank the members of my Graduate Advisory Committee, Drs.
Farhang Pourboghrat and Brian Feeny, for their helpful advice and suggestions. I especially wish
to thank my Dad for his courage and support, my Mom for her continued prayers and
encouragement, and my brother, Nduka for always being there for me in his special way, my
extended family and friends, especially Bola Ndifontah, for their faith, support, and
encouragement. Finally, and most of all, I thank my Lord and Redeemer, Jesus Christ, in Whom,
by Whom, and for Whom I and all things consist.

iii

Table of Contents

List of Tables ................................................................................................................................. vi

List of Figures ............................................................................................................................... vii

1.

Chapter 1: Introduction ........................................................................................................... 1

2.

Chapter 2: Theory ................................................................................................................... 6
2.1

VAWT............................................................................................................................. 6

2.1.1

Other Simulation Models .......................................................................................... 8

2.1.2

Cascade Theory as Applied for VAWT Simulation ............................................... 16

2.2

3.

FAST for HAWT .......................................................................................................... 35

Chapter 3: Vertical Axis Wind Turbines .............................................................................. 40
3.1

Validation ...................................................................................................................... 40

3.2

Results and Discussion ................................................................................................. 45

3.2.1
3.2.2

Effects of Reynolds number.................................................................................... 46

3.2.3

Torque Characteristics of VAWT ........................................................................... 60

3.2.4

Effects of Airfoil Shape .......................................................................................... 68

3.2.5

Effects of Solidity ................................................................................................... 73

3.2.6

4.

Tip Speed Ratio....................................................................................................... 45

Effects of Wind Shear ............................................................................................. 75

Chapter 4: VAWT with Variable Blade Pitch ...................................................................... 78
4.1
4.2

5.

Introduction ................................................................................................................... 78
Results and Discussion ................................................................................................. 79

Chapter 5: Horizontal Axis Wind Turbines .......................................................................... 87
5.1

Introduction ................................................................................................................... 87

5.2

Effects of Wind Speed .................................................................................................. 88
iv

5.3
5.4

Effects of Pitch Angle ................................................................................................... 98

5.5

6.

Effects of RPM ............................................................................................................. 93
Variation of Parameters along the Blade Span ........................................................... 103

Chapter 6: Summary and Conclusions ................................................................................ 115

Bibliography ............................................................................................................................... 120

v

List of Tables

Table 1-1: Comparing HAWT and VAWT by highlighting their advantages and disadvantages . 3

Table 3-1: The geometry and parameters of the simulated VAWT.............................................. 41

Table 3-2: Wind shear exponents for various terrains (The Engineering Toolbox: Wind Shear
n.d.) ............................................................................................................................................... 76

Table 4-1: Turbine efficiency for fixed and variable pitch turbines ............................................. 80

Table 5-1: The geometry and parameters of the simulated HAWT.............................................. 87

vi

List of Figures

Figure 2-1: The general configuration of a linear cascade model (Swamy and Kumaran 2009) ... 7

Figure 2-2: The orientation of blades in the linear cascade model for a VAWT............................ 8

Figure 2-3: The side view of the streamtube around the actuator disk for a HAWT (dashed line
drawn through the blade) and the velocity and pressure distribution upstream and downstream of
the disk .......................................................................................................................................... 10

Figure 2-4: A typical streamtube used in the analysis .................................................................. 11

Figure 2-5: Rotor element replaced by the actuator disk .............................................................. 12

Figure 2-6: Early development of flow around an airfoil ............................................................. 18

Figure 2-7: Development after the starting vortex has been formed ............................................ 18

Figure 2-8: Smooth flow over the airfoil ...................................................................................... 19

Figure 2-9: The curve C2 around the airfoil involving points abcda (Anderson Jr. 2001) ........... 20

Figure 2-10: The unwrapped configuration of the turbine in the cascade theory ......................... 22

Figure 2-11: The control surface around the reference blade ....................................................... 23

Figure 2-12: The path around the airfoil used to compute the circulation in the cascade theory
(Bhimarasetty and N. 2010) .......................................................................................................... 23

Figure 2-13: The blade forces on the airfoil ................................................................................. 27

vii

Figure 2-14 : Flow chart of simulations based on the cascade model. ......................................... 34

Figure 3-1: Power vs. V∞, obtained from the CARDAA (DSTM model) and the cascade model.
“For interpretation of the references to color in this and all other figures, the reader is referred to
the electronic version of this thesis.” ............................................................................................ 43

Figure 3-2: Cp vs. TSR, obtained by the CARDAA (DSTM) and cascade models. ..................... 43

Figure 3-3: The results from the CARDAA code compared with experimental results of a 17 m
SANDIA Darrieus VAWT ............................................................................................................ 44

Figure 3-4: Cp vs. Tip speed ratio for a 3m expereimental turbine .............................................. 44

Figure 3-5: Power coefficient, Cp variations with Re for the test case simulated using the cascade
model............................................................................................................................................. 49

Figure 3-6: CL vs. angle of attack for different Reynolds numbers .............................................. 51

Figure 3-7: CD vs. angle of attack for different Reynolds numbers .............................................. 51

Figure 3-8: Cp vs. TSR for Re calculated based on six different methods ................................... 52

Figure 3-9: Power vs. wind speed for Re calculated based on six different methods .................. 53

Figure 3-10: CL vs. alpha at a TSR of 4.4. The RPM simulated was fixed at 125. This figure is
plotted after convergence has been achieved for this TSR. This figure shows only the lift
coefficients and angle of attacks generated in the upstream of the turbine .................................. 54
Figure 3-11: Torque vs. Ѳ at a TSR of 4.4. This figure is plotted after convergence has been
achieved for this TSR. The RPM simulated was fixed at 125. This figure shows the torque
characteristics for both the up and downstream of the turbine ..................................................... 54

viii

Figure 3-12: Angle of attack vs. Ѳ at a TSR of 4.4. This figure is plotted after convergence has
been achieved for this TSR. The RPM simulated was fixed at 125. This figure shows the angles
of attack for both the up and downstream of the turbine .............................................................. 55

Figure 3-13: CL vs. angle of attack for the TSR of 4.4. The RPM simulated was fixed at 125 .... 58

Figure 3-14: CD vs. angle of attack for the TSR of 4.4. The RPM simulated was fixed at 125 ... 59

Figure 3-15: Power vs. wind speed for the TSR of 4.4. The RPM simulated was fixed at 125 ... 59

Figure 3-16: Cp vs. TSR for the turbine in Table 1, simulated with Re = ReT............................. 61
Figure 3-17: Torque vs. Ѳ for the TSR of 1.65 for the turbine with configuration in Table 3-1,
simulated with Ret. CL and CD coefficients are also plotted. For brevity only the upstream section
of the turbine has been shown ....................................................................................................... 62
Figure 3-18: Torque vs. Ѳ for the TSR of 3.09 for the turbine with configuration in Table 3-1,
simulated with Ret. CL and CD coefficients are also plotted. For brevity only the upstream section
of the turbine has been shown ....................................................................................................... 62
Figure 3-19: Torque vs. Ѳ for a TSR of 5.63 for the turbine with configuration in Table 3-1,
simulated with Ret. CL and CD coefficients are also plotted. For brevity only the upstream section
of the turbine has been shown ....................................................................................................... 63

Figure 3-20: RPM vs. time obtained from experiments of a VAWT (Dominy, et al. n.d.) .......... 66

Figure 3-21: Torque characteristic of a VAWT for the TSR of 1.65 ........................................... 67

Figure 3-22: Torque characteristic of a VAWT for the TSR of 1.65. Torques on both blades have
been plotted on a Ѳ axis going from 1˚ to 180˚ ............................................................................ 67

Figure 3-23: Torque characteristic of a VAWT for the TSR of 5.63. Torques on both blades have
been plotted on a Ѳ axis going from 1˚ to 180˚ ............................................................................ 68
ix

Figure 3-24: The shapes of symmetric airfoils studied – NACA 0012, NACA, NACA 0015, and
NACA 0021 (Malcolm's Human Powered Aircraft Site n.d.) ...................................................... 70
Figure 3-25: The shape of the asymmetric airfoil studied – NREL’s S809 (UIUC Airfoil Data
Site n.d.) ........................................................................................................................................ 70

Figure 3-26: Cp vs. TSR for the four different airfoils studied .................................................... 71

Figure 3-27: Power vs. wind speed V∞ for the four airfoils studied ............................................. 71
Figure 3-28: CL and CD vs. Ѳ for the NACA 0012 and NACA 0021 airfoils .............................. 72
Figure 3-29: The angle of attack vs. Ѳ for the NACA 0012 and NACA 0021 airfoils and two
different TSRs. The plots are for the upstream section only ........................................................ 72
Figure 3-30: CL/CD vs. Ѳ for the 0012 and 0021 airfoils and two different TSRs ....................... 73

Figure 3-31: Cp vs. TSR for turbines with different solidity........................................................ 74

Figure 3-32: Torque vs. TSR for turbines with different solidity ................................................. 75

Figure 3-33: Wind speed variations with the height for three different wind shear exponents .... 77

Figure 3-34: Cp vs. TSR for a turbine with and without wind shear. The wind shear exponents
used were 0.1 and 0.25 .................................................................................................................. 77
Figure 4-1: Torque vs. Ѳ for simulations with fixed and variable pitch blades. These plots are for
wind speed V∞ of 6 m/s................................................................................................................. 81
Figure 4-2: Torque vs. Ѳ for simulations with fixed and variable pitch blades. These plots are for
wind speed V∞ of 7 m/s................................................................................................................. 81
x

Figure 4-3: Torque vs. Ѳ for simulations with fixed and variable pitch blades. These plots are for
wind speed V∞ of 8 m/s ................................................................................................................. 82
Figure 4-4: The variation of angle of attack α against Ѳ using the optimization scheme. This plot
is for a wind speed V∞ of 6 m/s ..................................................................................................... 83

Figure 4-5: CL vs. alpha for Re = 80,000 ...................................................................................... 84
Figure 4-6: Pitch angle vs. Ѳ for the variable pitch blade ............................................................ 84
Figure 4-7: Angle of attack α vs. Ѳ for the variable pitch blades ................................................. 85
Figure 4-10: Pitch angle vs. Ѳ for the variable pitch blade with the tangent hyperbolic smoothing
being applied to optimized pitch results (V∞ = 6 m/s) .................................................................. 86

Figure 5-1: The variation of power with wind speed for three RPM values ............................. 89

Figure 5-2: Averaged CL at different wind speeds and RPM values ............................................ 90

Figure 5-3: Averaged CD at different wind speeds and RPM values ........................................... 90

Figure 5-4: The coefficient of power Cp at different wind speeds and RPM values .................... 91

Figure 5-5: The angle of attack vs. wind speed for different RPM valued ................................... 91

Figure 5-6: The dynamic pressure vs. wind speed for different RPM valued .............................. 92

Figure 5-7: Aerodynamic efficiency vs. wind speed for different RPM valued ........................... 92

Figure 5-8: The axial induction factor a vs. wind speed for different RPM valued ..................... 93
xi

Figure 5-9: Plot of power against RPM for different pitch angles ............................................... 95

Figure 5-10: CL vs. RPM for different RPM and pitch angles ...................................................... 95

Figure 5-11: CD vs. RPM for various RPMs and various pitch angles ......................................... 96

Figure 5-12: Cp vs. RPM for different pitch angles ..................................................................... 96
Figure 5-13: Plots of angle of attack, α, vs. RPM for different pitch angles ................................ 97

Figure 5-14: CL/CD ratio for various RPM values at different pitch angles ................................. 97

Figure 5-15: Power vs. pitch angle for different wind speeds ...................................................... 99

Figure 5-16: CL vs. blade pitch angle for different wind speeds................................................. 100

Figure 5-17: CD vs. the blade pitch angle for different wind speeds .......................................... 100

Figure 5-18: Cp vs. blade pitch angle for different wind speeds ................................................ 101

Figure 5-19: Angle of attack vs. blade pitch angle for different wind speeds ............................ 101

Figure 5-20: Dynamic pressure vs. pitch angle .......................................................................... 102

Figure 5-23: The CL/ CD ratio at the different sections of the blade ........................................... 103

Figure 5-24: Section pitch angle at the different sections of the blade ....................................... 104

Figure 5-25: The angle of attack at the different sections of the blade....................................... 104
xii

Figure 5-26: Re at the different sections of the blade ................................................................. 105

Figure 5-27: CD at the different sections of the blade ................................................................. 105

Figure 5-28: CL at the different sections of the blade ................................................................. 106

Figure 5-29: Normal force at the different sections of the blade ................................................ 106

Figure 5-30: Tangential force at the different sections of the blade ........................................... 107

Figure 5-33: Section pitch angle at the different sections of the blade ....................................... 107

Figure 5-34: Angle of attack at the different sections of the blade ............................................. 108

Figure 5-35: Re at the different sections of the blade ................................................................. 108

Figure 5-36: CL at the different sections of the blade ................................................................. 109

Figure 5-37: CD at the different sections of the blade ................................................................. 109

Figure 5-38: Normal force at the different sections of the blade ................................................ 110

Figure 5-39: Tangential force at the different sections of the blade ........................................... 110

Figure 5-42: Section pitch at the different sections of the blade ................................................ 111

Figure 5-43: Angle of attack at the different sections of the blade ............................................. 111

Figure 5-44: Re at the different sections of the blade ................................................................. 112

Figure 5-45: CL at the different sections of the blade ................................................................. 112
xiii

Figure 5-46: CD at the different sections of the blade ................................................................. 113

Figure 5-47: Normal force at the different sections of the blade ................................................ 113

Figure 5-48: Tangential force at the different sections of the blade ........................................... 114

xiv

1. Chapter 1: Introduction
Currently wind turbines generate less than 2% of the total electric energy in the US. The DOE
aims to produce 20% of electric energy used in the US from wind in the next few decades. This
represents more than a tenfold increase in production, and it will require extensive investments
as well as research and development in wind turbines, energy storage systems and electric grid
integration techniques. These machines have economic incentives, as well as reducing the US’s
energy dependence on fossil fuels which both pollutes the atmosphere, and are neither renewable
nor sustainable. Sustainability issues arise because as societies grow, there is a corresponding
increase in energy consumption. In addition, the supply of fossil fuels is subject to other external
factors such as politics and other world events, thus creating an issue of national security. In
order to meet this goal of energy independence, it is important to produce more efficient turbines
which can compete with other forms of energy production systems, and that is why this research
is necessary.
There are two general configurations of wind turbines: horizontal axis wind turbine (HAWT) and
vertical axis wind turbine (VAWT). Within these classifications there are various designs that
have been proposed over the years. As the name implies, the HAWT has its main shaft (i.e. the
shaft which connects the rotor to the generator) in the horizontal axis. These turbines are made of
several blades (two or more) erected atop a tower with all the main components (generator, gearbox, controls) sitting at the top of the tower. The HAWT can also be classified as either upwind
or downwind turbines. The difference being that the rotor blades are oriented either facing or
backing the wind. The VAWT configuration has its main shaft oriented in a vertical direction.
For this turbine, the main components are at ground level. Modern VAWTs have three main
types: Darrieus type, H-Rotor type, and Savonius type turbines. The Darrieus and H-Rotor type
1

turbines are lift force driven turbines, as they rely on the generation of a lift force on their blades
to generate torque which is converted to power by the generator. Savonius type turbines on the
other hand are drag force driven turbines. These turbines are essentially two cups or half-drums,
oriented in opposing directions and connected to a central shaft. One cup catches the wind,
rotates the shaft and brings the opposing cup into the wind’s direction which further turns the
shaft, thus forming a couple moment. As this process is repeated by the incoming wind, torque is
generated which can then be used to do work.
In general, lift type turbines are more efficient that drag type turbines. Power coefficient for lift
turbines vary between 30% and 45%, while drag type turbines are usually not above 25%. These
types of turbines are suitable for low torque applications and they are commonly used for wind
speed instruments such as wind vanes or anemometers (Kirke 1998)1; i.e. applications where
efficiency is not the primary concern.
Currently, HAWTs are the most prevalent type of wind turbines being used all over the world to
generate electricity (both onshore and offshore). VAWTs are the less commonly used systems;
they have primarily been used to generate small amounts of power (in the order of kilowatts).
The major reason for VAWTs being less prevalent is the fact that they are known to have lower
maximum efficiencies compared to HAWT. Table 1-1 compares the advantages and
disadvantages of HAWTs and VAWTs.

