AIAA JOURNAL
Vol. 54, No. 1, January 2016
Coriolis Effect on Dynamic Stall in a Vertical
Axis Wind Turbine
Hsieh-Chen Tsai∗ and Tim Colonius†
California Institute of Technology, Pasadena, California 91125
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DOI: 10.2514/1.J054199
The immersed boundary method is used to simulate the flow around a two-dimensional cross section of a rotating
NACA 0018 airfoil in order to investigate the dynamic stall occurring on a vertical axis wind turbine. The influence of
dynamic stall on the force is characterized as a function of tip-speed ratio and Rossby number. The influence of the
Coriolis effect is isolated by comparing the rotating airfoil to one undergoing an equivalent planar motion that is
composed of surging and pitching motions that produce an equivalent speed and angle-of-attack variation over the
cycle. Planar motions consisting of sinusoidally varying pitch and surge are also examined. At lower tip-speed ratios,
the Coriolis force leads to the capture of a vortex pair when the angle of attack of a rotating airfoil begins to decrease in
the upwind half cycle. This wake-capturing phenomenon leads to a significant decrease in lift during the downstroke
phase. The appearance of this feature depends subtly on the tip-speed ratio. On the one hand, it is strengthened due to
the intensifying Coriolis force, but on the other hand, it is attenuated because of the comitant decrease in angle of
attack. While the present results are restricted to two-dimensional flow at low Reynolds numbers, they compare
favorably with experimental observations at much higher Reynolds numbers. Moreover, the wake-capturing is
observed only when the combination of surging, pitching, and Coriolis force is present.
max
SPM
SSPM
sin
surge
VAWT
Nomenclature
CL
c
Em
k
p
R
Re
Ro
U∞
u^
u0
W
x^
α
α_
δ
λ
ν
ρ
θ
^
ω
ω0
Δt
Δx
Ω
l
=
=
=
=
=
=
=
=
=
=
=
=
=
=
=
=
=
=
=
=
=
=
=
=
=
=
lift coefficient
airfoil chord length
complete elliptic integral of the second kind
reduced frequency
pressure
radius of the turbine
Reynolds Number
Rossby Number
freestream velocity
velocity of the fluid in the rotating frame of reference
velocity introduced by the change of variables
incoming velocity
position vector in the rotating frame of reference
angle of attack
pitch rate
spatial distribution of the body-force actuation
tip-speed ratio
fluid kinematic viscosity
fluid density
azimuthal angle
vorticity of the fluid in the rotating frame of reference
vorticity introduced by the change of variables
time step
grid spacing
angular velocity of the turbine
ratio of the radius of the turbine to the chord length
=
=
=
maximum
sinusoidal pitching motion
sinusoidal surging–pitching motion
sinusoidal variation
surge velocity
vertical axis wind turbine
I.
V
Introduction
ERTICAL axis wind turbines (VAWT) offer several advantages
over horizontal axis wind turbines (HAWT), namely: their low
sound emission (consequence of their operation at lower tip-speed
ratios), their insensitivity to yaw wind direction (because they are
omnidirectional), and their increased power output in skewed flow
[1,2]. Dabiri et al. [3,4] showed that an array of counterrotating
VAWTs can achieve higher power output per unit land area and
smaller wind velocity recovery distance than existing wind farms
consisting of HAWTs. The aerodynamics of VAWTs are complicated
by inherently unsteady flow produced by the large variations in both
angle of attack and incident velocity magnitude of the blades, which
can be characterized as a function of tip-speed ratio. Typically,
commercial VAWTs operate at a tip-speed ratio around 2–5, which
produces an angle-of-attack variation with amplitude of 11.5–30°
and an incident velocity variation with amplitude of 21.5–49% of
its mean.
Aerodynamics of wings at low Reynolds numbers have been well
investigated due to the recent interest in the development of small
unmanned aerial vehicles and micro air vehicles. Morris and Rusak
[5] studied the onset of stall at low to moderately high Reynolds
number flows numerically and provided a universal criterion to
determine the static stall angle of thin airfoils. Taira and Colonius [6]
simulated three-dimensional flows around low-aspect-ratio flat-plate
wings at low Reynolds numbers, with a focus on the unsteady vortex
dynamics at poststall angles of attack. Choi et al. [7] numerically
investigated unsteady, separated flows around two-dimensional (2D) surging and plunging airfoils at low Reynolds numbers.
Dynamic stall refers to the delay in the stall of airfoils that are
rapidly pitched beyond the static stall angle, which is associated with
a substantially higher lift than is obtained quasi-statically, and has
been an active research topic in fluid dynamics for more than 60
years, largely because of the helicopter application [8]. Because of
large variations in angle of attack, dynamic stall occurs on VAWT
operating at low tip-speed ratios [9]. To study this phenomenon,
Wang et al. [10] introduced an equivalent planar motion (EPM),
which is composed of a surging and pitching motion that produces an
Subscripts
avg
EPM
inst
=
=
=
=
=
=
average velocity
equivalent planar motion
instantaneous velocity
Presented as Paper 2014-3140 at the 32nd AIAA Applied Aerodynamics
Conference, Atlanta, GA, 16–20 June 2014; received 16 January 2015;
revision received 13 June 2015; accepted for publication 19 July 2015;
published online 15 September 2015. Copyright © 2015 by Hsieh-Chen Tsai.
Published by the American Institute of Aeronautics and Astronautics, Inc.,
with permission. Copies of this paper may be made for personal or internal
use, on condition that the copier pay the $10.00 per-copy fee to the Copyright
Clearance Center, Inc., 222 Rosewood Drive, Danvers, MA 01923; include
the code 1533-385X/15 and $10.00 in correspondence with the CCC.
