-
Coriolis Effect on Dynamic Stall in a VerticalAxis Wind
Turbine
Hsieh-Chen Tsai∗ and Tim Colonius†
California Institute of Technology, Pasadena, California
91125
DOI: 10.2514/1.J054199
The immersed boundary method is used to simulate the flow around
a two-dimensional cross section of a rotating
NACA0018 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,
theCoriolis force leads to the capture of a vortex pairwhen 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.
Nomenclature
CL = lift coefficientc = airfoil chord lengthE�m� = complete
elliptic integral of the second kindk = reduced frequencyp =
pressureR = radius of the turbineRe = Reynolds NumberRo = Rossby
NumberU∞ = freestream velocityû = velocity of the fluid in the
rotating frame of referenceu 0 = velocity introduced by the change
of variablesW = incoming velocityx̂ = position vector in the
rotating frame of referenceα = angle of attack_α = pitch rateδ =
spatial distribution of the body-force actuationλ = tip-speed
ratioν = fluid kinematic viscosityρ = fluid densityθ = azimuthal
angleω̂ = vorticity of the fluid in the rotating frame of
referenceω 0 = vorticity introduced by the change of variablesΔt =
time stepΔx = grid spacingΩ = angular velocity of the turbinel =
ratio of the radius of the turbine to the chord length
Subscripts
avg = average velocityEPM = equivalent planar motioninst =
instantaneous velocity
max = maximumSPM = sinusoidal pitching motionSSPM = sinusoidal
surging–pitching motionsin = sinusoidal variationsurge = surge
velocityVAWT = vertical axis wind turbine
I. Introduction
V ERTICAL axis wind turbines (VAWT) offer several advantagesover
horizontal axis wind turbines (HAWT), namely: their lowsound
emission (consequence of their operation at lower tip-speedratios),
their insensitivity to yaw wind direction (because they
areomnidirectional), and their increased power output in skewed
flow[1,2]. Dabiri et al. [3,4] showed that an array of
counterrotatingVAWTs can achieve higher power output per unit land
area andsmaller wind velocity recovery distance than existing wind
farmsconsisting ofHAWTs. The aerodynamics of VAWTs are
complicatedby inherently unsteady flow produced by the large
variations in bothangle of attack and incident velocity magnitude
of the blades, whichcan be characterized as a function of tip-speed
ratio. Typically,commercial VAWTs operate at a tip-speed ratio
around 2–5, whichproduces an angle-of-attack variation with
amplitude of 11.5–30°and an incident velocity variation with
amplitude of 21.5–49% ofits mean.Aerodynamics of wings at low
Reynolds numbers have been well
investigated due to the recent interest in the development of
smallunmanned aerial vehicles and micro air vehicles. Morris and
Rusak[5] studied the onset of stall at low to moderately high
Reynoldsnumber flows numerically and provided a universal criterion
todetermine the static stall angle of thin airfoils. Taira and
Colonius [6]simulated three-dimensional flows around
low-aspect-ratio flat-platewings at low Reynolds numbers, with a
focus on the unsteady vortexdynamics at poststall angles of attack.
Choi et al. [7] numericallyinvestigated unsteady, separated flows
around two-dimensional (2-D) 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 witha substantially higher lift than is obtained
quasi-statically, and hasbeen an active research topic in fluid
dynamics for more than 60years, largely because of the helicopter
application [8]. Because oflarge variations in angle of attack,
dynamic stall occurs on VAWToperating 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
pitchingmotion that produces an
Presented as Paper 2014-3140 at the 32nd AIAA Applied
AerodynamicsConference, 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 internaluse, on
condition that the copier pay the $10.00 per-copy fee to the
CopyrightClearance Center, Inc., 222 Rosewood Drive, Danvers, MA
01923; includethe 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
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equivalent speed and angle-of-attack variation over the cycle.