2

Table 1-1: Comparing HAWT and VAWT by highlighting their advantages and disadvantages

Horizontal Axis Wind Turbine (HAWT)

Disadvantages



Higher efficiency due to variable blade pitching capability.



Advantages

Established manufacturing know-how and market acceptability.



Increased costs associated with transportation of tall towers and long
blades.



Installation requires very tall and expensive cranes, and skilled
operators.



Massive tower and heavy foundation required to support blades,
gearbox, and generator.



Complex design of twisted blades is difficult and expensive to
fabricate.



Large size may disrupt the landscape and create local opposition.



Requires yaw control to turn the blades and nacelle toward the wind.



A distance of 5 rotor diameters siting is needed in order to minimize
wake effects.



Difficult, expensive, and frequent maintenance required.

3

Table 1-1 (cont’d)
Vertical Axis Wind Turbine (VAWT)
Advantages



Smaller tower and support structure, since bearings are mounted
near the ground.



The generator and gearbox are installed near the ground which
reduces the cost of the foundation and support structure.



Suitable for low speed winds, due to lower wind startup speed.



May be built at locations where tall structures are prohibited.



Can take advantage of locations where landscape increases wind
speed near the ground.



May have a lower noise signature.



Needs less space than HAWT to generate the same amount of
power.



Straight blades are much easier and economical to fabricate or
extrude.




Lower aerodynamic efficiency compared to HAWT.



Disadvantages

Lower overall maintenance and transportation costs.

Blade fatigue failure due to change in stress sign during each
revolution.



May require dismantling the entire structure to fix the generator or
gearbox.

To enhance the performance of wind turbines, either a taller tower or a larger rotor must be used.
However, roughly speaking, tower costs increase with the height to the fourth power, while the
4

rotor costs increase with diameter cubed; meanwhile rotor power grows only with the diameter
squared (Jose 2010).2 The challenge is therefore to improve the efficiencies and power of the
current existing systems without growing the production, installation and transportation costs.
This research intends to explore the aerodynamics of wind turbines by using computer models.
The focus is on VAWTs. For aerodynamic analysis of HAWTs, FAST (Fatigue, Aerodynamics,
Structure and Turbulence) is used. This computer code was developed by National Renewable
Energy Laboratory (NREL) for studying two or three bladed turbines. For VAWTs, a
computational model I developed based on the cascade theory is used.
Chapter 2 describes the computational/ empirical models commonly used for the aerodynamic
analysis of wind turbines. They include the streamtube momentum models, the vortex model and
the cascade model. Details of the model used in this research are presented, and its
implementation is discussed. Chapter 3 presents the results involving the model/ code validation.
Additionally, in this chapter results describing the effects of various parameters on the
performance of VAWT are presented. The effects of parameters such as: Reynolds number,
airfoil shape, solidity, and wind shear are considered. Issues related to torque and self-start are
also discussed. Chapter 4 explores how a variable blade pitching technique can be used to
improve the turbine’s performance, as well as to resolve the problem of self-start associated with
VAWTs. A parametric study is also performed for the HAWTs and the results presented in
Chapter 5. Conclusions and recommendations concerning the operation, and optimization of
VAWT and HAWT are given in Chapter 6.

5

2. Chapter 2: Theory
2.1 VAWT
In this section the analytical/ computational models employed for VAWT and HAWT are
described.
The model used in this research for the aerodynamic analysis of VAWTs is based on the
Cascade Theory. This theory is widely used in the field of turbomachinery for calculating the
dynamic forces induced by the flow over blades in turbines and compressors. The cascade refers
to an array of blades around the hub of the rotor. The model assumes the airfoils to be
unwrapped into an array of blades having uniform pitch and spacing between them (see Figure
2-1). In turbomachinery applications, this modeling approach has been used to map pressure and
velocity distributions over the airfoils passages, and to obtain other basic quantities and
characteristics that are required in turbomachinery studies, such as: the inlet flow angle, mach
number, true Reynolds number, etc. Also since the unwrapped configuration simplifies the
problem, it is suitable for design purposes as well as carrying out parametric studies.
The use of cascade theory for vertical axis wind turbine simulation was first proposed by Hirsch
and Mandal (Hirsch and Mandal 1987).3Figure 2-2 shows the blade array configuration used to
simulate a three blade VAWT. Based on the theory, the particular type of cascade configuration
being used is a linear or rectilinear cascade. In computer simulation the cascade of blades are
represented as airfoils with the same shape, vis-à-vis the same aerodynamic properties such as
lift, drag and pitching moments.
The cascade theory as applied to VAWT further simplifies the problem by considering one of the
blades as the reference blade, and by assuming the flow conditions on other blades to be equal to
6

that on the reference blade. To investigate the relationship between blade forces (forces which
are generated due to energy exchange from the fluid to the turbine) and the flow properties, a
control surface is drawn around each airfoil with two identical streamlines along the longitudinal
lengths of the airfoil, and two parallel lines in front and behind the airfoil; the two parallel lines
form the upstream and downstream of the blade and represent the inlet and outlets of the control
surface. The continuity and momentum equations are applied to the control volume (Scholz.
n.d.). 4 Details of the theory are given in a later section.

Figure 2-1: The general configuration of a linear cascade model (Swamy and Kumaran 2009)5

7

Figure 2-2: The orientation of blades in the linear cascade model for a VAWT
2.1.1 Other Simulation Models
Blade Element Momentum (BEM) Model - A popular simulation model for VAWT is the
Blade Element Momentum (BEM) model. This model is the basis of various models such as:
Single Streamtube Model (Templin 1974)6, Multiple Streamtube Model (Wilson and Lissaman
1974),7 and the Double-multiple Streamtube Model (I. Paraschivoiu 1981).8In these models, the
flow velocities induced by the turbine blades in the flow’s path is obtained based on the principle
of conservation of momentum. The aerodynamic forces are tied to the momentum exchange and
are calculated as the rate of change of momentum of the fluid, i.e. the overall change in velocity
multiplied by the mass flow rate. The Actuator Disk Theory is then applied for computing the
turbine/ flow variables.
Actuator Disk Theory - The aim of a turbine in the wind’s path is to extract kinetic energy from
the wind. To model this extraction an actuator disk is placed in the flow’s path. The actuator disk
is simply a plane in the wind’s path over which changes in velocity and pressure can be
8

represented. The concept of the actuator disk theory is more intuitively understood for horizontal
axis wind turbines which when viewed from a plan or side view have the form of a “disk” in the
flow’s path (see Figure 2-3). This disk is surrounded by a streamtube which is an imaginary tube
around the disk; only the air flowing through the tube is considered in the analysis. Upstream of
the disk the streamtube has a cross-sectional area smaller than that of the disk and an area larger
than the disk in the downstream. The mass flow rate is the same everywhere, i.e.:
Eq. 2.1
In the above equation, subscripts “∞”, “d”, and “w” represent quantities upstream of the disk, at
the disk, and in the wake downstream of the disk. The velocity change experienced by the air as
it passes through disk is equated to the rate of change of momentum; this rate of change of
momentum is representative of the aerodynamic force used to rotate the turbine.
In the HAWT, this tube in broken up into various annular sections and analysis is carried out
over these sections. For applying the actuator disk theory to a VAWT, the disk is oriented in the
horizontal direction. The entire height of the turbine is broken into equal segments, and a
separate actuator disk is applied to each segment. A streamtube is also drawn around the circular
path traced by the turbine as shown in Figure 2-4. In the case of a multiple streamtube model the
streamtube surrounding the turbine, is divided into smaller rectangular “streamtubes” stretched
over the diameter of the turbine (Figure 2-5) (similar to the annular section of the HAWT). Here
the term streamtube refers to the rectangular strips. These streamtubes are fixed, and the turbine
blade is assumed to rotate around a circular path, cutting these streamtubes. Whenever the blade
cuts the stationary streamtube, the actuator disk theory is applied to solve for the momentum
change and force.
9

Figure 2-3: The side view of the streamtube around the actuator disk for a HAWT (dashed line
drawn through the blade) and the velocity and pressure distribution upstream and downstream of
the disk

10

Figure 2-4: A typical streamtube used in the analysis of VAWTs

11

Figure 2-5: Rotor element replaced by the actuator disk

12

The multiple streamtube model is a modification of an earlier single streamtube model, and it
was introduced because of the shortcomings of the earlier model. The earlier version was
sufficient in predicting the performance of a very lightly loaded wind turbine. However, it
always predicts higher power when compared to experimental results. It also does not adequately
predict the wind velocity variations across the rotor (Islam, S.K. and Ting 2008).9 It was
conceived that, by dividing the single streamtube into multiple streamtubes along the diameter of
the turbine, the predictive capabilities could be improved. From the results obtained, it has been
confirmed that better comparisons to experimental data were achieved by applying this model.
To perform the analysis via the double multiple streamtube model, the turbine is broken up into
two sections: the upstream and downstream sections. The free-stream velocity is the input for the
upstream section, and the wake velocity calculated from this section is used as the input to
calculate the variables in the downstream section. The general simulation procedure then follows
similar methodologies as with other streamtube models.
General Momentum Theory as Applied for Wind Turbines - The air that goes through the
streamtube undergoes an overall change in velocity, V∞ − Vw, and therefore the rate of change of
momentum is equal to this change in velocity multiplied by the mass flow rate.

Rate of change of momentum =

(V  Vw ) AcsVd

Eq. 2.2

where V with subscript "∞”, “d”, and “w” represents the velocity at points in the free-stream, at
the turbine, and in the wake of the flow; also Acs is the cross-sectional area of the disk and ρ is
the density of the fluid. The force causing this change of momentum comes entirely from the
pressure difference across the actuator disk because the system is otherwise completely
13

surrounded by air at atmospheric pressure which produces a net zero force (Burton, et al.
2001).10 Therefore,



( Pd  Pd ) Acs  (V  Vw ) AcsVd

Eq. 2.3

), the Bernoulli’s equation is applied for a streamline

To obtain the pressure difference(

over the length of a rectangular streamtube at its inlet and outlet. At the inlet we have,

1  V 2   gh  1  V 2  P    gh
 
d d
d
2  
2 d d

Eq. 2.4

Assuming the flow to be incompressible (ρ∞ = ρd) and horizontal (h∞ = hd) then,

1 V 2  P  1 V 2  P 

d
2 
2 d

Eq. 2.5

Similarly, at the outlet of the streamtube,

1 V 2  P  1 V 2  P 

d
2 w
2 d

Eq. 2.6

Subtracting these equations we obtain:


2
2
( Pd  Pd )  1  (V  Vw )
2

Eq. 2.7

1  (V 2  V 2 ) A  (V  V ) A V

w cs

w
cs d
2

Eq. 2.8

From Equation 2.3,

Therefore, we have:

Vw  (1  2a)V
14

Eq. 2.9

where the axial induction factor, a, is given as:

V
a 1 d
V

Eq. 2.10

The axial induction factor is a non-dimensional quantity which is used during simulation to
express the decrease in velocity from the free-stream at the rotor.
The force the turbine exerts on the air becomes:
2
F  ( P   P  ) Acs  2 AcsV a(1  a)
d
d

Eq. 2.11

This force is that at the actuator disk and is related to the power extraction at this location. The
force extraction is concentrated at the actuator disk, with the characteristic velocity Vd. After the
geometries and initial conditions have been specified an initial guess is made for the induction
factor, a, after which the various variables are calculated. Using an iterative approach the process
is repeated until the results converge. Details of the method can be found elsewhere in R.
Templin 1974, Strickland 1976, and I. Paraschivoiu 1981.11
Vortex Model - The vortex model is based on the potential theory of lifting bodies. The
potential theory is used to calculate the amount of circulation on the blade span, which by the
Kutta-Joukowski theorem (Equation 2.13) can be related to the lift on the blade; the theory
prescribes that the aerodynamic lift generated on the blade is due to the bound vorticity
distributions (Brownlee n.d.).12This model is based on the assumption that the induced velocity
field is due to the combined effects of the bound and wake vorticity distribution. The bound
vorticity distribution results from the distribution of circulation on the blade due to the vorticity
15

contained in the boundary layers of the wing. The wake vorticity distribution results from the
distribution of circulation in the wake of the blade section. The shed wake vorticity ultimately
stems from the bound vorticity distribution which flows from the blade tips. This model breaks
the entire turbine into nodes.
In applying the model, we start by finding the fluid’s relative velocity, ⃗ . This relative velocity
is due to the presence of three velocity components: the free-stream, ⃗

, the rotational velocity,

, and the induced velocity, ⃗ , produced by the bound and wake vorticity distribution:
⃗

⃗

⃗

Eq. 2.12

The first two velocities are known inputs, so the key to using this method is calculating the
induced velocity, ⃗ . At the first iteration, the induced velocity is set to zero at every node, and
the relative velocity is calculated based on the free-stream, and the rotational velocity. Along
with the relative velocity, the angle of attack is calculated, and by using the CL and CD data
obtained by experiment for a specific airfoil, the lift force can be calculated. Then by applying
Equation 2.13, the bound vorticity distribution is obtained and used to find the trailing wake
vorticity distribution. With these variables found, the induced velocity can be updated, and the
iteration proceeds until the convergence of vorticity distribution.