*Graduate Student, Mechanical Engineering. Student Member AIAA.
†
Professor, Mechanical Engineering. Associate Fellow AIAA.
216
217
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TSAI AND COLONIUS
equivalent speed and angle-of-attack variation over the cycle. They
further simplified the EPM to a sinusoidal pitching motion (SPM) to
investigate dynamic stall in a 2-D VAWT numerically. The results
matched the experiments done by Lee and Gerontakos [11].
Several attempts have been made to model a VAWT at
Re ∼ O105 , which is appropriate to the urban applications of
VAWTs. Reynolds-averaged Navier–Stokes (RANS) with different
turbulence models has been applied to a 2-D airfoil undergoing an
effective planar motion [12–14] and to a multibladed 2-D VAWT
[15,16]. Ferreira et al. [17] simulated dynamic stall in a section of a
VAWT using detached-eddy simulation at Re 50; 000 and
validated the results by comparing the vorticity in the rotor area with
particle image velocimetry (PIV) data. Duraisamy and Lakshminarayan [18] numerically analyzed interactions of VAWTs with various
configurations using RANS at Re 67; 000. Barsky et al. [19]
investigated the fundamental wake structure of a single VAWT
computationally by large-eddy simulation and experimentally
by PIV.
In this paper, a 2-D VAWT is investigated numerically at low
Reynolds numbers in order to understand qualitative features of the
flow field in a setting where a comparatively large region of
parameter space can be explored than could be for full-scale, threedimensional computations. To explore the parameter space in
relatively short computational time and have more understanding of
the details of the vortex dynamics, flows are simulated at low
Reynolds numbers, Re ∼ O103 . A major limitation of our present
approach is the restriction of flow to a 2-D cross section of an
otherwise planar turbine geometry. Comparisons with Ferreira et al.
[17] show qualitative agreement, but a precise accounting for threedimensional effects awaits future simulations.
We focus here on the Coriolis effect on dynamic stall by comparing
the rotating airfoil to one undergoing an EPM. The influence of
dynamic stall on forces is characterized as a function of tip-speed
ratio and Rossby number. Moreover, inspired by Wang et al. [10],
airfoils undergoing an SPM and a sinusoidal surging–pitching
motion (SSPM) are also compared to see if these two motions can be
an appropriate model for the VAWT. Furthermore, the coupling of the
Coriolis effect with the angle-of-attack and incoming velocity
variations is also examined.
II.
Methodology
A. Simulation Setup
Figure 1 shows a schematic of a VAWT with radius R rotating at an
angular velocity Ω with a freestream velocity U∞, coming from the
left. The chord length of the turbine blade is c. To systematically
investigate the aerodynamics of a VAWT, four dimensionless
parameters are introduced:
Tip-speed ratio∶
λ
ΩR
U∞
(1)
Radius-to-chord-length ratio∶ l Ro Rossby number∶
(2)
U∞ c
ν
(3)
U∞
l
1
2Ωc 2λ 4k
(4)
Re Renolds number∶
R
c
where ν is the kinematic viscosity of the fluid and k is the reduce
frequency, k Ωc∕2U∞ λ∕2l.
The instantaneous incoming velocity W inst and the angle of attack
α can then be characterized as a function of the tip-speed ratio λ and
the azimuthal angle θ:
αλ; θ tan−1
sin θ
λ cos θ
(5)
W inst λ; θ p2
1 2λ cos θ λ
U∞
(6)
Figure 2 shows the angle-of-attack variation and incoming
velocity variation of the VAWT at λ 2. From Eq. (5), the maximum
angle of attack
αmax λ tan−1
1
p
2
λ −1
(7)
occurs at θ cos−1 −1∕λ.
To isolate the Coriolis effect on dynamic stall, a moving airfoil
experiencing an equivalent incoming velocity and angle-of-attack
variation over a cycle is proposed. This EPM is composed of a
surging motion with a velocity W surge and a pitching motion around
_ The airfoil is undergoing the EPM
the leading edge with a pitch rate α.
in a freestream velocity W avg. W avg , W surge , and α_ are shown to be
W avg λ 1
2π
U∞
Z
2π W
0
inst λ;θ
U∞
21λ
E
dθ π
s!
4λ
1λ2
(8)
p
where the function Em ∫ π∕2
1 − m2 sin2 θ dθ is the complete
0
elliptic integral of the second kind,
W surge λ; θ W inst λ; θ − W avg λ
(9)
1
1 λ cos θ
2Ro 1 2λ cos θ λ2
(10)
_ θ αλ;
Moreover, due to the periodic oscillation of angle-of-attack and
incoming velocity variation, Wang et al. [10] studied dynamic stall in
a VAWT by investigating an airfoil undergoing a simplified SPM.
Inspired by their work, sinusoidal variations in the angle of attack and
incoming velocity, which are written as a function of the tip-speed
ratio λ and the azimuthal angle θ in Eqs. (11) and (12) and shown in
Fig. 2, are also considered:
Fig. 1 Schematic of a VAWT and the computational domain.
αsin λ; θ αmax λ sin θ
(11)
W inst;sin λ; θ
λ cos θ
U∞
(12)
218
TSAI AND COLONIUS
4
V.A.W.T.
sinusoidal motion
30
V.A.W.T.
sinusoidal motion
3.5
20
3
10
2.5
0
2
−10
1.5
1
−20
0.5
−30
0
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Fig. 2
60
120
180
240
300
360
0
0
60
120
180
240
300
360
a) Angle of attack variation
b) Incoming velocity variation
Comparison of angle-of-attack variation and incoming velocity variation between the VAWT and the sinusoidal motion at λ 2.