Theyfurther simplified the EPM to a sinusoidal pitching motion
(SPM) toinvestigate dynamic stall in a 2-D VAWT numerically. The
resultsmatched the experiments done by Lee and Gerontakos
[11].Several attempts have been made to model a VAWT at
Re ∼O�105�, which is appropriate to the urban applications
ofVAWTs. Reynolds-averaged Navier–Stokes (RANS) with
differentturbulence models has been applied to a 2-D airfoil
undergoing aneffective planar motion [12–14] and to a multibladed
2-D VAWT[15,16]. Ferreira et al. [17] simulated dynamic stall in a
section of aVAWT using detached-eddy simulation at Re � 50; 000
andvalidated the results by comparing the vorticity in the rotor
area withparticle image velocimetry (PIV) data. Duraisamy and
Lakshminar-ayan [18] numerically analyzed interactions of VAWTs
with variousconfigurations using RANS at Re � 67; 000. Barsky et
al. [19]investigated the fundamental wake structure of a single
VAWTcomputationally by large-eddy simulation and experimentallyby
PIV.In this paper, a 2-D VAWT is investigated numerically at
low
Reynolds numbers in order to understand qualitative features of
theflow field in a setting where a comparatively large region
ofparameter space can be explored than could be for full-scale,
three-dimensional computations. To explore the parameter space
inrelatively short computational time and have more understanding
ofthe details of the vortex dynamics, flows are simulated at
lowReynolds numbers, Re ∼O�103�. A major limitation of our
presentapproach is the restriction of flow to a 2-D cross section
of anotherwise planar turbine geometry. Comparisons with Ferreira
et al.[17] show qualitative agreement, but a precise accounting for
three-dimensional effects awaits future simulations.We focus here
on theCoriolis effect on dynamic stall by comparing
the rotating airfoil to one undergoing an EPM. The influence
ofdynamic stall on forces is characterized as a function of
tip-speedratio and Rossby number. Moreover, inspired by Wang et al.
[10],airfoils undergoing an SPM and a sinusoidal
surging–pitchingmotion (SSPM) are also compared to see if these two
motions can bean appropriatemodel for theVAWT. Furthermore, the
coupling of theCoriolis effect with the angle-of-attack and
incoming velocityvariations is also examined.
II. Methodology
A. Simulation Setup
Figure 1 shows a schematic of a VAWTwith radiusR rotating at
anangular velocity Ω with a freestream velocity U∞, coming from
theleft. The chord length of the turbine blade is c. To
systematicallyinvestigate the aerodynamics of a VAWT, four
dimensionlessparameters are introduced:
Tip-speed ratio∶ λ � ΩRU∞
(1)
Radius-to-chord-length ratio∶ l � Rc
(2)
Renolds number∶ Re � U∞cν
(3)
Rossby number∶ Ro � U∞2Ωc� l
2λ� 1
4k(4)
where ν is the kinematic viscosity of the fluid and k is the
reducefrequency, k � Ωc∕2U∞ � λ∕2l.The instantaneous incoming
velocityWinst and the angle of attack
α can then be characterized as a function of the tip-speed ratio
λ andthe azimuthal angle θ:
α�λ; θ� � tan−1�
sin θ
λ� cos θ
�(5)
Winst�λ; θ�U∞
���������������������������������������1� 2λ cos θ� λ2
p(6)
Figure 2 shows the angle-of-attack variation and
incomingvelocity variation of the VAWTat λ � 2. FromEq. (5),
themaximumangle of attack
αmax�λ� � tan−1�
1�������������λ2 − 1p
�(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-attackvariation over a cycle is proposed. This EPM is
composed of asurging motion with a velocityWsurge and a pitching
motion aroundthe leading edgewith a pitch rate _α. The airfoil is
undergoing theEPMin a freestream velocityWavg.Wavg,Wsurge, and _α
are shown to be
Wavg�λ�U∞
� 12π
Z2π
0
Winst�λ;θ�U∞
dθ�2�1�λ�π
E
����������������4λ
�1�λ�2
s !(8)
where the function E�m� � ∫ π∕20���������������������������1 −m2
sin2 θp
dθ is the completeelliptic integral of the second kind,
Wsurge�λ; θ� � Winst�λ; θ� −Wavg�λ� (9)
_α�λ; θ� � 12Ro
�1� λ cos θ
1� 2λ cos θ� λ2�
(10)
Moreover, due to the periodic oscillation of angle-of-attack
andincoming velocity variation,Wang et al. [10] studied dynamic
stall ina VAWT by investigating an airfoil undergoing a simplified
SPM.Inspired by their work, sinusoidal variations in the angle of
attack andincoming velocity, which are written as a function of the
tip-speedratio λ and the azimuthal angle θ in Eqs. (11) and (12)
and shown inFig. 2, are also considered:
αsin �λ; θ� � αmax�λ� sin θ (11)
Winst;sin�λ; θ�U∞
� λ� cos θ (12)Fig. 1 Schematic of a VAWT and the computational
domain.