L  hV 

Eq. 2.13

2.1.2 Cascade Theory as Applied for VAWT Simulation
The cascade theory at its core is based on the momentum principle; it however incorporates some
of the ideas of potential theory. Recall that a turbine is used to extract energy from the flowing
16

fluid in order to utilize this energy to do useful work, i.e. generate electricity. Drag forces on
aerodynamic bodies tend to reduce the amount of energy that is transferred from the fluid to the
body hence making them less efficient; as a result the work generated by such forces (drag) are
considered losses. Therefore for lift driven turbines the aim is to reduce this energy loss on the
rotating aerodynamic components that aid power generation. For such systems we always want
to foster a large enough lift force which produces torque.
To achieve this objective most modern wind turbines are made of airfoils. When the airfoil
moves relative to air, the air follows the curvature of the airfoil’s profile and is accelerated
downwards. The result of this downward acceleration is the pressure increases under the airfoil
and the pressure decreases above it. At this early development of the flow, the fluid mass tries to
curl around the pointed trailing edge from the bottom surface to the top surface as shown in
Figure 2-6.

A consideration of inviscid, incompressible flow indicates that the theoretical

velocity becomes very large at sharp corners. A stagnation point develops on the upper surface of
the profile towards the trailing edge, along with a very strong vortical structure, known as the
starting vortex, see Figure 2-7. The strength of this structure is obtained from the sharp velocity
gradients being formed due to the tendency of the flow to curl upwards, and the net downward
flow over the airfoil. However, these conditions cannot exist for an extended period of time, so
in an actual flow the stagnation point on the top surface moves further towards the trailing edge.
In addition to eliminating the starting vortex from the profile, a circulation, , must be created
about the airfoil profile which has equal strength but opposite in direction when compared to the
starting vortex. According to Thomson’s theory, the circulation does not change in time. This
circulation is important so that the bifurcation streamline is displaced until it occurs at the
trailing edge. The condition that allows smooth flow to form around the airfoil’s trailing edge is
17

known as the Kutta’s trailing edge condition. This stage of flow development is illustrated in
Figure 2-8. The final step in lift generation comes after the initial transient process dies out, and
the steady flow is reached. The circulation which causes this kind of flow also increases the flow
velocity at the upper airfoil surface, and a corresponding reduction in the lower surface’s
velocity; in accordance with Bernoulli’s equation this corresponds to a low pressure distribution
on the top surface and higher pressures on the lower surface, so that a net upward resultant force
with a line of action perpendicular to the relative flow direction is formed. This resultant force is
the lift that is used to drive the turbine. The lift generated by the airfoil can be related to the
amount of circulation via the Kutta – Jourkowski theorem (as given in Equation 2.13):

L  hV 

Eq. 2.14

The flow visualizations obtained from PIV data (Anderson Jr. 2001) confirm the evolution of
flow illustrated in Figure 2-6 to Figure 2-8.

Figure 2-6: Early development of flow around an airfoil

Figure 2-7: Development after the starting vortex has been formed
18

Figure 2-8: Smooth flow over the airfoil
A key point in understanding the cascade model is to understand how circulation is generated.
Consider an arbitrary inviscid, incompressible flow over an airfoil starting from rest (hence no
initial aerodynamic forces). The circulation around path C1 at a given time t1 is defined as

 
    V  ds . As the fluid elements within C1 move downstream, at a later time, t2, they
1
C1

 

form another curve, C2, around which the circulation is    V  ds . These fluid elements

2
C2
have the same amount of energy and circulation at t1 and t2. A flow at rest has zero velocity
everywhere, and hence circulation around curve C1 is zero. Figure 2-9 shows path C2 at a later
time downstream of the flow field after a smooth steady flow over the airfoil has been
established. The fluid elements within C2 are the same as those in C1, however, because of the
influence of the flow field, the path C2 is different and is made of points abcda as shown in
Figure 2-9. Kelvin’s theorem gives the rate of change of circulation of a path from one location
to another. According to this theorem, the circulation,

, within C2 (which forms a closed path

around the airfoil and the starting vortex) is the same as those in C1. Since the airfoil was at rest
19

initially, i.e. a non-lifting surface with a circulation of zero around C1; then based on Kelvin’s
theorem,

. Now let C2 be divided into two parts: C4 a curve formed by points abda

(around the airfoil) and C3 a curve formed by tracing points bcdb (around the starting vortex).
The circulation around the starting vortex is
observation, the direction of

, and

is the circulation around the airfoil. By

is in the counterclockwise direction to the circulation around the

airfoil. The circulation in C4 is what we are interested in, and is the one that generates the lift.
However, since C3 and C4 are made from one curve, C2, the value of
from the equation:
conclude that

. Since it has been established that

can be simply found
is equal to zero, we can

, i.e. the circulation around the airfoil is equal and opposite to the that

around the starting vortex.
In summary, a circulation equal in magnitude and opposite in direction to the circulation around
the starting vortex has to be formed around the airfoil to ensure that the overall value of
circulation is maintained.

Figure 2-9: The curve C2 around the airfoil involving points abcda (Anderson Jr. 2001)
20

The first step taken in modeling using the cascade theory is that, for any turbine with N number
of blades, a reference blade is selected, and based on the free-stream velocity, RPM, and angular
position, the angle of attack, , and local relative velocity, W0, are calculated as:





sin  cos 
1 

 0  tan
  R Vd 

  cos  
/
 


  V V 


W0 Vd

V V

Eq. 2.15

2

  R Vd 

  cos    sin 2  cos 2 

/
 

  V V 


Eq. 2.16

Vd in the equations above represents the induced velocity at the turbine blade, ω is the rotational
speed of the turbine, R is the radius at the equatorial plane of the turbine, µ is the nondimensional turbine radius (r/R) (r is the radius from the turbine axis to the local blade element),
Ѳ is the angular position of the blade element, and ɸ is the blade angle normal to the equatorial
plane. The calculations are performed on one of the blades (the reference blade), after the turbine
has been unwrapped into the cascade configuration as shown in Figure 2-10.
The characteristics of blades 2 to N are assumed to be similar to those of the reference blade
(assume blade one). For the control surface around the reference blade (Figure 2-11), the inflow,
W1, and outflow, W2, velocities are the velocities calculated from W0 and
circulation). We use W1 and W2 to compute the tangential force:

21

(velocity due to



F   mW1 cos1  W2 cos 2 

Eq. 2.17

Figure 2-10: The unwrapped configuration of the turbine in the cascade theory
Equation 2.17 is simply an expression of the momentum balance which relates the force to the
change in velocity. In addition, the elemental mass flow rate balance is given as:



 m  tW1 sin 1s  tW2 sin  2s  tWxs .

“t” represents the blade spacing, i.e. the

ratio between the number of blades and the radius of the turbine (2πr/N) and δs represents the
incremental height of the blade element under consideration.
Circulation around the blade profile is defined as the integral of the velocity vector:

  wds
s

22

Eq. 2.18

where “w” represents the velocity vector. The cascade theory models the circulation around the
airfoil as the integral around the path abcd (Figure 2-12).

Figure 2-11: The control surface around the reference blade

Figure 2-12: The path around the airfoil used to compute the circulation in the cascade theory
(Bhimarasetty and N. 2010)

23

The net circulation around the path abcd with parallel lines bc and ad and ab and cd is:

  t (W1 cos 1  W2 cos  2 )

Eq. 2.19

where α1 and α2 are angles that W1 and W2 make with the horizontal axis, and t is the spacing
distance of the two streamlines along the longitudinal length of the profile. Magnitudes of W1
and W2 must be different since they represent the inlet and outlet flow velocities, and since the
turbine extracts kinetic energy from the flow, the outflow velocity will certainly be lower than
the inlet velocity. Hence the circulation around the path can be represented by (from Figure
2-12):
∫ ⃗⃗⃗

∫ ⃗⃗⃗

∫ ⃗⃗⃗

∫ ⃗⃗⃗

Eq. 2.20

Since the second and fourth terms in the RHS of Equation 2.20 cancel each other the circulation
around the blade is represented by:
∫ ⃗⃗⃗

∫ ⃗⃗⃗

Eq. 2.21

A comparison between Equations 2.17 and 2.19 indicated that the elemental tangential force can
be expressed as a function of the circulation as: Ft  Wxs .
Alternatively, the flow through the control surface generates a net normal and tangential force on
the airfoil which is found by applying the momentum conservation in the y direction, and are
equal to the difference in momentum of the flow at the inlet and outlet of the cascade. This is
represented by the following equations:

24





Ft  m1 W1 cos 1  m 2 W2 cos  2


Eq. 2.22



Fn  m1 W1 sin 1  m 2 W2 sin  2  t ( P  P2 )s
1

Eq. 2.23



where the elemental mass flow rate is given as:  m  tW1 sin 1s  tW2 sin  2s  tWxs
thus, δFt and δFn can be expressed as:

Ft  t[W1 sin 1W1 cos 1]s  t[W2 sin  2W2 cos  2 ]s

Eq. 2.24

Fn  t[W1 sin 1W1 sin 1]s  t[W2 sin  2W2 sin  2 ]s  t ( P  P2 )s Eq. 2.25
1
The pressure difference also generates a force which is normal to the control surface. If the
overall cascade loss is accounted for by a total pressure loss term,

, then for an

incompressible flow, the Bernoulli equation can be applied to obtain an expression for this
pressure difference between the cascade inlet and outlet:

P P 
1
2

2
W2  W 2
1  P
ov
2

Eq. 2.26

Therefore,
W cos 1  W2 cos  2 
P  P2   (W2 cos  2  W1 cos 1 )  1
1
  Pov
2



Eq. 2.27

By simplifying Equations 2.24 and 2.25 via Equations 2.19 and 2.27, the expressions for the
normal and tangential forces are obtained as:

Ft  Wxs
25

Eq. 2.28

and
W cos 1  W2 cos  2 
Fn     1
ov

s  tP s


Eq. 2.29



2

The formulations in Equations 2.28 and 2.29 indicates that the tangential force does not depend
on the pressure loss

, and the normal force is made of two parts, the part due to inviscid

flow, and the part due to pressure loss.
W cos 1  W2 cos  2 
Fn,id     1

s


2

Eq. 2.30



and

Fn,V  tP s
ov

Eq. 2.31

The lift force of a frictionless cascade flow, Lid, is computed by the following equation:

Lid 

2

2
2  W1 cos 1  W2 cos  2  
Ft  Fn,id    Wx  
 s
2


 
2



Eq. 2.32



The velocity squares under the square root are the components of W0. Hence the validity of the
Kutta-Joukowski theorem (

) for the blade in a cascade is proved.

W0 is considered to be the “free-stream” velocity with respect to the cascade and is made of the
fluid velocity, V∞, and Rω, the rotational velocity of the turbine. The flow velocities upstream of
the cascade,

(

)⁄
26

⁄ , and downstream of the cascade,

(

)⁄

⁄ , have been superimposed on the free-stream velocity

in order to calculate the inlet and outlet velocities of the cascade (W1 and W2). The ratio ⁄ ,
has the units of m/s, and hence is referred to as the circulation velocity

. Based on the vector

diagram of the flow velocities in Figure 2-11, the resulting force diagram is of the form shown in
Figure 2-13.

Figure 2-13: The blade forces on the airfoil

27

In the above force diagram, the force components

and

are normal to the direction of W0

and are computed from the following equations:

D  Fn,V sin  0  tP V sin  0s
0

Eq. 2.33

and

LV  Fn,V cos  0  tP V cos  0s
0
Evidently, the pressure loss,

, contributes to drag the,

, and lift force loss,

Eq. 2.34

. In an

accelerating cascade the pressure energy is transformed into kinetic energy of the rotating blades.
The total lift on the blades is given by the relation:

L  Lid  LV  W0s  tP0V cos 0s

Eq. 2.35

On the other hand, the lift force can be expressed as (Figure 2-13):

L 

By introducing

⁄

Ft
 D cot  0
sin  0

and substituting

Eq. 2.36

into Equation 2.36, one can

express the elemental lift force as:



L  W0
s
1   cot  0 

Eq. 2.37

Also recalling the definition of lift coefficient:

L 

1
2
C L W0 Cs
2
28

Eq. 2.38

From Equations 2.37 and 2.38, one can obtain an expression for the circulation and hence the
circulation velocity:

V
1
 NC  1 W0
1   cot  0 

CL 

V 8
 R   V

Eq. 2.39

The expressions for W1 and W2 and their corresponding angles α1 and α2 are found, from at
Figure 2-13,

W12
2
V



2
W2

2
Wx
2
V



2
V

W y  V 2


2
Wx
2
V



W y  V 2









Wx / V

 W y  V V 



 2  tan 1 

Eq. 2.41

2
V


Wx / V

 W y  V V 



1  tan 1 

Eq. 2.40

2
V





Eq. 2.42

Eq. 2.43

The elemental tangential and normal forces are represented as a function of W1, W2, α1 and α2
(similar to Equations 2.22, and 2.23)


Ft   mW1 sin 1  W2 cos  2 

Eq. 2.44



Fn   mW1 sin 1  W2 sin  2   t ( P  P2 )s
1
29

Eq. 2.45

Additionally, the normal force due to drag is:

Fn,v 
which is related to the pressure loss,

D
sin  0 ,

Eq. 2.46

, as (Equation 2.31):

Fn,v  tP V s
0

Eq. 2.47

Also recalling the definition of drag coefficient, the elemental drag force, δD can be represented
as (CD represents the drag coefficient):

1

2
D  C D W0 Cs
2

Eq. 2.48

Therefore the pressure loss term can be reexpressed as:
2
C D  NC  W0
1




2
2
4 sin  0  R
 V
V

P0V

Eq. 2.49

To complete the formulation, we need an expression for the wake velocity. For this, the
Bernoulli equation is used to find the absolute velocity at inlet (V∞) and outlet (wake, Vw) of the
cascade. Here the assumption is made that the pressure difference is the same as calculated
before. Also it is assumed that the static pressure is unchanged with elevation, and therefore
cancels each other. Consequently,

P P2 
1

2
2
V  Vw 
2



30

Eq. 2.50

Substituting Equations 2.27 and 2.49 into the Equation 2.50, the wake velocity Vw, can be
expressed as:
2
 W 2 W 2  1 NC
Vw

 C D W0
 2  1 
 1


 V 2 V 2  2  R
2
V

 sin  0 V
 
 

Eq. 2.51

The cascade model is an iterative model. The induced velocity Vd of the wind turbine is what is
iterated for and it is based on the parameters like: the solidity, wind speed, number of blades, the
blade spacing, RPM, the height and radius of the turbine, lift and drag coefficients, etc. From one
iteration to the next, Vd is updated using the following a relationship between the free-stream
velocity and wake velocity:

Vd
V

K
 Vw  i
 
V 
 

Eq. 2.52

where Ki is found from experimental data, and is a function of solidity,  :