Although the sinusoidal motion shares the same amplitude, it
overestimates the angle of attack in the upstroke phase,
underestimates the incoming velocity in the downstroke phase, and
slightly underestimates the instantaneous velocity over the entire half
cycle. To search for the most appropriate model for a VAWT, we
introduce two additional motions: SPM and SSPM. Airfoils
undergoing both the SPM and SSPM pitch with the sinusoidal angleof-attack variation described in Eq. (11) in a freestream velocity
W avg;sin λU∞ . Airfoils undergoing the SSPM also surge with a
velocity W surge;sin U∞ cos θ.
NACA 0018 airfoils are used as blades in the present study. The
ratio of the radius of the turbine to the chord length l depends on
the choice of the tip-speed ratio λ and the Rossby number Ro.
In preliminary simulations of a three-bladed VAWT, as well as in
previous studies [17], vorticity–blade interaction is only observed in
the downwind half of a cycle. Because only the flow in the upwindhalf cycle is important to torque generation, we save computational
time by modeling a single-bladed turbine. We compute about five
periods of motion, which is equal to 5π∕Ro convective time units, to
remove transients associated with the startup of periodic motion. For
the largest Rossby number we examined, the starting vortex
propagates far enough into the wake to have an insignificant effect on
the forces on the blades after five periods. An additional five periods
of nearly periodic stationary-state motion were then computed and
analyzed below.
B. Numerical Method
The immersed boundary projection method (IBPM) developed by
Colonius and Taira [20,21] is used to compute 2-D incompressible
flows in an airfoil-fixed reference frame with appropriate forces added
to the momentum equation to account for the noninertial reference
frame. The equations are solved on multiple overlapping grids that
become progressively coarser and larger (greater extent). The
dimensionless momentum equation in the rotating frame of reference is
∂u^
1 2
dΩ
u^ · ∇u^ −∇p ∇ u^ −
× x^ − 2Ω × u^ − Ω
∂t
Re
dt
^
× Ω × x
(13)
where u^ and x^ are the fluid velocity and the position vector in the
rotating frame of reference and Ω 1∕2Ro is the dimensionless
angular velocity of rotating frame of reference. We then introduce the
change of variables
^ ∇ × u^ is the vorticity field in the rotating frame of
where ω
reference. By taking the divergence and the curl of Eq. (13), we have
the following vorticity and pressure equation
∂ω 0
1
∇ × ∇ × ω 0 ∇ × ^u × ω 0 −
Re
∂t
(16)
1 2 1
^ − jΩ × xj
^ 2 ∇ · u^ × ω 0 ∇2 p juj
2
2
(17)
Because the flow is incompressible and 2-D, the first term on the
right-hand side of Eq. (16) is just the advection of vorticity with
^ Therefore, in the body-fixed frame of reference, the Coriolis
velocity u.
force does not generate vorticity except on the boundary so that it only
changes the way vorticity propagates in free space. Moreover, because
the magnitude of Coriolis force is proportional to the magnitude of
velocity, fluid with high velocity will be deflected more rapidly.
C. Verification and Validation
The IBPM has been validated and verified by Colonius and Taira
[20,21] and others for problems such as three-dimensional flows
around low-aspect-ratio flat-plate wings [6,22], optimized control of
vortex shedding from an inclined flat plate [23,24], 2-D flows around
a NACA0018 airfoil with a cavity [25,26], and 2-D flows around
surging and plunging flat-plate airfoils [7].
As noted in Sec. II.B, the Navier–Stokes equations are solved on
multiple overlapping grids. Based on the analysis by Colonius and
Taira [21], the IBPM is estimated to have an O4−Ng convergence
rate, where N g is the number of the grid levels. Six grid levels are used
for the present computations in order to have the leading-order error
dominated by the truncation error arising from the discrete delta
functions at the immersed boundary and the discretization of the
Poisson equation. The coarsest grid extends to 96 chord lengths in
both the transverse and streamwise directions of the blade.
To show grid convergence, we examine a single-bladed VAWT
with l 1.5 rotating at λ 2. The velocity field in the streamwise
direction is compared with one on the finest grid with Δx 0.00125
at t 1. Figure 3 shows the spatial convergence in the L2 norm. The
rate of decay for the spatial error is about 1.5, which agrees with Taira
and Colonius [20]. All ensuing computations use a 600 × 600 grid,
which corresponds to Δx 0.005. The time step Δt is chosen to
make Courant–Friedrichs–Lewy number less than 0.4.
III.
Results
A. Qualitative Flow Features in a VAWT
u 0 u^ Ω × x^
(14)
^ 2Ω
ω0 ω
(15)
We begin by examining flows at low tip-speed ratio, λ 2,
Ro 1.5, and Re 1500, which gives a maximum amplitude of
30° in angle-of-attack variation and a reduced frequency k 1∕6.
Figure 4 shows the vorticity field generated by the blade at different
azimuthal angles over a cycle. Negative and positive vorticity are
TSAI AND COLONIUS
−1
10
−2
10
−3
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10 −3
10
−2
10
Fig. 3 The L2 norm of the error of the velocity field in the streamwise
direction in a single-bladed VAWT with l 1.5 rotating at λ 2 at t 1.
Fig. 4
219
plotted in blue and red contour levels, respectively, and all vorticity
contour plots use the same contour levels.
At the beginning of a cycle (Fig. 4a), the airfoils are just returning
to zero angle of attack, and there are still the remnants of earlier vortex
shedding in the wake. The flow reattaches by the time airfoil reaches
α 5∘ (Fig. 4b). When the angle of attack increases further, the wake
behind the airfoil starts to oscillate and vortex shedding commences.
