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Multiple Streamtube Approximation of Flow Induced Forces on a Savonius Wind Turbine
Kevin Pope
1
and Greg F. Naterer
2
University of Ontario Institute of Technology, Oshawa, Ontario, Canada
Abstract
This paper develops a new approximate model to predict the pressure and momentum
forces on a Savonius style VAWT (vertical axis wind turbine). Flow distributions through and
around the turbine are examined for analytical predictions of the torque and power output, at all
rotor angles. A new approximate streamtube method is developed to predict the momentum, lift
and drag forces on the rotor surfaces by the air stream, based on an integral force balance on the
turbine blades. Unlike other past analytical methods, the technique predicts both momentum and
pressure forces imposed on the rotor surface during operation. The calculated results are carefully
compared with numerical predictions from CFD (computational fluid dynamics).
Nomenclature
a radius of semi-cylindrical blade [m]
A area per unit turbine height [m
2
/m]
C
p
power coefficient
d distance to origin [m]
g acceleration of gravity [m/s
2
]
p pressure [Pa]
1
PhD candidate. Faculty of Engineering and Applied Science University of Ontario Institute of Technology, 2000
Simcoe St, Oshawa, Ontario, Canada, L1H 7K4 , kevin.pope@mycampus.uoit.ca
2
Associate Dean and Canada Research Chair Professor. Faculty of Engineering and Applied Science, University of
Ontario Institute of Technology, 2000 Simcoe Street North, Oshawa, Ontario, Canada, L1H 7K4, greg.naterer@uoit.ca
2
Q torque [N-m]
R turbine radius [m]
S solidity
t time [s]
u x-component of velocity [m/s]
U free-stream velocity [m/s]
v y-component of velocity [m/s]
V mean wind velocity [m/s]
W
x
x-component of relative wind velocity [m/s]
Greek
rotor overlap [m]
rotor position [radians]
λ tip speed ratio
μ dynamic viscosity of air [Pa-s]
ρ air density [kg/m
3
]
semi-cylindrical coordinate
stream function
Ω
rotor velocity [radians/s]
Subscripts
m momentum
p pressure
x drag
y lift
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1. Introduction
Wind power is becoming an increasingly significant source of sustainable power
generation, for example growing at 23.6% in 2010 [1], and providing a valuable synergy with other
carbon-free technologies (such as solar, nuclear and geothermal) [2]. Wind power can supplement
current power generation methods while mitigating environmental degradation and resource
depletion [3]. Although horizontal axis wind turbines (HAWTs) represent the majority of the
installed wind capacity, many opportunities exist for small vertical axis wind turbine (VAWT)
installations to achieve significant future growth. A wide variety of small wind turbine applications
can be fulfilled by VAWTs. Advantages of a VAWT over a HAWT include a simpler design and
omni-directional operation (omitting the need for a yaw mechanism). These attributes reduce
maintenance and installation expenses, due to the reduced complexity of the system, compared to
a HAWT.
Several types of VAWT designs are known, each with unique benefits and drawbacks. The
most common are the Darrieus, H-rotor, and Savonius VAWTs. The Savonius turbine has the
lowest maximum efficiency of these three VAWTs; however, it has several useful attributes that
motivate further development. Compared to the other VAWTs, this design can more effectively
operate in turbulent conditions. For example, it has been shown in several studies that its
performance is independent of the Reynolds number of the air stream [4, 5]. The L-sigma criterion
[6], which considers the frontal area of a turbine and the mechanical stresses during operation,
identified the Savonius VAWT as advantageous over other types of turbines (including HAWTs).
It was also reported to have better start-up capabilities and lower tip-speed-ratios (λ).
There is significant potential to combine a Savonius design with a Darrius or H-rotor design
to effectively utilize the benefits of each geometry in a hybrid type VAWT [7, 8]. Reducing the
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environmental impact of residential, commercial, and industrial buildings can substantially reduce
overall carbon emissions, and small wind turbines are a promising technology for renewable power
production in a populated environment [9]. A HAWT typically provides poor potential in this
environment [10], but a small hybrid VAWT can provide significant power generation. Recent
advances in generator technology have identified radial coreless synchronous generators as a
potential candidate for small VAWT installations, where its small size and low start-up speed
coincide with the requirements of a Savonius VAWT [11].
The performance of any wind turbine is highly sensitive to the wind conditions at the
installation location. Wind conditions can be highly non-uniform and the turbine must operate
through a wide range of tip-speed-ratios [12], no single blade geometry gives optimal performance
for all values of λ [13], and the power output of a Savonius VAWT is highly sensitive to the blade
geometry [14]. In contrast to other methods, where a single maximum power coefficient is defined,
at one particular wind speed [15, 16], to achieve optimal performance, the blade geometry should
be designed to effectively match the operating conditions of the installation site. Considering the
effects of various wind velocity distributions and their effects on λ and the system’s start-up
capabilities, these factors can significantly improve the total power output. Even sites without
sufficient wind speed data can utilize spatial estimation techniques to accurately predict the
conditions at the installation site [17]. The turbine operation involves complicated flow patterns
that vary with time throughout the rotation of the rotor. Transient numerical simulations can
provide a useful tool to provide insight about the operation of a Savonius turbine, for a wide range
of λ, as an alternative to wind tunnels, or field testing of prototype turbines. Transient aerodynamic
loading is a key issue that influences the overall cost of wind power generation, because it affects
the operation, availability, power quality, and energy yield of each installation [18].
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Previous research has determined that the predictive accuracy of wind turbine installations
are more closely linked to the accuracy of the aerodynamic model than the electromechanical
model [19]. VAWT models can be categorized into three main divisions: (i) momentum
(streamtube), (ii) discrete vortex, and (iii) cascade models [20]. However, these current predictive
VAWT models cannot be effectively applied to drag type turbines (such as a Savonius VAWT).
Steamtube techniques use static airfoil lift and drag coefficients, with local angles of attack and
relative velocities [21]. The operation of a Darrieus VAWT can be represented well by this
technique, as corresponding airfoil data is available and there is no significant interaction between
the rotor blades. A single streamtube was successfully applied to a single blade Savonius VAWT
[22]. However, it cannot represent the operation when a second blade is added. The discrete vortex
method has been applied to a Savonius rotor [23 - 25], but a complex iterative solution was required
to obtain a solution. This method could not accurately represent a stationary rotor (important for
start-up), and it required calculations to be performed at much higher Reynolds numbers than flow
conditions of available experimental data [24]. The third approach (cascade model) represents the
rotation of rotors as a linear series of blades [26]. This technique cannot represent the flow
interaction between blades or the changing blade profile throughout the rotation. As a result, it
does not accurately represent the operation of a Savonius turbine.
In this paper, a new predictive model is developed to analyze the momentum, lift and drag
forces created from the flow fields generated during the operation of a Savonius VAWT. In
contrast to previous techniques, where only the momentum force is represented, this model
considers momentum, lift and drag forces to develop a transient model to predict the operation of
a Savonius wind turbine. The vector flow field is represented by the superposition of approximate
flow fields from potential flow theory, to analyze the pressure force, induced by the velocity