K i  0.425  0.332

Eq. 2.53

The values of Ft , Fn , Vw, α0, etc are obtained at each angular position for all angles between Ѳ
= 0˚ and 360˚. Finally, the torque is computed from Ft by averaging over all angular or azimuthal
positions for all blades,

H 360

 N

0

 Ft dds

0
31

Eq. 2.54

The power and power coefficient (Cp) are then calculated as:

Power    RPM
Cp 

Power

Eq. 2.55

Eq. 2.56

3
0.5V HD

The cascade model is formulated in such a way that the entire turbine is broken up into two
segments (two half circles), the upstream half and the downstream half. The model is formulated
to be applied separately for the two segments of the turbine. Even though the formulations for
these two segments are similar, there are some differences.
First, in simulating the downstream half of the turbine, there are three possibilities for the inflow
velocity:
1. The free-stream velocity. This is not a reasonable assumption because it will overestimate the amount of energy available to generate power in this segment of the turbine.
There has to be a balance between the energy transferred to the wind turbine from the
wind to that removed from the flow. Thus, the wake velocity for any segment should be
less than that of inflow velocity.
2. Use the wake velocity calculated for the upstream as the inflow velocity for the
downstream segment. This is done in most of our simulations and is a logical choice. At
every segment the wake velocity is obtained from analysis of the blade at that upstream
segment and then used for the corresponding downstream segment.
3. Use the averaged upstream wake velocity of flow past over blades for the downstream
segment of the turbine. Although this approach may also lead to either an over- or
32

underestimation of results when compared to an actual turbine, it was employed to ensure
numerical stability when performing simulations at low free-stream velocities or high tip
speed ratios (

⁄

). Although the cascade model handles turbines at higher

TSRs better than BEM models, because it is still based on momentum principles, thus has
similar issues. One of these issues is the non-convergence of the iteration for Vd. In cases
where the inflow velocity is low, (e.g. V∞ = 3 m/s) at some point the lowest wake
velocity may reach to 1 m/s at a certain angle. If this velocity is applied as the inflow
velocity for the downstream segment, the iteration may not converge. By averaging the
upstream wake velocities, the overall input velocity would be higher and convergence
problems can be avoided.
The results (for efficiency, power and torque) that are obtained with the averaged wake velocity
are within 5% of those predicted by the original method. The basic elements of the cascade
model are shown in Figure 2-14, where the flowchart used in the simulation is shown.

33

Figure 2-14 : Flowchart of simulations based on the cascade model.

34

2.2 FAST for HAWT
As stated previously, FAST stands for: Fatigue, Aerodynamics, structures, and Turbulence; it is a
code developed by the National Renewable Energy Laboratory (NREL) for wind turbine
analysis. It can be used to model both two and three bladed, horizontal-axis wind turbines. FAST
models the wind turbine as a combination of rigid and flexible bodies. The aerodynamic module
or subroutine in FAST is AeroDyn, and it is used in calculating the aerodynamic loads. Some
key points about AeroDyn are:


AeroDyn performs aerodynamic calculations on an element-by-element basis along the
blade span for wind turbines using either: Generalized Dynamic Wake (GDW) model or
the Blade Element Momentum (BEM) method. The GDW method takes considerably less
time to run compared to the BEM method because the BEM method iterates for the
induction factors used in calculating the loads on the blade element.



The outputs from AeroDyn are: Velocity (x, y, and z components at the outmost blade
element), angle of attack for each element, the dynamic pressure at the center of each
element, CL, CD, CNorm, CTang, blade pitch, axial induction factor, the tangential
induction factor, and Reynolds Number.



The accuracy of AeroDyn/ FAST prediction is highly dependent on the accuracy of the
CL and CD values used. The values of CL and CD for every angle of attack are obtained
from experimental data.



AeroDyn uses a blade element method (BEM) similar to that described above for a
VAWT. This BEM method is however customized for horizontal axis turbines. Like the
35

BEM method for VAWTs, this technique also uses an iterative process in which two
variables: a and a’ (axial and tangential induction factors) are iterated.


When AeroDyn is using the BEM method it accounts for losses due to tip-loss, hub-loss,
and losses due to a > 0.4 by applying the appropriate loss factor. However, in the GDW
method these losses have already been factored into the calculations. An advantage of the
GDW is that the induced velocities in the rotor plane are determined from a set of firstorder differential equations, which are solved by a non-iterative technique; AeroDyn uses
the fourth-order Adams-Bashford-Moulton (Press et al. 1982) predictor-corrector method.



AeroDyn accepts either of two wind models: Hub-height wind profile, or a Full-Field
Turbulence model.

Some key variables/ parameters of AeroDyn/ FAST are:
2

1. Dynamic pressure (½ρV ) - Defined based on ρ, the air density, and V is the total
velocity at the element.
2. Reynolds Number - Defined based on the chord length and relative absolute local
velocity.
3. Element or Section Pitch - The angle between the element’s chord and the plane of
rotation. It is a combination of the element twist angle and the blade pitch angle.
4. Axial Induction Factor - The extraction of kinetic energy from the air that passes over
the wind turbine causes a reduction in the velocity. This factor is used to calculate the
Betz limit, and the BEM equations are derived as a function of this factor. The ratio of
36

this reduction of velocity at the rotor plane to that of the air velocity far away from the
wind turbine is defined as:

V
a 1 d
V
where

(

),

Eq. 2.57

is the induced velocity at the rotor. As the induction factor

increases from zero, the wind speed behind the rotor slows down.
5. Tangential Induction Factor - This similar to the axial induction factor. However, it is
used to represent the effects of the induced rotational velocity in the wake as a result of
the interaction of the free-stream with the rotating blade. After the wind passes through
the rotating blade there is some induced rotation in the flow in an opposite direction. The
tangential induction factor is the ratio of the, ω, induced rotational velocity in the wake to
the rotational velocity of the blade, Ω:

a' 




Eq. 2.58

From this we can see that the induced velocity of the flow across the blade is a’Ωr (r is
the rotor location).
A 2-D wind turbine can be represented as having two velocity components: axial velocity at the
turbine, Vd and rotational or tangential Vrot. These two velocities can be represented by a single
velocity Vrel which has an inflow angle ɸ:

Vd
Vrot
37

tan  

Eq. 2.59

The axial velocity is the velocity of the air at the blade, and is computed from the free-stream
and induced velocities as:

Vd  1  a V

Eq. 2.60

The rotational velocity is obtained from the turbine blade and wake rotation:

Vrot    r

Eq. 2.61

Vrot  1  a'r

Eq. 2.62

Equation 2.58 suggests that:

Hence the inflow angle is:

tan  

1  a V
1  a'r

Eq. 2.63

The BEM method used in FAST iterates for a and a’ until convergence is achieved.
The BEM method has some limitations. It assumes that the airflow field around the airfoil is
always in equilibrium and that the passing flow accelerates instantaneously to adjust to the
changes in the vorticity in the wake. It also assumes that the forces acting on the blade element
are essentially two-dimensional. It does not model the blades experiencing large deflections out
of the rotor plane. One of the major limitations of the original BEM theory is that there is no
influence of vortices shed from the blade tips on the induced velocity. Lastly, when the induction
factor is greater than about 0.4, the basic theory becomes invalid. This occurs with turbines
operating at high tip speed ratios.

38

The GDW method also has its limitations. As with most wake models, the generalized dynamic
wake was developed for lightly loaded rotors and assumes that the induced velocities are small
relative to the mean flow. This assumption leads to instability of the method at low wind speeds
when the turbulent wake conditions are approached.

39

3. Chapter 3: Vertical Axis Wind Turbines
3.1 Validation
To validate the cascade model for VAWT, two cases were simulated, and their results compared
against those presented by other researchers. First, the results from this current study are
compared with those generated from simulations done by the group at the Institut de Recherche
d’Hydro-Quebec (IREQ) led by Ion Paraschivoiu. Those simulations are based on the doublemultiple streamtube model and were conducted with the software called CARDAA (I.
Paraschivoiu 1981).13 Figure 3-1 and Figure 3-2 summarize the results of this comparison. They
show power versus V∞, and Cp versus TSR for a 6 m straight blade VAWT. The parameters of
this turbine are listed in Table 3-1 below.
There is good correlation between the results, especially at low wind speeds or low TRSs. It was
also observed that the Reynolds number (Re) is important in these simulations because it
determines the lift and drag coefficients used, and hence the amount of torque generated by the
turbine blades. This study used tabular interpolation of experimental values of CL and CD as a
function of angle of attack (α) and Re to calculate the different variables.
The Reynolds number in this study is based on the local flow conditions at the blade, i.e. the
relative velocity based on the rotational speed and the free-stream velocity and is defined as:

Re 

CW



where W is the relative velocity and is defined as:

40

Eq. 3.1

2

  R V 

  cos   sin 2  cos 2 
W  V  
/


  V V 

,

Eq. 3.2

and C represents the chord length of the airfoil and ν is the kinematic viscosity.
Table 3-1: The geometry and parameters of the simulated VAWT
Parameter

Value

Rotor diameter

6.0 m

Rotor Height

6.0 m

Blade length

6.0 m

Blade chord length

0.2 m
NACA 0015

Airfoil type

2

Number of blades

3.0 m

Rotor ground clearance

125

Simulation RPM

Since Re is based on the local conditions, for W in the formulation above, V is the induced
velocity calculated at every azimuth angle.

41

The code by Paraschivoiu has been validated against experimental data (mainly for Darrieus
shaped turbines, see Figure 3-3) and therefore its results can be considered suitable for
comparison.
The next comparison was done with the results presented by Migliore et. al., who conducted
experiments on a 3 m straight blade turbine at West Virginia University (Migliore, W.P. and J.B.
1980).14 Figure 3-4 shows Cp versus TSR for this turbine. The author states that, this experiment
was carried out based on a Reynolds number of 131,000; however it is unclear how a single fixed
value was used to characterize the TSRs experienced by the turbine. According to the author, Re
for this experiment is defined as Re = RωC/ν. It is also stated that Re given is “for Re at which
max Cp occurred.” This and other geometric information are the only available information
about these tests. From this information, the RPM was calculated and set constant and the wind
velocity was calculated for each TSR. However, since only the Re at maximum Cp was
presented for the entire test case, without having the instantaneous V∞ or ω, at best any
simulation to match these results will be dependent on selecting the right Reynolds number
which characterizes the average flow conditions experienced by this turbine. Figure 3-4 shows
that by using a Reynolds number of 69,000, the results become very close.
The code developed here based on cascade theory offers an efficient way of studying the effects
of various parameters on the power output of VAWTs as well as their efficiencies. A turbine
with input parameters as in Table 3-1 is analyzed in the following sections. The goal is to gain a
better understanding of the characteristics of a straight bladed VAWT and to give insight as to
how they can be optimized.

42

12000

Power (Watts)

10000
8000
(DMST) model

6000
4000

Cascade Re: Rel.
vel.

2000
0
0

10
20
Vinf (m/s)

30

Figure 3-1: Power vs. V∞, obtained from the CARDAA (DSTM model) and the cascade model.
“For interpretation of the references to color in this and all other figures, the reader is referred to
the electronic version of this thesis.”

0.5
0.4
0.3
Cp

(DMST) model

0.2
Cascade Re: Rel.
vel.

0.1
0
0

5
λ: Tip Speed Ratio

10

Figure 3-2: Cp vs. TSR, obtained by the CARDAA (DSTM) and cascade models.

43

60

Power (kW)

50
40
30
CARDAAV

20
10
0
0

5

10
TSR

15

20

Figure 3-3: The results from the CARDAA code compared with experimental results of a 17 m

Cp

SANDIA Darrieus VAWT

0.5
0.45
0.4
0.35
0.3
0.25
0.2
0.15
0.1
0.05
0

Re Experimental
Data
Re - 69000:
Cascade
Model
Re - 131000:
Cascade
Model
0

2

4
6
λ: Tip Speed Ratio

8

Figure 3-4: Cp vs. Tip speed ratio for a 3 m expereimental turbine

44

3.2 Results and Discussion
In this section the effects of various parameters on the VAWT performance are discussed.
3.2.1 Tip Speed Ratio
The tip speed ratio (TSR), λ, is a nondimensional quantity which gives the ratio of rotational
speed to the free-stream velocity V∞:



R ,
V

Eq. 3.3

where R is the radius of the turbine, ω is the turbine’s rotational speed, in radians/second, and
V∞ is the free-stream velocity of the wind. In a Darrieus turbine with curved blades, the radius is
taken at the turbine equator (the section along the span with the largest radius), λeq.

The coefficient of power, Cp is simply calculated by dividing the power extracted by the turbine
in the wind’s path with the total power available in the wind.

Cp 

P
Turbine
1
3
V HD
2

Eq. 3.4

Plots of Cp versus λ are good indicators of how a particular turbine performs under various
inflow and rotational conditions and are shown in the following sections. This becomes
particularly important in experiments because the rotational velocity and the wind velocity are
constantly changing, and since both quantities are inherently tied to each other (i.e. a change in
the wind velocity may change the RPM if no load is applied to the system), the tip speed ratio
provides an effective way of summarizing their effects on the power output of the turbine.
45

3.2.2 Effects of Reynolds number
The importance of selecting the right Re cannot be over-emphasized. In this section the global
effects of Re will be studied based on how it affects power generation for the turbine. The classic
definition of Re takes into account the flows density, the velocity of the flow, a length scale
which is descriptive of the surface over which the fluid flows, and the fluid’s viscosity. The
characteristic length takes on various forms depending on the shape of the body, and what
sections interacts with the fluid, in the case of a wind turbine this characteristic length can be the
chord length. The wind turbine community has several methods of calculating the Re:

1. Some researchers have used the Reynolds number based on wind speed, Rew, defined as:

V C
Re w  



Eq. 3.5

This is a common definition used for flows over an airfoil. This definition was used in
the Sandia experiment to generate lift and drag coefficients data, and this data was used
in our study. For VAWT simulations, characterizing the flow based only on the freestream velocity, viscosity, and the chord length may not provide an adequate description
of the flow because it does not consider the rotational velocity.
2. Another definition of Reynolds number specific to wind turbines, known as the turbine
speed Reynolds number, Ret, uses the chord length and rotational velocity:

Re t 

46

RC


Eq. 3.6

3. The definition based on the local relative velocity and chord length is, ReV
:
relative
ReVrelative 

WC



Eq. 3.7

W is the relative velocity defined as:
2

  R V 

2
2
W  V  
 V / V   cos    sin  cos 

   


Eq. 3.8

and V is the induced relative velocity calculated at every azimuth angle.
An argument can be presented that any of these definitions is in-line with how the airfoil
experiments are conducted, or the turbine operated. The results obtained with different Re are
presented next.
Figure 3-5 shows Cp versus TSR for three Re values. In these simulations, the RPM was fixed at
three values of 105, 115, and 125; and the wind velocities were varied between 4.5 to 26 m/s (10
to 60 mph); the RPM is changed so that Ret = 4.5E+5, 4.9E+5, and 5.4E+5. This method was
also used by scientists at the Sandia National Laboratories when reporting experimental effects
of Re on a 17 m Darrieus turbine (Worstell 1978). Simulation results presented in Figure 3-5
show that for the chosen values of Re, there is not much difference in Cp at very low TSRs.
However, at TSR

2.0 the efficiency of the turbine increases with Ret. The results in Figure 3-5

also suggest that for a given turbine, at a certain inflow velocity, the turbine can be adjusted for
the peak performance by varying the RPM to attain the optimum TSR (or vice versa if V∞ is
fixed).
47

As has been emphasized earlier, the accuracy of any empirical model is dependent on the CL and
CD data used. Although the CL and CD data were from experimental testing of the actual NACA
0015 airfoil, they represent static test results and do not include the rotational effects experienced
*

by the turbine (Migliore, W.P. and J.B. 1980),15nor the effects of dynamic stall resulting from
the constantly changing angle of attack. These factors drastically change the “effective” angle of
attack and hence the lift and drag coefficients which ultimately affect the torque generated at
each azimuth angle.
The three definitions of Re indeed yield very different results. We use different methods in
which the flow is simulated, either with Re averaged over all TSRs, found at each angular
position, or use Reynolds number calculated at every TSR: (1) Ret, (2) Rew averaged over all
TSR, (3) Rew changed at every TSR, (4) Re based on the local flow velocities, (5) Constant Rew
using the V∞ value which gives maximum Cp, (6) Re based on the absolute relative velocity:

ReV

absolute



2
C R 2  V .


For sake of comparison, the case conducted by Paraschivoiu will be taken as a reference case.