Dynamic stall then takes place, and is marked by the growth, pinchoff, and advection of a leading edge vortex (LEV) on the suction side
of the airfoil (Figs. 4c–4e). The vortices generated will propagate
downstream into the wake of the VAWT or interact with the blades in
the downwind half of a cycle.
When the angle of attack starts to decrease, a trailing edge vortex
(TEV) develops (Fig. 4f). Bloor instability [27] occurs in the trailingedge shear layer at this Reynolds number, which resembles the
convectively unstable Kelvin–Helmholtz instability observed in
plane mixing layers. This TEV couples with an LEV to form a vortex
pair that travels downstream together with the airfoil (Figs. 4g–4i).
Vorticity field for a (clockwise rotating) VAWT at various azimuthal angles at λ 2, Ro 1.5, and Re 1500.
220
TSAI AND COLONIUS
B. Comparison of VAWT and EPM
In this section, we compare flows around an airfoil undergoing the
EPM and a single-bladed VAWT at λ 2, Ro 1.5, and
Re 1500. We are interested in the tangential force response of the
blade over a cycle because the power output is proportional to the
tangential force acting on the blade when VAWTs operate at a
constant tip-speed ratio. The tangential force acting on the blade can
be written as a linear combination of lift and drag:
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This vortex pair interacts with the airfoil in the downwind half of a
cycle (Figs. 4j–4l), which was also observed by Ferreira et al. [17].
When the blade rotates in the downwind half of a cycle, the angle of
attack becomes negative. Vortices are now generated on the other side
of the airfoil and shed into the wake of the VAWT (Figs. 4j–4p). For a
multibladed VAWT, when a blade is traveling in the downwind half of
a cycle, it interacts with vortices generated upstream from other
blades or from the wake it generated at an earlier time (Fig. 4o).
Fig. 5
Vorticity field for EPM and VAWT and the Coriolis force for VAWT at various azimuthal angles.
221
TSAI AND COLONIUS
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CT CL sin α − CD cos α CL sin α −
1
cos α
CL ∕CD
Figure 5 shows the comparison of VAWT and EPM motion in the
surging–pitching configuration. Negative and positive vorticity are
plotted in blue and red contour levels, respectively, and all vorticity
contour plots are using the same contour levels. In the Coriolis force
plots, black arrows show the direction of velocity, blue arrows point
the direction of the Coriolis force, and the color contour plots the
magnitude of the Coriolis force. Since the frame of reference is
rotating clockwise, the Coriolis force deflects the fluid in the
clockwise direction. Figure 6 shows a comparison of the lift
coefficient against dimensionless time and angle of attack for a single
rotation and for the average of both lift coefficients over five cycles.
Although there are still the remnants of earlier vortex shedding in the
wake when the airfoil just returns to zero angle of attack (Fig. 5a), the
flow reattaches by α 5° (Fig. 5b), which leads to a smoothly
increasing lift coefficient at low angle of attack. The differences in the
lift coefficient between the EPM and VAWT are small (Fig. 6). As the
angle of attack increases, dynamic stall commences (Figs. 5c–5e),
which leads to rapidly increasing lift. EPM- and VAWT-induced
(18)
where α is the angle of attack of the blade. From preliminary
simulations, a three-bladed VAWT with l 4 will be free-spinning
with a time-averaged tip-speed ratio λ 0.95 at Re 1500, so that in
the flow we are examining the average tangential force is expected to be
negative. However, as the Reynolds number increases to the range
where commercial VAWTs usually operate, Re ∼ O105 − 106 , drag
coefficient drops dramatically while the change in the lift coefficient is
small. This leads to a large increase in the lift-to-drag ratio CL ∕CD [28].
Therefore, the contribution of lift to the tangential force dominates at
high Reynolds numbers. Moreover, the power of a VAWT is generated
mostly in the upwind half cycle because large vorticity–blade
interactions cancel out the driving torque in the downwind half cycle
[17]. Therefore, in this study we will focus on the lift in the upwind half
of a cycle.
3
3
VAWT
EPM
3
VAWT
EPM
2
2
2
1
1
1
0
0
0
−1
−1
−1
−2
0
−2
30
60
90
120
150
180
VAWT
EPM
−2
0
5
10
15
20
25
30
0
30
60
90
120
150
180
c)The average lift coefficients over five cycles.
a) The lift coefficients over a cycle against
b) The lift coefficients over a cycle against
dimensionless time.
angle of attack.
Fig. 6 Comparing CL;VAWT and CL;EPM at λ 2, Ro 1.5, and Re 1500.
3
3
2
VAWT
EPM
SPM
SSPM
2
1
1
0
0
−1
−1
−2
0
1
5
10
15
20
25
30
VAWT
EPM
SPM
SSPM
−2
0
1
0.5
0.5
0
0
−0.5
−0.5
0
2
4
6
8
10
12
14
VAWT
EPM
SPM
SSPM
0
5
10
15
20
25
30
VAWT
EPM
SPM
SSPM
2
4
6
8
10
12
14
Fig. 7 Lift coefficients of VAWT, EPM, SPM, and SSPM over a cycle at λ 2 and 4, Ro 0.75, and Re 1000.
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222
TSAI AND COLONIUS
overestimated in the downstroke phase. Moreover, when this vortex
pair travels downstream, it interacts with the airfoil in the downwind
half of a cycle. This leads to a lift coefficient with large fluctuations
and small mean, as was also observed by Ferreira et al. [17].
We can see from the second and third columns in Fig. 5 that the
Coriolis force deflects the flow around the rotating airfoil in the
clockwise direction. The magnitude of the Coriolis force acting on
the background fluid decreases as the azimuthal angle increases.