*

In the current analysis the flow is considered as rectilinear, however, according to Migliore et
el, from experimental work done on a straight bladed VAWT, the curvilinear effects on the
turbine blade significantly affect the behavior of the turbine when compared to one immersed in
a rectilinear flow. Their experiments show that the turbine blade relative inflow velocity and
angle of attack are unique everywhere on the chord.
48

0.45
0.4
0.35
0.3
Cp

0.25
0.2

Ret: 4.5E+05 (105 RPM)

0.15

Ret: 4.9E+05 (115 RPM)

0.1

Ret: 5.3E+05 (125 RPM)

0.05
0
-0.05
0

2

4

6

8

10

TSR

Figure 3-5: Power coefficient Cp variations with Re for the test case simulated using the cascade
model
The Reynolds number used in this study is based on a linear interpolation method which
calculates CL and CD from tables. The range of Re available from experiment is 10,000 to
2,000,000. To show the significant differences derived from using the property tables of different
Re, Figure 3-6 and Figure 3-7 are plotted which show CL and CD versus α for different Re
obtained directly from experimental data (and tabular interpolation). Figure 3-8 and Figure 3-9
show Cp for different methods of calculating Re; the RPM was fixed at 125, and V∞ varied
between 4.5 to 26 m/s. Figure 3-8 indicates that the simulations based on

and

adequately predict the performance of the turbine at low to moderate TSRs,
however tends to over-predict it at high TSRs. While from low to mid TSRs, results based on
49

Rew averaged under-predicts the turbine efficiency, and for mid to high TSRs the performance is
over-predicted. The case using Rew under-predicts Cp in for all TSRs. It is interesting to note
that Cp based on Ret at the largest TSR simulated is seven times larger than that based on Rew.

Figure 3-10 shows CL versus α for a TSR of 4.4, this TSR gives the maximum Cp of all plots in
Figure 3-8. The results in this figure are all based on the same flow conditions; the only
difference is how the Reynolds number is calculated. As can be seen in Figure 3-10, irrespective
of the simulation Re used, for the same flow conditions and turbine configuration, the computed
flow parameters such as the: relative velocity, α, azimuthal torques calculated, etc., are the same
before stall develops (see Figure 3-11 for torque versus Ѳ for various Re). In Figure 3-12, at Ѳ
angles of 0˚, 180˚, and 360˚ the flow is not stalled. This characteristic of momentum models was
also observed in the results obtained by the National Renewable Energy Laboratory (NREL),
“When used with accurate 2-D airfoil wind-tunnel data at angles of attack below stall, BEM
[Blade-Element Momentum] provides reasonably accurate performance prediction (Tangler and
Kocurek 2005).”16

50

CL

1.4
1.2
1
0.8
0.6
0.4
0.2
0
-0.2
-0.4

Re: 10000
Re: 100000
Re: 120000
Re: 150000

Re: 2000000
0

20
40
Alpha (degrees)

60

Figure 3-6: CL vs. angle of attack for different Reynolds numbers

1.4
1.2

CD

1

Re: 10000

0.8

Re: 100000

0.6

Re: 120000

0.4

Re: 150000

0.2

Re: 2000000

0
0

20
40
Alpha (degrees)

60

Figure 3-7: CD vs. angle of attack for different Reynolds numbers

51

0.6

0.5
(DMST) model

Cascade Re: Ret =
537,943

0.3

Cascade ReW: 189,913
(Averaged)

0.2

Cascade Re: Rew
(Changed w/ TSR)

0.1

Cascade Re: Local relative
vel.

Cp

0.4

Cascade Re: - Absolute
relative vel.

0
0
-0.1

2

4

6

8

10

TSR

Figure 3-8: Cp vs. TSR for Re calculated based on six different methods

52

12000
10000

Power (Watts)

8000

(DMST) model

6000

Cascade Re: Ret =
537,943

4000

Cascade ReW: 189,913
(Averaged)

2000

Cascade Re: Rew
(Changed w/ TSR)
Cascade Re: Local relative
vel.

0

Cascade Re: Absolute
relative vel.

-2000
-4000
0

10

20

30

Vinf (m/s)

Figure 3-9: Power vs. wind speed for Re calculated based on six different methods

53

1.2
1

Ret: 537,943

CL

0.8
0.6

ReW: 189,913
(Averaged)

0.4
0.2

Rew (Changed w/
TSR): 123,288

0
0

5
10
Alpha (degrees)

15

Figure 3-10: CL vs. alpha at a TSR of 4.4. The RPM simulated was fixed at 125. This figure is
plotted after convergence has been achieved for this TSR. This figure shows only the lift

Torque (Nm)

coefficients and angle of attacks generated in the upstream of the turbine

700
600
500
400
300
200
100
0
-100

Ret: 537,943

ReW: 189,913
(Averaged)

0

100
200
300
Theta (degrees)

400

Rew (Changed w/
TSR): 123,288

Figure 3-11: Torque vs. Ѳ at a TSR of 4.4. This figure is plotted after convergence has been
achieved for this TSR. The RPM simulated was fixed at 125. This figure shows the torque
characteristics for both the up and downstream of the turbine
54

Alpha (degrees)

15
Ret: 537,943

10
5
0

ReW: 189,913
(Averaged)

-5
-10
-15
0

100
200
300
Theta (degrees)

400

Rew (Changed w/
TSR): 123,288

Figure 3-12: Angle of attack vs. Ѳ at a TSR of 4.4. This figure is plotted after convergence has
been achieved for this TSR. The RPM simulated was fixed at 125. This figure shows the angles
of attack for both the up and downstream of the turbine

Corrections to CL and CD Data
Several corrections that have been proposed to account for deficiencies of using static property
data derived from experiments in simulations. The effects of two of these corrections will be
observed in this section.
The first correction corrects the experimental data to account for the effects of a finite wing. In
experiments to obtain CL and CD, a wind tunnel is used with the airfoil’s span length usually the
width of the wind tunnel. This wing section is classified as a semi-infinite wing. The relevance of
this classification is to make the approximation that the effects of the flow in the z direction are
negligible. In addition, this classification makes the effects of the wing-tip vorticity negligible
(Anderson Jr. 2001).17 In a finite wing, the tip vorticity induces some drag, known as the induced
drag, CD,i, and with this there is also an induced angle of attack, αi. Modifications to the data are
55

made based on the theory of Elliptical Lift Distribution which was developed by Ludwig Prandtl
and is part of his classic lifting-line theory. For the unstalled conditions, the equations that
modify α and the CD are:

C

 New     i    L
AR

C D, New  C D  C D, i  C D 

2
CL

AR

Eq. 3.9

Eq. 3.10

where AR is the aspect ratio, AR = H/C, H is the height, and C is the chord length.

Due to the nonlinear nature of the CL and CD curves at stall conditions, the Elliptical Lift
Distribution theory becomes inadequate and should not be used to modify the property data in
this region. For post-stall data, a method proposed by Viterna and Corrigan provides a
convenient global approach to relate the post-stall CL and CD to the overall blade geometry and
flow conditions. However, although this method has been used for predicting post-stall airfoil
characteristics, an incomplete understanding of the stall process existed at the time of its
development (Tangler and Kocurek 2005).18The Viterna equations were developed for αstall up
to 90˚:
if AR < 50, CD, max  1.11  0.18 AR and if AR > 50, C D, max  2.01 ,

CD  B1 sin 2   B 2 cos 
where B1 and B2 are:
56

Eq. 3.11

B1  CD, max

Eq. 3.12

C D, stall  C D, max sin 2  stall
B2 
cos  stall

Eq. 3.13

cos 2 
CL  A sin 2  A2
1
sin 

Eq. 3.14

where A1 and A2 are:

B
A  1
1

Eq. 3.15

2

A2  C L, stall  C D, max sin  stall cos  stall 

sin  stall

Eq. 3.16

cos  stall
2

It should be noted that for the cases simulated so far none of these corrections have been
implemented to modify the property tables.
To study the effects of these modifications on the data, CL and CD versus α are plotted in Figure
3-13 and Figure 3-14 applying the different corrections. These figures were plotted for Ret. For
the plots without correction and that with Prandtl’s correction there is only a slight difference;
the average difference between their α values is 0.0074, and the difference in values of CD is
0.0060. On the hand, when the Viterna correction is applied, there is a significant reduction in
the amount of stalling that occurs as can be observed in Figure 3-13 and Figure 3-14. It also
reduces the lift coefficient at higher (α > 24˚) angles of attack. The effect of these corrections on
57

the turbine performance is shown in Figure 3-15 which shows power versus V∞. As should be
expected, there is only a slight difference in the results for the case using Prandtl’s correction and
the case with no corrections. However, when the Viterna correction is applied, the power is overpredicted for high wind velocities.

1.2
1

CL

0.8

No correction

0.6
Prandtl correction

0.4

Viterna
Correction

0.2
0
0

50
Alpha (degrees)

100

Figure 3-13: CL vs. angle of attack for the TSR of 4.4. The RPM simulated was fixed at 125

58

2
1.5
CD

No correction
1
Prandtl correction
0.5

Viterna
Correction

0
0

50
Alpha (degrees)

100

Figure 3-14: CD vs. angle of attack for the TSR of 4.4. The RPM simulated was fixed at 125

16000

Power (Watts)

14000
12000
No Correction:
ReT

10000
8000

Prandtl
Correction: ReT

6000
4000

Viterna
Correction: ReT

2000
0
0

10
20
Vinf (m/s)

30

Figure 3-15: Power vs. wind speed for the TSR of 4.4. The RPM simulated was fixed at 125

59

3.2.3 Torque Characteristics of VAWT
Figure 3-16 shows four points on the Cp versus TSR curve for the turbine in Table 3-1, and
Figure 3-17, Figure 3-18 and Figure 3-19 present the torque characteristics of these points (for
brevity only the upstream is shown, the downstream section has similar behavior). The CL and
CD coefficients are also plotted versus azimuthal angles to show how they affect the torque
produced. The result of the last marked point was not included due to convergence issues for
such a heavily loaded turbine and high TSR.
Comparing the torque plots for different TSRs, it can easily be seen why a TSR of 1.65 has the
poorest efficiency: the very low to negative torque values at most angular positions which is
caused by high amounts of drag generates low power. We can observe in this figure that the
torque peaks coincides with the low drag coefficient and with relatively high lift coefficients.
Around azimuthal angle of 50˚ stall occurs, resulting in a rapid drop in lift coefficient and a
significant increase in drag. Over the remainder of the upstream section, the turbine exhibits a
poor lift to drag ratio and aerodynamic efficiency. The plots for TSR of 3.09 follow a similar
trend, even though that the aerodynamic efficiencies are better. At TSR of 5.63, the aerodynamic
efficiency is superior with almost zero drag and high lift coefficients over the entire upstream
section.

60

0.5
0.45
0.4
0.35

Cp

0.3
0.25
0.2
0.15
0.1
0.05
0
0

2

4

6
TSR

8

10

1.46939
1.64898
1.73061
1.9102
2.05714
2.20408
2.35102
2.54694
2.61224
2.7102
2.80816
2.92245
2.98776
3.08571
3.18367
3.29796
3.41224
3.55918
3.65714
3.73878
3.91837
4.08163
4.35918
4.60408
4.88163
5.09388
5.38776
5.63265
5.86122
6.07347
6.25306
6.51429
7.02041
7.34694
7.60816
7.86939
8.11429
8.39184
8.68571

Figure 3-16: Cp vs. TSR for the turbine in Table 1, simulated with Re = Ret.

61

Torque (kNm), CL and CD

1.6
1.4
1.2
1
0.8
0.6
0.4
0.2
0
-0.2
-0.4
-0.6

Torque (kNm)
CL
CD
0

50

100

150

200

Theta

Figure 3-17: Torque vs. Ѳ for the TSR of 1.65 for the turbine with configuration in Table 3-1,
simulated with Ret. CL and CD coefficients are also plotted. For brevity only the upstream

Torque (kNm), CL and CD

section of the turbine has been shown

1.2
1
0.8
0.6
0.4
0.2
0
-0.2
-0.4

Torque (kNm)
CL
CD
0

50

100
Theta

150

200

Figure 3-18: Torque vs. Ѳ for the TSR of 3.09 for the turbine with configuration in Table 3-1,
simulated with Ret. CL and CD coefficients are also plotted. For brevity only the upstream
section of the turbine has been shown

62

Torque (kNm), CL and CD

1
0.8
0.6
Torque (kNm)
0.4

CL

0.2

CD

0
-0.2

0

50

100

150

200

Theta

Figure 3-19: Torque vs. Ѳ for a TSR of 5.63 for the turbine with configuration in Table 3-1,
simulated with Ret. CL and CD coefficients are also plotted. For brevity only the upstream
section of the turbine has been shown
The self-starting behavior of VAWTs is strongly related to the torque which is generated at each
azimuthal angle in the upstream and downstream sections. It is dependent on turbine parameters
such as: the airfoil shape, number of blades, solidity, the location of the blade, the blade
incidence angle in relation to the wind velocity, and the pitch angle with respect to the wind
direction. The term self-starting can has been defined in a number of ways (Dominy, et al. n.d.):


Ebert and Wood define the starting process as being complete when significant power
extraction can be derived from the turbine (Ebert and Wood 1997).19



Kirke says a turbine has self-started if it can accelerate from rest to the point where it
starts to produce a useful output (Kirke 1998).20



Lunt adopted a more specific definition which states that a machine has started if it has
accelerated from rest to conditions where the blades operate at a steady rotational speed
63

that exceeds the wind speed, i.e. the tip speed ratio, TSR 1.0 (Lunt 2005).21
As has been mentioned previously, one of the draw backs of a VAWT is its inability to self-start
even in the presence of high wind speeds (Yao and Habbtamu n.d.).22 Self-starting is a transient
process, and therefore requires a transient analysis of the turbine. The model used in this research
is limited in its ability to show the self-starting problems of a VAWT because it models the
turbine in steady-state conditions, i.e. the effects of the initial net negative torques that are
generated have already been overcome. Figure 3-20 shows the rotational speeds versus time for a
VAWT starting from rest. This figure shows the transient nature of a turbine which starts from
rest, and accelerates to a steady rotational speed. This tells us that before reaching steady-state
the turbine initially goes through several values of TSR, some of which can have very poor
torque characteristics. These unfavorable conditions can send the turbine into several states: the
turbine blades can experience flow reversal, or it can cause the turbine to oscillate back and forth
without actually rotating (in this case the turbine “rocks” back and forth) (Dominy, et al. n.d.).23
This last state was observed in an actual preliminary prototype test conducted here at Michigan
State University. At low TSRs, considerably low (often negative) azimuthal torque values are
generated. This can be observed in Figure 3-21 for the TSR of 1.65 (and wind speed of 23.8 m/s
and an RPM of 125). To understand this behavior, the torque plots are studied closely and the
turbine is divided into two sections considered as the locations of blade one and two of a two
bladed turbine. For example when blade one is located at 1˚ (upstream), blade two will be 180˚
apart, at 181˚ (downstream). For a TSR of 1.65, at angles of 1˚ and 181˚, the torque is negative,
and the turbine starting from rest will accelerate (see Figure 3-21), albeit in the "wrong"
direction. Let it be assumed it was designed to rotate in the anticlockwise direction. The torque
increases on both blades up to a point when both become positive. When this happens we can
64

assume that the turbine decelerates from its initial rotational speed. If this condition persists, the
angular acceleration in the opposite direction will cause the turbine to rotate in the “correct”
clockwise direction. However, blade two soon becomes negative around 213˚, while blade one
continues to generate positive torques which peaks at about 33˚. Due to high torques generated
by blade one, even with the negative torques on blade two, the turbine may still rotate in the
positive clockwise direction.
As blade one approaches 134˚ (a point of negative torque after it had peaked at about 32˚), blade
two will be at 315˚ (the peak of blade two’s generated torque). A case now exists where the
initial rotational speed starts to decelerate because of a net negative torque. The turbine slows
down, and with persistence of this condition it can start to rotate in the opposite direction. When
this process is repeated again and again, the turbine experiences the "rocking" phenomenon.
To further illustrate this point, the torques generated on blades one and two are plotted in Figure
3-22 versus Ѳ ranging from 1˚ to 180˚. This figure shows that from rest both blades generate
negative torques which increases to become positive. Then at about 14˚, blade one continues to
generate positive torque, but blade two starts generating negative torques, although this torque is
not so large. At this angle, there is a net positive torque (Figure 3-22) and the turbine will
continue to accelerate. At about 32˚, blade one’s torque begins to drop rapidly, and because
negative torque is also being generated by blade two, the turbine will decelerate. This
deceleration continues up to around 50˚, where negative torque is being generated on both
blades. Around 130˚, blade two starts generating positive torque and blade one is still generating
negative torque, which will cause the turbine to start decelerating again. It keeps on decelerating
until the torque on blade two is large enough that the net torque generated from both blades is
positive. If flow conditions persist and TSR stays low, this process is repeated and hence a
65

rocking back and forth of the turbine. Start-up is a lesser issue at mid-range TSRs because both
blades one and two are always generating positive or negative torques (see Figure 3-23). This
explanation assumes this plot is an instantaneous snapshot of what happens during the transient
process.
This “rocking” phenomenon was observed in the fluids laboratory at Michigan State University
while conducting initial wind tunnel tests on a small straight bladed prototype turbine with two
blades. During these tests, the turbine started from very low RPM and TSR for a wind speed of
about 7 m/s. With a small push, the turbine started to rotate, but then slowed down, and rotated
very slightly in the opposite direction. However, when half of the wind tunnel was blocked the
turbine started to rotate consistently, overcoming its initial startup problem. It then accelerated
until it reached steady-state conditions. When half of the tunnel was blocked the flow was
blocked from seeing the second blade. This reduced the effect of negative torque generated on
the second blade.
600
500

RPM

400
300

Experiment
(Chau 2002)

200
100

0
0

100

200

300

400

Time (s)

Figure 3-20: RPM vs. time obtained from experiments of a VAWT (Dominy, et al. n.d.)

66

Torque (Nm)

1600
1400
1200
1000
800
600
400
200
0
-200
-400
-600

Torque
(kNm)

0

100

200
Theta

300

400

Figure 3-21: Torque characteristic of a VAWT for the TSR of 1.65

2000
Torque (Nm)

1500
Blade 1

1000
500

Blade 2

0

Net torque
(fixed)

-500
-1000
0

100
Theta

200

Figure 3-22: Torque characteristic of a VAWT for the TSR of 1.65. Torques on both blades have
been plotted on a Ѳ axis going from 1˚ to 180˚

67

Torque (Nm)

700
600
500
400
300
200
100
0
-100

Blade 1
Blade 2
Net torque
(fixed)
0

50

100
Theta

150

200

Figure 3-23: Torque characteristic of a VAWT for the TSR of 5.63. Torques on both blades have
been plotted on a Ѳ axis going from 1˚ to 180˚
3.2.4 Effects of Airfoil Shape
This section investigates the effects of various airfoil shapes on turbine performance. Four
different airfoils are considered, three of them are symmetric airfoils, and one asymmetric airfoil.
Figure 3-24 and Figure 3-25 show the geometry of the four different airfoils studied labeled as:
NACA 0012, NACA 0015, NACA 0021, and S809. The data for the symmetric airfoil comes
from experiments at Sandia National Laboratories. The asymmetric airfoil studied is one that has
been optimized for HAWT at the National Renewable Energy Laboratory (NREL).
Figure 3-26 and Figure 3-27 show the performance of VAWTs using different airfoils. It shows
that with an increase in the airfoil thickness there is an increase in the efficiency of the system at
lower TSRs; however, for mid to high TSRs, the performance of the turbine is better for thinner
airfoils. The performance of the asymmetric airfoil overall was inferior, even if a higher
simulation Re was used (the experimental data available was only for a Re of 750,000 as
68

opposed to 537,943 used with the symmetric turbines). The maximum thickness of asymmetric
airfoil, S809, is comparable to that of the symmetric 0021 airfoil, yet the performance of the
asymmetric airfoil is lower than that of other airfoils.

In Figure 3-28 the raw CL and CD data for the 0012 and 0021 airfoils are shown. Figure 3-29
shows the variation of angle of attack with Ѳ for TSRs of 2.71 and 5.63 in the upstream. These
plots are presented to explain why turbines with different airfoils can have significantly different
performance. In Figure 3-29, it is observed that there is only a slight difference between the α
values for the same TSRs. The CL/CD ratio in Figure 3-30 help to explain why at low TSRs,
turbines with the 0012 airfoil perform better than that with 0021, and vice-versa at mid to high
TSRs. Figure 3-30 and Figure 3-29 indicate that when stall angles are approached, the
aerodynamic efficiency of the turbine drops significantly. At TSR of 2.71, the induced flow over
the airfoil is in stall range, explaining why the performance of the turbine at this TSR is poor
compared to that at TSR of 5.63. The results for TSRs of 2.71 and 5.63 over the mid-range of Ѳ
show that there is a direct correlation between the Cp and the overall aerodynamic efficiency.
The consideration to use a thinner or thicker airfoil may be controlled by, and based on the blade
weight. This also determines the base support required to holdup the entire structure. From a
manufacturing stand point, it is easier to produce a symmetric airfoil instead of a non-symmetric
one.

69

y/c

0.15
0.1
0.05
0
-0.05
-0.1
-0.15

NACA 0012
NACA 0015
NACA 0021
0

0.2

0.4

0.6
x/c

0.8

1

1.2

Figure 3-24: The shapes of symmetric airfoils studied – NACA 0012, NACA, NACA 0015, and

y/c

NACA 0021 (Malcolm's Human Powered Aircraft Site n.d.)24

0.15
0.1
0.05
0
-0.05
-0.1
-0.15

NREL's S809

0

0.2

0.4

0.6
x/c

0.8

1

1.2

Figure 3-25: The shape of the asymmetric airfoil studied – NREL’s S809 (UIUC Airfoil Data
Site n.d.)25

70

With Prandtl corrections
0.5
NACA 0012

0.4
Cp

0.3
NACA 0015

0.2
0.1

NACA 0021

0
-0.1
0

2

4

6

8

10

S809 (Re =
750,000)

TSR

Figure 3-26: Cp vs. TSR for the four different airfoils studied

With Prandtl corrections

Power (Watts)

15000
NACA 0012

10000
5000

NACA 0015

0
NACA 0021

-5000
-10000
0

5

10 15 20
Vinf (m/s)

25

30

S809 (Re =
750,000)

Figure 3-27: Power vs. wind speed V∞ for the four airfoils studied

71

CL and CD

1.6
1.4
1.2
1
0.8
0.6
0.4
0.2
0

CL - 0012
CD - 0012
CL - 0021
CD - 0021
0

20

40
60
Alpha (degree)

80

Figure 3-28: CL and CD vs. Ѳ for the NACA 0012 and NACA 0021 airfoils

Alpha (degrees)

25
20
15

0012 TSR = 2.71

10

0021 TSR = 2.71
0012 TSR = 5.63

5

0021 TSR = 5.63

0
0

50
100
150
Theta (degrees)

200

Figure 3-29: The angle of attack vs. Ѳ for the NACA 0012 and NACA 0021 airfoils at two
different TSRs. The plots are for the upstream section only

72

CL/CD

45
40
35
30
25
20
15
10
5
0

0012 TSR = 2.71
0021 TSR = 2.71
0012 TSR = 5.63
0021 TSR = 5.63
0

50
100
150
Theta (degrees)

200

Figure 3-30: CL/CD vs. Ѳ for the NACA 0012 and NACA 0021 airfoils at two different TSRs
3.2.5 Effects of Solidity
Solidity is the ratio of the total turbine planform area to the total swept area. This parameter has a
significant effect on the turbine’s performance. Solidity, σ, of a VAWT can be defined as:



NCH NC

RH
R

Eq. 3.17

Designers commonly change the solidity of a turbine by changing the number of blades. High
solidity turbines are used for high torque applications. These turbines generally rotate more
slowly. Figure 3-32 illustrates that a turbine with high solidity has a high initial starting torque at
low TSR. However, in high speed wind applications turbines which operate at the lowest
possible torques are more desirable. The reason for this is easier startup and cheaper gearboxes.
Figure 3-31 shows the Cp versus TSR for a turbine simulated with a chord length of 0.10 m, and
different solidities. The solidity was changed by varying the number of blades in the turbine.
There is a key observation that can be made from Figure 3-31 (Burton, et al. 2001): a low
73

solidity turbine produces a broad, flat performance curve, i.e. Cp will change very little over a
wide range of TSRs; however, the maximum Cp is lower because the drag losses are higher
(drag losses are roughly proportional to the cube of the tip speed ratio). In turn, turbines with a
high solidity produce a narrower performance curve with a sharper peak, hence making the
turbine very sensitive to changes in TSR.
A heavily loaded turbine is said to have a high solidity and for the same value of solidity, a
heavily loaded turbine is considered to have a high TSR. As mentioned earlier, the performance
of momentum models for heavily loaded turbines is poor. This affects the code’s ability to
produce a converged solution for higher values of solidity and/ or TSR.

0.6
0.5

Cp

0.4

No of Blades: #2
(Solidity = 0.067)

0.3

No of Blades: #3
(Solidity = 0.100)

0.2

No of Blades: #4
(Solidity = 0.133)

0.1
0
0

2

4

6

8

10

TSR

Figure 3-31: Cp vs. TSR for turbines with different solidity

74

700

Torque (Nm)

600
500

No of Blades: #2
(Solidity = 0.067)

400
300

No of Blades: #3
(Solidity = 0.100)

200

No of Blades: #4
(Solidity = 0.133)

100
0
0

2

4

6

8

10

TSR

Figure 3-32: Torque vs. TSR for turbines with different solidity
3.2.6 Effects of Wind Shear
The increase in wind velocity with elevation, which is characterized by the wind shear pattern
due to the atmospheric boundary layer must be considered for large turbines which extend
several meters above ground. A simple model commonly used to simulate the wind shear is the
power law model:





V H   VEQ H H EQ 

Eq. 3.18

The subscript “EQ” are used for variables at the equator (mid-span) of the turbine, and ε
represents the Wind Shear Exponent. This exponent determines the slope of the wind profile.
Figure 3-33 shows the wind distribution at different heights with an equatorial speed of 26.72
m/s.
Figure 3-34 shows the variation of Cp with TSR for the turbine simulated with and without wind
shear. The conditions for open water and building landscape are compared to a case without
75

wind shear. From this figure we can conclude that with the presence of wind shear the amount of
power available to the turbine increases. Also it can be seen that the turbine performs better if it
is placed in a more open terrain (e.g. a smooth, level, grass-coverage) as opposed to one around a
lot of obstacles such as buildings. This is why wind farms are often located in open fields far
away from city areas. Table 3-2 presents some of the wind model’s exponents that are typically
used for various terrains. These values are obtained modeling.
Table 3-2: Wind shear exponents for various terrains (The Engineering Toolbox: Wind Shear
n.d.)26
Terrain

Wind Shear Exponent (ε)

Open Water

0.10

Smooth, level, grass-covered

0.15

Row crops

0.20

Low bushes with a few trees

0.20

Heavy trees

0.25

Several buildings

0.25

Hilly, mountainous terrain

0.25

76

10.0

Height (m)

8.0
6.0
ε = 0.10
4.0

ε = 0.20

2.0

ε = 0.25

0.0
0

10

20
Vinf (m/s)

30

40

Figure 3-33: Wind speed variations with the height for three different wind shear exponents

0.6
0.5

Cp

0.4

W/out Wind
shear

0.3

ε = 0.10

0.2

ε = 0.25

0.1
0
0

2

4

6

8

10

TSR

Figure 3-34: Cp vs. TSR for a turbine with and without wind shear. The wind shear exponents
used were 0.1 and 0.25