Therefore, the Coriolis force acting on the fluid around vortices
becomes relatively important in the downstroke phase. A stronger
Coriolis force is exerted on the fluid around the vortex pair, which
deflects the fluid in such a way that the vortex pair travels with the
airfoil (Figs. 5f–5h).
flows are quite similar, with just a small phase difference when the
airfoils pitch up. They result in comparable lift throughout the
upstroke phase.
When the downstroke phase starts (Figs. 5e and 5f), the
development of a TEV leads to a decrease in lift. The aforementioned
Bloor instability in the shear layer at the trailing edge produces highfrequency fluctuations in the lift coefficient. For EPM, the TEV sheds
into the wake and a secondary vortex [29] appears as the angle of
attack decreases further (Fig. 5g), which results in a sudden increase
in the lift coefficient. On the other hand, for VAWT, as described in
Sec. III.A, this TEV couples with the LEVand forms a vortex pair that
travels together with the airfoil (Figs. 5g and 5h). This generates high
pressure on the suction side and further decreases the lift. This vortex
pair is “captured” by the rotating airfoil. By analogy with flow
observed in insect flight by Dickinson et al. [30], we refer to this
phenomenon as the wake-capturing of a vortex pair in VAWT. The
wake-capturing occurs at a slightly different phase in each cycle and
leads to a significant decrease in the average lift in the downstroke
phase. In general, the lift of an airfoil undergoing the EPM is
C. Comparison with an Airfoil Undergoing a Sinusoidal Motion
Flows around an airfoil undergoing SPM and SSPM introduced in
Sec. II.A are compared with one undergoing EPM and in a VAWT. A
comparison of the lift response at λ 2 and 4, Ro 1.5, and
Re 1000 is shown in Fig. 7.
3
3
3
2
2
2
1
1
1
0
0
0
−1
−1
−1
−2
−2
−2
−3
0
30
60
90
120
150
180
−3
0
30
60
90
120
150
180
−3
0
3
3
3
2
2
2
1
1
1
0
0
0
−1
−1
−1
−2
−2
−2
−3
0
30
60
90
120
150
180
−3
0
30
60
90
120
150
180
−3
0
3
3
3
2
2
2
1
1
1
0
0
0
−1
−1
−1
−2
−2
−2
−3
0
30
60
90
120
150
180
−3
0
30
60
90
120
150
180
−3
0
Ro=0.75, VAWT
Ro=0.75, EPM
Ro=1.00, VAWT
Ro=1.00, EPM
Ro=1.25, VAWT
Ro=1.25, EPM
30
60
90
120
150
180
30
60
90
120
150
180
30
60
90
120
150
180
Fig. 8 Comparing lift coefficients of VAWT and EPM with Ro 0.75, 1.00, and 1.25 at λ 2, 3, and 4 and Re 500, 1000, and 1500.
TSAI AND COLONIUS
considered in this study. However, it overestimates the lift
coefficients in the downstroke phase due to its inability to predict the
wake-capturing phenomenon.
D. Effect of Tip-Speed Ratio, Rossby Number, and Reynolds Number
In this section, the effect of tip-speed ratio, Rossby number, and
Reynolds number on the flow in a VAWT is investigated to
understand when wake-capturing will occur. We compare the
simulations of a rotating wing with a wing undergoing EPM. We
examine the flows at tip-speed ratios λ 2, 3, and 4, and Reynolds
numbers Re 500, 1000, and 1500. The corresponding lift
coefficients with Rossby numbers Ro 0.75, 1.00, and 1.25 are
shown in Fig. 8.
As the tip-speed ratio increases, the amplitudes of angle-of-attack
variation and the corresponding lift decrease. Because the maximum
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At lower tip-speed ratio, λ 2, in the upstroke phase, we can see
that only CL;EPM is close to CL;VAWT at low angle of attack. CL;SPM
and CL;SSPM overestimate the lift due to the overestimation of the
pitch rate. In the downstroke phase, none of CL;EPM , CL;SPM , and
CL;SSPM matches the behavior of CL;VAWT because of the strong effect
on lift of the wake-capturing that occurs in the flows. At higher tipspeed ratio, λ 4, CL;SPM and CL;SSPM still overestimate the lift at the
beginning of the upstroke phase. Nevertheless, as the angle of attack
increases, and after vortex shedding starts, differences between the
four lift coefficients are relatively small. In the downstroke phase,
behaviors of CL;EPM , CL;SPM , and CL;SSPM are close to that of
CL;VAWT due to the low angle of attack.
We can see that, among all simplified motions, an airfoil
undergoing the EPM is the best approximation to a rotating airfoil in a
VAWT in the upstroke phase for the subscale Reynolds numbers
223
Fig. 9 The comparison of the vorticity fields of the LEV-filtered (a–f) and TEV-filtered (g–i) phase-averaged PIV data (gray scale), and of the
corresponding simulations (color scale) at λ 2 and Ro 1.
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224
TSAI AND COLONIUS
angle of attack is slightly above the static stall angle of a NACA 0018
airfoil predicted by Morris and Rusak [5], the lift coefficients of EPM
are close to that of VAWT at λ 4 for all Rossby numbers and
Reynolds numbers examined. Therefore, EPM motion is a good
approximation of VAWT at larger tip-speed ratios due to the low
angle of attack. However, at lower tip-speed ratios, CL;EPM remains
close to CL;VAWT only in the upstroke phase. In the downstroke
phase, the discrepancy in lift coefficients due to the wake-capturing
effect becomes larger as Rossby number decreases and Reynolds
number increases. As the VAWT rotates faster, on the one hand, the
wake-capturing effect is strengthened due to the intensifying Coriolis
force, which corresponds to decreasing Rossby number; on the other
hand, it is attenuated because of the decreasing amplitude of the
angle-of-attack variation due to the increasing tip-speed ratio.
Therefore, the growth of the discrepancy depends subtly on the
increase of the rotating speed of the VAWT.
To probe the existence of wake-capturing at higher Reynolds
numbers, the vorticity field in VAWT (Re 1500) for a single period
is compared with phase-averaged PIV data from Ferreira et al. [17]
(Re ≈ 105 ) at λ 2 and Ro 1 (l 4) in Fig. 9. The contours in
gray are the phase-averaged vorticity field taken from the
experiments. In Figs. 9a–9f, their phase-averaged field was filtered
to plot only the LEV generated around θ 72° and the plot represents
a composite of overlaid fields from various azimuthal angles.
Similarly, in Figs. 9g–9i, the contours in the gray scale show the
filtered, phase-averaged TEV evolution. To make qualitative
comparisons, our vorticity fields at the corresponding azimuthal
angles are overlaid in the color scale on top of the results from the
experiments. Our blue contours correspond to negative vorticity,
which should be compared with the LEV-filtered PIV data in
Figs. 9a–9f, while the red contours correspond to positive vorticity
that should be compared with the TEV-filtered PIV data in
Figs. 9g–9i.
The trajectories of the LEVand TEV from Ferreira et al. [17] seem
to be reasonably captured by the simulation in the upwind half of a
cycle. The disagreement in Figs. 9f and 9i may be due to strong
vortex–blade and vortex–vortex interactions in the downwind half of
a cycle. An LEV is generated around θ 72∘ and wake-capturing
occurs around θ 90° , which forms a vortex pair traveling with the
blade (Figs. 9a–9c). The vortex pair then detaches around θ 133°
and propagates downstream (Figs. 9d–9f). The location of the vortex
pair composed of the phase-averaged LEV and TEV agrees with the
current simulation, especially at θ 158° (Figs. 9e and 9h). The
qualitative agreement in the upwind half of a cycle suggests that
wake-capturing may also be occurring in Ferreira et al. [17]
experiment.
E. Decoupling the Effect of Surging, Pitching, and Rotation
The flow around a rotating airfoil in a VAWT is complicated not
only by the Coriolis effect but also because the angle of attack and
incoming velocity vary simultaneously. It is interesting to understand
whether the Coriolis effect has strong coupling with the angle-of-
1.5
VAWT
Equiv. motion
3
VAWT
Equiv. motion
2.5
2
1.5
1
0.5
0
−0.5
−1
0
30
60
90
120
150
Fig. 11 Comparing lift responses of airfoils undergoing only pitching
motion at λ 2, Ro 1.5, and Re 1500.
attack or incoming velocity variations. Therefore, we independently
examine airfoils undergoing the decoupled pitching and surging
motion associated with the EPM.
1. Airfoil Undergoing Only Surging Motion
We examine a surging motion with fixed angles of attack of 15°
and 30° at λ 2, Ro 1.5 (k 1∕6), and Re 1500. A rotating
airfoil undergoing only the surging motion of a VAWT is achieved by
pitching the airfoil around the leading edge simultaneously as it
rotates so that the angle of attack is fixed with respect to the incoming
velocity. For an airfoil surging at an angle of attack of 15°, lift
coefficients are shown in Fig. 10a. We can see that dynamic stall is
relatively stable and no wake-capturing phenomenon is observed.
Moreover, from the analysis by Choi et al. [7], when the reduced
frequency is low enough, the flow can be approximated as quasisteady, which results in both lift coefficients for VAWT and EPM
fluctuating about a slowly increasing mean value. For the case of
α 30°, lift coefficients are shown in Fig. 10b. The flow is well
separated so that there is no stationary vortex shedding. Moreover, no
wake-capturing phenomenon is observed in the flow.
2. Airfoil Undergoing Only Pitching Motion
We consider a pitching motion in a freestream velocity W avg λU∞ at λ 2, Ro 1.5, and Re 1500. A rotating airfoil
undergoing only the pitching motion in a VAWT is achieved by
rotating an airfoil in a VAWT without the external freestream and
simultaneously pitching it around the leading edge with the exact
angle-of-attack variation. The corresponding lift coefficients are
shown in Fig. 11. We can see dynamic stall in both lift coefficients as
angle of attack increases. However, there is no wake-capturing.
The wake-capturing effect is therefore only present when pitching,
surging, and the Coriolis force are all present.
3.5
VAWT
Equiv. motion
3
2.5
1
2
1.5
0.5
1
0.5
0
0
30
60
90
120
150
180
180
0
0
30
60
90
120
150
180
b) α = 30°
a) α = 15°
Fig. 10 Comparing lift responses of airfoils undergoing only surging motion at λ 2, Ro 1.5, and Re 1500.
TSAI AND COLONIUS
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IV.
Conclusions
In simulating the flow around a single-bladed vertical axis wind
turbine (VAWT), an interesting wake-capturing phenomenon that
occurs during the pitch-down portion of the upstream, lift-generating
portion of the VAWT cycle was observed. This phenomenon leads to
a substantial decrease in lift coefficient due to the presence of a vortex
pair traveling together with the rotating airfoil. Our results show that
this flow feature persists and grows stronger as tip-speed ratio and
Rossby number are reduced and Reynolds number is increased.