77

4. Chapter 4: VAWT with Variable Blade Pitch
4.1 Introduction
All simulations carried out so far have been for a turbine with fixed blade pitch set to zero
degree. A pitch angle of zero degree corresponds to the chord line of the airfoil being aligned
with the tangent of the circular path the airfoil makes as it rotates. Based on the flow conditions
and geometry of the turbine and blades, the torque characteristic and hence power and
performance can be negatively affected. This section aims to study how the performance of a
turbine can be controlled by varying the blade pitch. Also by varying the pitch angle, we can
potentially solve the self-starting problem that hinders the operation of VAWT. To carry out this
study with the cascade model, the equations of W0u, α0u, W0d, and α0d must be modified to
include the pitch angle:



Vn 
 R tan   Vd sin    cos  
 0  tan 1    tan 1 

Vc 


   R cos    Vd cos    

 


2
W0  Vc2  Vn

Eq. 4.1

Eq. 4.2

In the above equations, variables with subscripts “c” and “n” represent the upstream and
downstream values of the chord-wise and normal velocity components. For β equal to zero, the
original equation is recovered.
Other researchers have used various methods to optimize the turbine by varying the pitch. This
includes the work by I. Paraschivoiu which optimizes the turbine by creating the best variation in
pitch angle through a genetic algorithm (Paraschivoiu, Trifu and Saeed 2009). The second
78

approach optimizes the turbine by restricting the turbine’s blade to have a pre-specified angle of
attack (Staelens, Saeed and Paraschivoiu 2003). These two methods have their merits; however,
the current study uses a unique optimization approach by finding the best pitch angle at every
azimuthal angle which gives the turbine the best torque.
4.2 Results and Discussion
Before the results of the current research are presented, it is important to understand how the
optimization is performed. This optimization technique uses the same cascade model used in
previous section. In this section only a one blade, straight-blade wind turbine is considered. The
reason for choosing this setup is to simplify the aerodynamic problem, and limit the errors that
can be associated with other blades interacting with each other. This is especially important due
to simplicity of the model and its inability to capture the flow physics. During simulations, for
every azimuthal angle, the blade pitch is varied from 0˚ to a maximum specified angle of 90˚ and
the results recorded. With every pitch angle selection, the iteration is done until convergence is
achieved before advancing to the next Ѳ angle. For example, if Ѳ = 1˚, at the start of iteration, β
= 0˚ is selected, and with a guess of Vd, the steps explained in the previous chapter are taken
until there is convergence in Vd. Once convergence is reached for the chosen Ѳ, β is increased by
one degree and the process is repeated until β = 90˚. After this, Ѳ is advanced by one degree, and
then the pitch angle variation is repeated; this is done for all Ѳ from 0˚ to 360˚. At every level of
convergence, all variables are stored in an array to be used later for optimization purposes. After
this raw data has been obtained, the optimization is performed based on the tangential force or
torque. For every azimuthal angle, the pitch angle that generated the highest tangential force/
torque is selected, and the corresponding values of α, β, normal force, CL, and CD are recorded.
79

For this analysis, three wind speeds are used: 6, 7, and 8 m/s. These velocities have been selected
because they represent typical velocities experienced by wind turbines in the field. All
calculations are conducted with fixed Reynolds number of Re = 80,000.
The torque characteristics for the fixed and variable pitch turbines for the three wind speeds are
plotted in Figure 4-1, Figure 4-2, and Figure 4-3. As expected, variable pitch turbines generate
significantly more torque throughout the turbine’s operation. The main observations are the
higher values of positive torques for variable pitch blades, and the fact that the turbine which
varies its blade orientation also has the advantage of self-starting without any difficulties. These
plots represent the best torque that the optimization process was able to obtain, within the
limitations of the code. Figure 4-3 shows the torque variation with angular position for the
variable pitch case to have some spikes; this is due to some convergence issues.
Based on the optimizations performed, the improvements in Cp for these three wind speeds are
computed and shown in Table 4-1.
Table 4-1: Turbine efficiency for fixed and variable pitch turbines
V∞

Cp (fixed)

Cp (variable)

6 m/s

0.2896

0.7944

7 m/s

0.2502

0.4951

8 m/s

0.0487

0.2092

From this table it can be seen that there are definite improvements in efficiency of the system
when the blade pitching is optimized.
80

500

Torque (N.m)

400
300
200

Fixed Trq.

100

Variable Trq.

0
-100
0

100
200
300
Theta (degrees)

400

Figure 4-1: Torque vs. Ѳ for simulations with fixed and variable pitch blades. These plots are for
wind speed V∞ of 6 m/s

500

Torque (N.m)

400
300
200

Fixed Trq.

100

Variable Trq.

0
-100
0

100
200
300
Theta (degrees)

400

Figure 4-2: Torque vs. Ѳ for simulations with fixed and variable pitch blades. These plots are for
wind speed V∞ of 7 m/s

81

Torque (N.m)

500
400
300
200
100
0
-100
-200
-300
-400

Fixed Trq.
Fixed Trq.

0

100
200
300
Theta (degrees)

400

Figure 4-3: Torque vs. Ѳ for simulations with fixed and variable pitch blades. These plots are for
wind speed V∞ of 8 m/s

By observing CL and CD against angle of attack, α, it is possible to infer what angles the airfoil’s
relative velocity must make to get the optimal performance: from basic aerodynamics it is known
that CL increases linearly with α until stall. Figure 4-4 shows the angle of attack variations with
the angular positions of the turbine for a wind speed of 6 m/s. At this wind speed relatively high
efficiencies are achieved, even for a turbine with fixed pitch. For a turbine to give the optimal
performance, it will be essential that it avoids stall conditions and be at an angle of attack that
gives the best lift based on the input conditions (typically occurs just before the stall). From the
simulation, for all Ѳ and β combinations, the maximum angle of attack was 32.34˚ and the
minimum was -30.88˚ (based on the simulated flow condition, and geometry of turbine).
Looking at Figure 4-4 it can be observed that the angle of attack varies between 8.08˚ and 9.34˚
(at the upstream section), and -9.34˚ and -8.04˚ (at the downstream section) for the variable blade
pitch case. Figure 4-5 shows the plot of α against CL for Re of 80,000. This figure has been
82

plotted within the range of maximum and minimum α observed in our simulation for the given
flow conditions and geometry. It can be observed that for positive α angles, between 7˚ and 9˚
reasonably high values of lift are generated that will allow the turbine to generate high powers. A
similar conclusion can also be made for negative α angles, between -7˚ and -9˚ which also
generates reasonably high lifts.

15
Alpha (degrees)

10
5
0
-5
-10
-15
0

100

200
Theta

300

400

Figure 4-4: The variation of angle of attack α against Ѳ using the optimization scheme. This plot
is for a wind speed V∞ of 6 m/s

83

0.8
0.6
0.4
0.2

CL

0
-0.2
-0.4
-0.6
-0.8
-1

-30

-20

-10

0
10
Alpha (degrees)

20

30

Figure 4-5: CL vs. angle of attack α for Re = 80,000

Pitch angle (degrees)

15
10
5
0

Vinf = 7 m/s

-5

Vinf = 8 m/s

-10
-15
0

100
200
300
Theta (degrees)

400

Figure 4-6: Pitch angle vs. Ѳ for the variable pitched blade at two wind speeds to show the
differences in pitch angle needed to optimize the turbine

84

15
Alpha (degrees)

10

5
0

Vinf = 7 m/s

-5

Vinf = 8 m/s

-10
-15
0

100

200
Theta

300

400

Figure 4-7: Angle of attack α vs. Ѳ for the variable pitch blades
For the optimized variable pitch turbine there is a sharp (step) change in angles between the
upstream and downstream sections of the turbine (Figure 4-6 and Figure 4-7). These step
changes may be mechanically unfeasible to achieve; therefore, in order to soften this transition a
tangent hyperbolic smoothing technique has been applied to the pitch variations. Figure 4-8
shows the results obtained with this modification.
In summary, it is believed that a variable pitch system should be used to start the vertical axis
wind turbines, but once steady-state conditions has been achieved, it may be “turned-off” to
reduce the number of moving parts, which can constitute further maintenance issues. If moving
mechanisms for variable pitching is not an issue, the variable pitch system may be operated for
all operating conditions to always obtain higher power outputs and improved efficiencies. To
achieve this however, a control mechanism which monitors and reacts to inflow conditions must
be developed.

85

Pitch angle (degree)

15
10
Pitch angle
variation

5
0

Mod. Pitch
angle #1

-5

Mod. Pitch
angle #2

-10
-15
0

100

200
300
Theta (degree)

400

Figure 4-8: Pitch angle vs. Ѳ for the variable pitch blade with the tangent hyperbolic smoothing
being applied to optimized pitch results (V∞ = 6 m/s)

86

5. Chapter 5: Horizontal Axis Wind Turbines
5.1 Introduction
This chapter is focused on the results we have obtained from simulations using FAST for
horizontal axis wind turbines (HAWT). The specifications for the simulated HAWT are given in
Table 5-1.
Table 5-1: The geometry and parameters of the simulated HAWT
Parameter

Value

Rotor diameter

10.00 m

Hub Height

12.19 m

Rated Power

20 kW

Blade chord length

Tapered
S809

Airfoil type

2

Number of blades
Blade Length

5.03 m

Initial blade pitch

4.815˚

Simulation RPM

71.9

87

The effects of tip losses as well as hub losses were included in these calculations. As mentioned
before, the subroutine, AeroDyn, performs the aerodynamic calculations on a spanwise basis.
Here the plots are shown for the mid-blade element (9

th

out of 18 sections). Even though each

individual section has a different twist angles, in this chapter, the blade pitch represents that of
the entire blade section. Lastly, note that the Reynolds numbers shown in the results are to the
sixth power.
5.2 Effects of Wind Speed
From the FAST simulations and results in Figure 5-1 to Figure 5-8 we observe that:


As the wind speed increases, the power also increases until the speed reaches a peak
value, after which the power starts to reduce. Furthermore, as the RPM increases, the
peaks are reached at higher wind speeds.



The average (steady-state) CL typically increases with wind speed. However, CL values
are generally lower for higher RPMs.



The average (steady-state) CD value increased with wind speed. The magnitude of CD
also reduces with increasing RPM.



Cp values reduced with increasing wind speeds, and the magnitude of the Cp increased
with increasing RPM.



The angle of attack increased with the increasing wind speed, and the magnitudes
reduced with increasing RPMs.



As the wind speed increased, the dynamic pressure also increased, and its magnitude
also increased with increasing RPM.
88



Figure 5-7 illustrates the aerodynamic efficiency, CL/CD, of the turbine. The turbine is
most efficient at lower wind speeds, and reduces in efficiency with increasing wind
speed.

Note: The graphs below simulate a wind turbine with blade pitch angle of 4.815˚ and a nacelle
yaw angle of 0˚. The graphs show different parameters at three at three different RPM values:

Power (kW)

50, 71.9 and 85.

20
18
16
14
12
10
8
6
4
2
0

RPM: 50
RPM: 71.9
RPM: 85

0

5
10
Wind speed (m/s)

15

Figure 5-1: The variation of power with wind speed for three RPM values

89

1.2
1

CL

0.8
RPM: 50

0.6

RPM: 71.9

0.4

RPM: 85

0.2
0
0

5
10
Wind speed (m/s)

15

Figure 5-2: Averaged CL at different wind speeds and RPM values

0.6
0.5

CD

0.4
0.3

RPM: 50

0.2

RPM: 71.9
RPM: 85

0.1
0
0

5
10
Wind speed (m/s)

15

Figure 5-3: Averaged CD at different wind speeds and RPM values

90

0.4
0.35
0.3
Cp

0.25
RPM: 50

0.2
0.15

RPM: 71.9

0.1

RPM: 85

0.05
0
0

5
10
Wind speed (m/s)

15

Figure 5-4: The coefficient of power Cp at different wind speeds and RPM values

35
Alpha (degrees)

30
25
20

RPM: 50

15

RPM: 71.9

10

RPM: 85

5
0
0

5
10
Wind speed (m/s)

15

Figure 5-5: The angle of attack vs. wind speed for different RPM valued

91

Dynamic pressure (Pa)

450
400
350
300
250
200
150
100
50
0

RPM: 50
RPM: 71.9
RPM: 85

0

5
10
Wind speed (m/s)

15

Figure 5-6: The dynamic pressure vs. wind speed for different RPM valued

60
50
CL/ CD

40
30

RPM: 50

20

RPM: 71.9
RPM: 85

10
0
0

5
10
Wind speed (m/s)

15

Figure 5-7: Aerodynamic efficiency vs. wind speed for different RPM valued

92

Axial Induction factor

0.25
0.20
0.15
RPM: 50
0.10

RPM: 71.9

0.05

RPM: 85

0.00
0

5
10
Wind speed (m/s)

15

Figure 5-8: The axial induction factor a vs. wind speed for different RPM valued
We expect that as the wind speed increases, the angle of attack, α, should also increase.
Generally α increases linearly between 0˚ and 15˚ - 20˚ after which stalling occurs. At stall, there
is significant increase in drag with a reduction in lift. The plots above helps to illustrate this
general trend. Studies have shown that during regular operation a wind turbine, they often
operate in a dynamic stall state. One effect of dynamic stall is to delay the onset of stalling to a
higher α value than for static aerodynamic conditions (Leishman 2002).27 FAST uses a semiempirical model: the Beddoes-Leishman model which was initially developed to analyze
dynamic stall in helicopters.
5.3

Effects of RPM

From the FAST simulations and results in Figure 5-9 to Figure 5-14 we observe that:


Power increased with increase in RPM, and the magnitude increased with increasing
pitch angles. However, from my simulations, at a pitch of 10˚ there was a significant drop
in power after an RPM of about 50.
93



For low RPMs, more favorable CL results were obtained at a mid-range pitch angle, and
at higher RPMs. A lower angle generated better results. At pitch angle of 10˚ as the RPM
increased there is a significant drop in CL.



The CD reduced with increasing RPM, but approached a constant value at high RPMs.
Also, it should be noted that the magnitude of CD reduced with increasing pitch angle.



Cp varied in a similar manner to the power.



The angle of attack, α, reduced with increasing RPM, and the magnitude of α reduced
with increase in the pitch angle.



The CL/CD ratio is also plotted for a fixed wind speed, and pitch. The changes in α are
achieved by changing the RPM of the wind turbine. Similar trends to those observed in
Section 5.2 are seen.

Note: The graphs below simulate a wind turbine with steady inflow wind speed of 7m/s and a
nacelle yaw angle of 0 deg. The graphs show the effects of RPM on different parameters at three
different pitch angles: 4.815, 6 and 10˚.