Therefore, the growth of this features depends subtly on the increase
of the rotating speed of the VAWT, which, on the one hand, is
strengthened due to the intensifying Coriolis force. On the other
hand, it is attenuated because of the decreasing amplitude of the
angle-of-attack variation. Moreover, although our study is restricted
to 2-D flow at relatively low Reynolds numbers, the qualitative
agreement of the leading edge vortex and trailing edge vortex
evolutions with Ferreira et al. [17] experiment suggests that this
feature may persist in real applications. The corresponding decrease
in efficiency could be improved by implementing flow control (e.g.,
blowing) to remove this flow feature [31].
An equivalent planar surging–pitching motion was introduced in
order to isolate the Coriolis effect on dynamic stall in a VAWT.
Simplified planar motions consisting of sinusoidally varying pitch
and surge were also examined. Except at the beginning of the pitch-up
motion, all of the simplified motions are good approximations to
VAWT motion at sufficiently high tip-speed ratios because the
corresponding maximum angle of attack is close to or lower than the
stall angle of the blade. However, at low tip-speed ratios, while the
equivalent planar motion captures the pitch-up part of the cycle, all
the motions show significant differences in forces during the pitchdown motion. The results show that the equivalent motion is a good
approximation to a rotating airfoil in a VAWT in the upstroke phase
where the Coriolis force has a relatively small effect on vortices.
However, it overestimates the average lift coefficient in the
downstroke phase by eliminating the aforementioned wakecapturing.
The flow by decomposing the planar motion into surging- and
pitching-only motions was further investigated. Wake-capturing was
observed only when the combination of surging, pitching, and rotation
are present, which suggests that this feature is associated with an
unique combination of angle-of-attack variation, instantaneous
velocity variation, and the Coriolis effect.
Acknowledgments
This project is sponsored by the Caltech Field Laboratory for
Optimized Wind Energy with John Dabiri as principal investigator
under the support of the Gordon and Betty Moore Foundation. We
would like to thank John Dabiri, Beverley McKeon, Reeve Dunne,
and Daniel Araya for their helpful comments on our work. The
parametric study in this work used the Extreme Science and
Engineering Discovery Environment (XSEDE), which is supported
by National Science Foundation grant number ACI-1053575.
[5]
[6]
[7]
[8]
[9]
[10]
[11]
[12]
[13]
[14]
[15]
[16]
[17]
[18]
[19]
[20]
References
[1] Mertens, S., van Kuik, G., and van Bussel, G., “Performance of an HDarrieus in the Skewed Flow on a Roof,” Journal of Solar Energy
Engineering, Vol. 125, No. 4, 2003, pp. 433–440.
doi:10.1115/1.1629309
[2] Ferreira, C. S., van Bussel, G. J., and van Kuik, G. A., “Wind Tunnel
Hotwire Measurements, Flow Visualization and Thrust Measurement of
a VAWT in Skew,” Journal of Solar Energy Engineering, Vol. 128,
No. 4, 2006, pp. 487–497.
doi:10.1115/1.2349550
[3] Dabiri, J. O., “Potential Order-of-Magnitude Enhancement of Wind
Farm Power Density via Counter-Rotating Vertical-Axis Wind Turbine
Arrays,” Journal of Renewable and Sustainable Energy, Vol. 3, No. 4,
2011, Paper 043104.
doi:10.1063/1.3608170
[4] Kinzel, M., Mulligan, Q., and Dabiri, J. O., “Energy Exchange in an
Array of Vertical-Axis Wind Turbines,” Journal of Turbulence, Vol. 13,
[21]
[22]
[23]
[24]
225
No. 38, 2012, pp. 1–13.
doi:10.1080/14685248.2012.712698
Morris, W. J., and Rusak, Z., “Stall Onset on Aerofoils at Low to
Moderately High Reynolds Number Flows,” Journal of Fluid
Mechanics, Vol. 733, Oct. 2013, pp. 439–472.
doi:10.1017/jfm.2013.440
Taira, K., and Colonius, T., “Three-Dimensional Flows Around LowAspect-Ratio Flat-Plate Wings at Low Reynolds Numbers,” Journal of
Fluid Mechanics, Vol. 623, March 2009, pp. 187–207.
doi:10.1017/S0022112008005314
Choi, J., Colonius, T., and Williams, D. R., “Surging and Plunging
Oscillations of an Airfoil at Low Reynolds Number,” Journal of Fluid
Mechanics, Vol. 763, Jan. 2015, pp. 237–253.
doi:10.1017/jfm.2014.674
Corke, T. C., and Thomas, F. O., “Dynamic Stall in Pitching Airfoils:
Aerodynamic Damping and Compressibility Effects,” Annual Review of
Fluid Mechanics, Vol. 47, No. 1, 2015, pp. 479–505.
doi:10.1146/annurev-fluid-010814-013632
Ferreira, C. S., van Kuik, G., van Bussel, G., and Scarano, F.,
“Visualization by PIV of Dynamic Stall on a Vertical Axis Wind
Turbine,” Experiments in Fluids, Vol. 46, No. 1, 2009, pp. 97–108.
doi:10.1007/s00348-008-0543-z
Wang, S., Ingham, D. B., Ma, L., Pourkashanian, M., and Tao, Z.,
“Numerical Investigations on Dynamic Stall of Low Reynolds Number
Flow Around Oscillating Airfoils,” Computers & Fluids, Vol. 39, No. 9,
2010, pp. 1529–1541.
doi:10.1016/j.compfluid.2010.05.004
Lee, T., and Gerontakos, P., “Investigation of Flow over an Oscillating
Airfoil,” Journal of Fluid Mechanics, Vol. 512, Aug. 2004, pp. 313–341.
doi:10.1017/S0022112004009851
Paraschivoiu, I., and Allet, A., “Aerodynamic Analysis of the Darrieus
Wind Turbines Including Dynamic-Stall Effects,” Journal of Propulsion
and Power, Vol. 4, No. 5, 1988, pp. 472–477.
doi:10.2514/3.23090
Allet, A., Hallé, S., and Paraschivoiu, I., “Numerical Simulation of
Dynamic Stall Around an Airfoil in Darrieus Motion,” Journal of Solar
Energy Engineering, Vol. 121, No. 1, 1999, pp. 69–76.
doi:10.1115/1.2888145
Paraschivoiu, I., and Beguier, C., “Visualization, Measurements and
Calculations of Dynamic Stall for a Similar Motion of VAWT,”
Proceedings of the European Community Wind Energy Conference,
Commission of the European Communities, Luxembourg, 1998,
pp. 197–201.