94

7

Power (kW)

6
5
4

Pitch: 4.815 deg

3

Pitch: 6.0 deg

2

Pitch: 10 deg

1
0
35

45

55
65
RPM

75

Figure 5-9: Plot of power against RPM for different pitch angles

1.2
1

CL

0.8
Pitch: 4.815 deg

0.6
0.4

Pitch: 6.0 deg

0.2

Pitch: 10 deg

0
30

40

50

60
RPM

70

80

Figure 5-10: CL vs. RPM for different RPM and pitch angles

95

CD

0.45
0.4
0.35
0.3
0.25
0.2
0.15
0.1
0.05
0

Pitch: 4.815 deg
Pitch: 6.0 deg
Pitch: 10 deg

0

25

50
RPM

75

100

Figure 5-11: CD vs. RPM for various RPMs and various pitch angles

0.4
0.35
0.3
Cp

0.25
Pitch: 4.815 deg

0.2
0.15

Pitch: 6.0 deg

0.1

Pitch: 10 deg

0.05
0
0

50
RPM

100

Figure 5-12: Cp vs. RPM for different pitch angles

96

Alpha (degrees)

25
20
15
Pitch: 4.815 deg
10

Pitch: 6.0 deg

5

Pitch: 10 deg

0
0

50
RPM

100

Figure 5-13: Plots of angle of attack, α, vs. RPM for different pitch angles

70
60

CL/CD

50
40

Pitch: 4.815 deg

30

Pitch: 6.0 deg

20

Pitch: 10 deg

10
0
20

40

60

80

RPM

Figure 5-14: CL/CD ratio for various RPM values at different pitch angles

97

From the actuator disk theory for HAWTs, the torque on each annular ring is given by the
change in angular momentum. The change in angular momentum is obtained by multiplying the
mass flow rate, the change of tangential velocity and the radius of the turbine. The toque per
annular ring is represented by δQ and is calculated as:

Q  AcsV 1  a 2a' r 2

Eq. 5.1

From this equation we see that the torque of a wind turbine should increase linearly with the
RPM. The power generated relates to torque as:
ower = orque

2π

R

Eq. 5.2

The power per annular ring is δ , then,

P  AcsV 1  a 2 2 a' r 2

Eq. 5.3

This tells us that the power (and Cp) increases with the squared of RPM. This we seen in Figure
5-9.
5.4

Effects of Pitch Angle

From the FAST simulations and results in Figure 5-15 to Figure 5-20 we can observe that:


Power reduced with increasing pitch angles. The magnitude increased with increase in
wind speed. However, at a wind speed of 11 m/s the power did increase with increasing
pitch angles, until about 10˚, before it leveled out and began to decrease.



CL reduced for increasing pitch angle, and the magnitude increased with increasing wind
speeds. There was an exception for the wind speed at 11 m/s, where the CL increased
98

with increasing pitch angle.


CD reduced with increasing pitch angle and the magnitude increased with increase in
wind speed.



Cp reduced with increase in pitch angle. Again an exception was at the wind speed of 11
m/s were Cp increased with increasing pitch.

The angle of attack decreased with increasing pitch angle. Note: The graphs below simulate a
wind turbine with a constant R

of 71.9 and a nacelle yaw angle of 0˚. he graphs show the

Power (kW)

effects of pitch angle on different parameters at three different wind speeds: 6, 9, and 11 m/s.

16
14
12
10
8
6
4
2
0
-2

Wd Spd: 6
Wd Spd: 9
Wd Spd: 11

0

5
10
Pitch angle (degree)

15

Figure 5-15: Power vs. pitch angle for different wind speeds

99

1.2
1
0.8
Wd Spd: 6

0.4

CL

0.6

Wd Spd: 9
Wd Spd: 11

0.2
0
0

5
10
Pitch angle (degree)

15

Figure 5-16: CL vs. blade pitch angle for different wind speeds

0.4
0.35

0.3
CD

0.25
0.2

Wd Spd: 6

0.15

Wd Spd: 9

0.1

Wd Spd: 11

0.05
0
0

5
10
Pitch angle (degree)

15

Figure 5-17: CD vs. the blade pitch angle for different wind speeds

100

0.4
0.3

Cp

0.2
Wd Spd: 6
0.1

Wd Spd: 9
Wd Spd: 11

0
-0.1
0

5
10
Pitch angle (degree)

15

Figure 5-18: Cp vs. blade pitch angle for different wind speeds

Alpha (degrees)

25
20
15

Wd Spd: 6
10

Wd Spd: 9

5

Wd Spd: 11

0
0

5
10
Pitch angle (degree)

15

Figure 5-19: Angle of attack vs. blade pitch angle for different wind speeds

101

Dynamic pressure (Pa)

350
300
250
200

Wd Spd: 6

150

Wd Spd: 9

100

Wd Spd: 11

50
0
0

5
10
Pitch angle (degree)

15

Figure 5-20: Dynamic pressure vs. pitch angle

102

5.5 Variation of Parameters along the Blade Span
This section presents the results of the spanwise variation of the aerodynamic parameters.
The effects of blade pitch angle:
The results of the effects of pitch angle on the spanwise parameters are presented in Figure 5-21
to Figure 5-28.

70
60

CL/CD

50
40

Pitch of 4.815 deg

30

Pitch of 6 deg

20

Pitch of 10 deg

10
0
0

10
Section #

20

Figure 5-21: The CL/ CD ratio at the different sections of the blade

103

Section Pitch (degrees)

35
30
25
20

Pitch of 4.815 deg

15

Pitch of 6 deg

10

Pitch of 10 deg

5
0
0

10
Section #

20

Figure 5-22: Section pitch angle at the different sections of the blade

70
Alpha (degree)

60
50
40

Pitch of 4.815 deg

30

Pitch of 6 deg

20

Pitch of 10 deg

10
0
0

10
Section #

20

Figure 5-23: The angle of attack at the different sections of the blade

104

Reynolds number

1.2
1
0.8
0.6

Pitch of 4.815 deg

0.4

Pitch of 6 deg
Pitch of 10 deg

0.2
0
0

10
Section #

20

Figure 5-24: Re at the different sections of the blade

0.35
0.3

CD

0.25
0.2

Pitch of 4.815 deg

0.15

Pitch of 6 deg

0.1

Pitch of 10 deg

0.05
0
0

10
Section #

20

Figure 5-25: CD at the different sections of the blade

105

1.6
1.4
1.2
CL

1
Pitch of 4.815 deg

0.8

Pitch of 6 deg

0.6

Pitch of 10 deg

0.4
0.2
0
0

10
Section #

20

Figure 5-26: CL at the different sections of the blade

Normal Force (N)

60
50
40
30

Pitch of 4.815 deg

20

Pitch of 6 deg
Pitch of 10 deg

10
0
0

10
Section #

20

Figure 5-27: Normal force at the different sections of the blade

106

Tangential Force (N)

25
20
15
Pitch of 4.815 deg

10

Pitch of 6 deg

5

Pitch of 10 deg

0
-5
0

10
Section #

20

Figure 5-28: Tangential force at the different sections of the blade
The Effects of Wind speed:
The results of the effects of wind speed on the spanwise parameters are presented in Figure 5-29
to Figure 5-35.

Section pitch(degrees)

30

25
20
15

Wd Spd of 7m/s

10

Wd Spd of 9m/s
Wd Spd of 11m/s

5
0
0

5

10
15
Section #

20

Figure 5-29: Section pitch angle at the different sections of the blade
107

70
Alpha (degrees)

60
50
40

Wd Spd of 7m/s

30

Wd Spd of 9m/s

20

Wd Spd of 11m/s

10
0
0

5

10
15
Section #

20

Figure 5-30: Angle of attack at the different sections of the blade

1.2
1.0

Re

0.8

0.6

Wd Spd of 7m/s

0.4

Wd Spd of 9m/s

0.2

Wd Spd of 11m/s

0.0
0

5

10
15
Section #

20

Figure 5-31: Re at the different sections of the blade

108

CL

1.8
1.6
1.4
1.2
1.0
0.8
0.6
0.4
0.2
0.0

Wd Spd of 7m/s
Wd Spd of 9m/s
Wd Spd of 11m/s

0

5

10
15
Section #

20

Figure 5-32: CL at the different sections of the blade

0.50
0.40

CD

0.30
0.20

Wd Spd of 7m/s

0.10

Wd Spd of 9m/s
Wd Spd of 11m/s

0.00
-0.10
0

5

10
15
Section #

20

Figure 5-33: CD at the different sections of the blade

109

70
Normal Force (N)

60
50
40

Wd Spd of 7m/s

30

Wd Spd of 9m/s

20

Wd Spd of 11m/s

10
0
0

5

10
15
Section #

20

Figure 5-34: Normal force at the different sections of the blade

Tangential Force (N)

30
25
20
15

Wd Spd of 7m/s

10

Wd Spd of 9m/s

5

Wd Spd of 11m/s

0
-5
0

5

10
15
Section #

20

Figure 5-35: Tangential force at the different sections of the blade

110

Effects of RPM:
The results of the effects of RPM on the spanwise parameters are presented in Figure 5-36 to
Figure 5-42.

Section Pitch (degrees)

30
25
20
15

RPM of 50

10

RPM of 70
RPM of 80

5
0
0

5

10
Section #

15

20

Figure 5-36: Section pitch at the different sections of the blade

70
Alpha (degrees)

60
50
40

RPM of 50

30

RPM of 70

20

RPM of 80

10
0
0

5

10
Section #

15

20

Figure 5-37: Angle of attack at the different sections of the blade
111

1.2
1.0

Re

0.8
0.6

RPM of 50

0.4

RPM of 70
RPM of 80

0.2
0.0
0

5

10
Section #

15

20

Figure 5-38: Re at the different sections of the blade

1.6
1.4
1.2
CL

1.0

0.8

RPM of 50

0.6

RPM of 70

0.4

RPM of 80

0.2
0.0
0

5

10
Section #

15

20

Figure 5-39: CL at the different sections of the blade

112

CD

0.5
0.4
0.4
0.3
0.3
0.2
0.2
0.1
0.1
0.0
-0.1

RPM of 50
RPM of 70
RPM of 80

0

5

10
Section #

15

20

Figure 5-40: CD at the different sections of the blade

Normal Force (N)

50

40
30
RPM of 50
20

RPM of 70

10

RPM of 80

0
0

5

10
Section #

15

20

Figure 5-41: Normal force at the different sections of the blade

113

Tangential Force (N)

14
12
10
8
6

RPM of 50

4

RPM of 70

2

RPM of 80

0
-2
0

5

10
Section #

15

20

Figure 5-42: Tangential force at the different sections of the blade

114

6. Chapter 6: Summary and Conclusions
This research is focused on the aerodynamic analysis of horizontal and vertical axis wind
turbines. The effects of the parameters that affect the performance of VAWTs has been studied
both globally and in-depth to understand some physical features of how these turbines operate.
Some key points have been made on how the torque characteristics affect the startup behavior
and power production.
As part of this research, a computational model has been developed to qualitatively study, the
behavior of VAWTs; the results presented are based on the outputs of this model. The model was
developed and applied to study VAWTs in order to enhance the knowledge and understanding of
their aerodynamics and how various operating parameters will affect the efficiency and power
produced. Both fixed pitch and variable pitch blades are considered. From the variable pitch
results we can conclude that the problem of self-start can be solved by developing new
mechanisms. This research has provided a new and unique way of achieving the first step in
developing efficient VAWTs. A variable pitch system, both for improved self-start capabilities
and improved power generation will help the wind industry to advance turbine designs. Some of
the conclusions made from the results are presented below:


The results show that more research is needed to characterize the Reynolds number for
wind turbines. The selection of the Re significantly affects the accuracy of the simulation
results, whether a momentum or vortex model is used, as CL and CD data are very
sensitive to Re. For example, in the equation of Re, the numerator has a length quantity
(the characteristic length of the flow), flow visualization experiment is needed to

115

understand what length (blade chord length, or rotor diameter, etc.) best represents that of
a turbine.


Empirical models which take into consideration the transient nature of wind turbines are
necessary to better understand the turbine's torque performance. Knowing how much
torque is generated at every azimuthal angle with time will help in understanding how
turbines start, and will assist in the development of proper control systems for the
optimized machines.



The success of any empirical model for wind turbines depends on the accuracy of lift and
drag property data for the flow conditions experienced. More experiments are needed for
variable angle of attack blades. Currently, most empirical models use property data from
steady-state static experiments; however, the flow conditions seen by a VAWT are
significantly different because of the turbine's rotational velocity, its blade's orientation
and how it interacts with the wind, and the fact that at every azimuthal angle, the blade
angle of attack changes. The results presented in this thesis, although they consider
changes in flow conditions, the underlying data tables they use were unchanged which
may lead to errors in results. Results from the variable pitch blade will especially benefit
unsteady CL and CD data because of the very complicated flow structures that occurs by
blade pitching. The results presented here may over-predict the power and efficiencies
that are experienced by an actual turbine as the rapid changes which affect the flow
structure may reduce the actual amount of lift and torque generated by a highly unsteady
flow over the blade when compared to a static case.



The cascade model considers circulation as one of its calculated variables; however, the
assumption applied to simplify the problem may be inadequate to represent the physics
116

of circulation around the airfoil’s profile. In essence, circulation in this model is
calculated as a consequence of the inflow and outflow velocities parallel to the
longitudinal section of the airfoil. What is therefore proposed is to use a similar
formulation as presented in the theory section above; however, after the initial guess is
made for Vd and the iteration process proceeds, instead of calculating the circulation
(which is ultimately used to find the component of circulation velocity on the airfoil
profile) based on those two velocity components, the panel method could be used to
calculate it. In order to implement this, at every iteration of Vd a panel method subroutine
may be called to calculate the circulation around the airfoil, given the flow conditions;
the resulting value of circulation which satisfies the Kutta-Joukowski theorem is then
applied to more accurately calculate

, the velocity due to circulation in the cascade.

This approach is proposed because the panel method would serve as a better model in
predicting the circulation, thus improving the overall performance of the cascade theory.
A drawback to this approach will be the poor performance of the panel method at stall
angles.


The model used in analysis does not consider the interactions of one blade on another, hence
the assumption of a reference blade being selected and the other blades are assumed to have
similar aerodynamic characteristics. It is believed that for a more complete analysis the
effects of one blade which alters the flow conditions for other blades needs to be considered.



Ultimately, in order to better understand the basic flow physics for wind turbines, a
computational fluid dynamics (CFD) approach is necessary.

117

HAWT, FAST results show significant effects of the flow/ turbine parameters on turbine power
and efficiency.

118

Bibliography

119

Bibliography

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Brownlee, Baron Gene. A Vortex Model for Vertical Axis Wind Turbine. Thesis, n.d.

Burton, Andy, David Sharpe, Nick Jenkins, and Ervin Bossanyi. "Wind Energy Handbook." 1.
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turbine." Proceedings of the Institute of Mechanical Engineers, Power and Energy,
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