Hansen, M., and Søresen, D., “CFD Model for Vertical Axis Wind
Turbine,” Proceedings of the 2001 European Wind Energy Conference
and Exhibition, WIP–Renewable Energies, Munich, 2001.
Nobile, R., Vahdati, M., Barlow, J., and Mewburn-Crook, A., Dynamic
Stall for a Vertical Axis Wind Turbine in a Two-Dimensional Study,”
World Renewable Energy Congress, Linköping, Sweden, 2011.
Ferreira, C. S., van Bussel, G., and Van Kuik, G., “2-D CFD Simulation
of Dynamic Stall on a Vertical Axis Wind Turbine: Verification and
Validation with PIV Measurements,” 45th AIAA Aerospace Sciences
Meeting and Exhibit, AIAA Paper 2007-1367, 2007.
Duraisamy, K., and Lakshminarayan, V., “Flow Physics and
Performance of Vertical Axis Wind Turbine Arrays,” 32nd AIAA
Applied Aerodynamics Conference, AIAA Paper 2014-3139, 2014.
Barsky, D., Posa, A., Rahromostaqim, M., Leftwich, M. C., and Balaras,
E., “Experimental and Computational Wake Characterization of a
Vertical Axis Wind Turbine,” 32nd AIAA Applied Aerodynamics
Conference, AIAA Paper 2014-3141, 2014.
Taira, K., and Colonius, T., “The Immersed Boundary Method: A
Projection Approach,” Journal of Computational Physics, Vol. 225,
No. 2, 2007, pp. 2118–2137.
doi:10.1016/j.jcp.2007.03.005
Colonius, T., and Taira, K., “A Fast Immersed Boundary Method Using a
Nullspace Approach and Multi-Domain Far-Field Boundary Conditions,” Computer Methods in Applied Mechanics and Engineering,
Vol. 197, No. 25, 2008, pp. 2131–2146.
doi:10.1016/j.cma.2007.08.014
Taira, K., and Colonius, T., “Effect of Tip Vortices in Low-ReynoldsNumber Poststall Flow Control,” AIAA Journal, Vol. 47, No. 3, 2009,
pp. 749–756.
doi:10.2514/1.40615
Joe, W. T., Colonius, T., and MacMynowski, D. G., “Optimized Control
of Vortex Shedding from an Inclined Flat Plate,” AIAA Paper 20094027, 2009.
Joe, W. T., Colonius, T., and MacMynowski, D. G., “Optimized
Waveforms for Feedback Control of Vortex Shedding,” Active Flow
Control II, edited by King, R., Vol. 108, Notes on Numerical Fluid
226
[25]
[26]
[27]
Downloaded by CALIFORNIA INST OF TECHNOLOGY on June 22, 2016 | http://arc.aiaa.org | DOI: 10.2514/1.J054199
[28]
TSAI AND COLONIUS
Mechanics and Multidisciplinary Design, Springer–Verlag, Berlin,
2010, pp. 391–404.
Olsman, W. F. J., Willems, J., Hirschberg, A., Colonius, T., and Trieling,
R., “Flow Around a NACA0018 Airfoil with a Cavity and Its Dynamical
Response to Acoustic Forcing,” Experiments in Fluids, Vol. 51, No. 2,
2011, pp. 493–509.
doi:10.1007/s00348-011-1065-7
Olsman, W. F. J., and Colonius, T., “Numerical Simulation of Flow over an
Airfoil with a Cavity,” AIAA Journal, Vol. 49, No. 1, 2011, pp. 143–149.
doi:10.2514/1.J050542
Bloor, M. S., “The Transition to Turbulence in the Wake of a Circular
Cylinder,” Journal of Fluid Mechanics, Vol. 19, No. 02, 1964, pp. 290–
304.
doi:10.1017/S0022112064000726
Lissaman, P., “Low-Reynolds-Number Airfoils,” Annual Review of
Fluid Mechanics, Vol. 15, No. 1, 1983, pp. 223–239.
doi:10.1146/annurev.fl.15.010183.001255
[29] McCroskey, W., Carr, L., and McAlister, K., “Dynamic Stall
Experiments on Oscillating Airfoils,” AIAA Journal, Vol. 14, No. 1,
1976, pp. 57–63.
doi:10.2514/3.61332
[30] Dickinson, M. H., Lehmann, F.-O., and Sane, S. P., “Wing Rotation and
the Aerodynamic Basis of Insect Flight,” Science, Vol. 284, No. 5422,
1999, pp. 1954–1960.
doi:10.1126/science.284.5422.1954
[31] Tsai, H.-C., and Colonius, T., “Coriolis Effect on Dynamic Stall in a
Vertical Axis Wind Turbine at Moderate Reynolds Number,” 32nd AIAA
Applied Aerodynamics Conference, AIAA Paper 2014-3140, 2014.
Z. Rusak
Associate Editor