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Although the sinusoidal motion shares the same amplitude,
itoverestimates the angle of attack in the upstroke
phase,underestimates the incoming velocity in the downstroke phase,
andslightly underestimates the instantaneous velocity over the
entire halfcycle. To search for the most appropriate model for a
VAWT, weintroduce two additional motions: SPM and SSPM.
Airfoilsundergoing both the SPM and SSPM pitch with the sinusoidal
angle-of-attack variation described in Eq. (11) in a freestream
velocityWavg;sin � λU∞. Airfoils undergoing the SSPM also surge
with avelocityWsurge;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
onthe choice of the tip-speed ratio λ and the Rossby number Ro.In
preliminary simulations of a three-bladed VAWT, as well as
inprevious studies [17], vorticity–blade interaction is only
observed inthe downwind half of a cycle. Because only the flow in
the upwind-half cycle is important to torque generation, we save
computationaltime by modeling a single-bladed turbine. We compute
about fiveperiods of motion, which is equal to 5π∕Ro convective
time units, toremove transients associated with the startup of
periodic motion. Forthe largest Rossby number we examined, the
starting vortexpropagates far enough into thewake to have an
insignificant effect onthe forces on the blades after five periods.
An additional five periodsof nearly periodic stationary-state
motion were then computed andanalyzed below.
B. Numerical Method
The immersed boundary projection method (IBPM) developed
byColonius and Taira [20,21] is used to compute 2-D
incompressibleflows in an airfoil-fixed reference framewith
appropriate forces addedto the momentum equation to account for the
noninertial referenceframe. The equations are solved on multiple
overlapping grids thatbecome progressively coarser and larger
(greater extent). Thedimensionlessmomentumequation in the rotating
frameof reference is
∂û∂t� �û · ∇�û � −∇p� 1
Re∇2û −
dΩdt
× x̂ − 2Ω × û −Ω
× �Ω × x̂� (13)
where û and x̂ are the fluid velocity and the position vector
in therotating frame of reference and Ω � 1∕2Ro is the
dimensionlessangular velocity of rotating frame of reference. We
then introduce thechange of variables
u 0 � û�Ω × x̂ (14)
ω 0 � ω̂� 2Ω (15)
where ω̂ � ∇ × û is the vorticity field in the rotating frame
ofreference. By taking the divergence and the curl of Eq. (13), we
havethe following vorticity and pressure equation
∂ω 0
∂t� ∇ × �̂u × ω 0� − 1
Re∇ × �∇ ×ω 0� (16)
∇2�p� 1
2jûj2 − 1
2jΩ × x̂j2
�� ∇ · �û ×ω 0� (17)
Because the flow is incompressible and 2-D, the first term on
theright-hand side of Eq. (16) is just the advection of vorticity
withvelocity û. Therefore, in the body-fixed frame of reference,
theCoriolisforce does not generate vorticity except on the boundary
so that it onlychanges theway vorticity propagates in free space.
Moreover, becausethe magnitude of Coriolis force is proportional to
the magnitude ofvelocity, 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
flowsaround low-aspect-ratio flat-plate wings [6,22], optimized
control ofvortex shedding from an inclined flat plate [23,24], 2-D
flows arounda NACA0018 airfoil with a cavity [25,26], and 2-D flows
aroundsurging 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
andTaira [21], the IBPM is estimated to have an O�4−Ng�
convergencerate, whereNg is the number of the grid levels. Six grid
levels are usedfor the present computations in order to have the
leading-order errordominated by the truncation error arising from
the discrete deltafunctions at the immersed boundary and the
discretization of thePoisson equation. The coarsest grid extends to
96 chord lengths inboth 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
streamwisedirection is compared with one on the finest grid withΔx
� 0.00125at t � 1. Figure 3 shows the spatial convergence in the L2
norm. Therate of decay for the spatial error is about 1.5, which
agrees with Tairaand Colonius [20]. All ensuing computations use a
600 × 600 grid,which corresponds to Δx � 0.005. The time step Δt is
chosen tomake Courant–Friedrichs–Lewy number less than 0.4.
III. Results
A. Qualitative Flow Features in a VAWT
We begin by examining flows at low tip-speed ratio, λ � 2,Ro �
1.5, and Re � 1500, which gives a maximum amplitude of30° in
angle-of-attack variation and a reduced frequency k � 1∕6.Figure 4
shows the vorticity field generated by the blade at
differentazimuthal angles over a cycle. Negative and positive
vorticity are
0 60 120 180 240 300 360
−30
−20
−10
0
10
20
30 V.A.W.T.sinusoidal motion
a) Angle of attack variation
0 60 120 180 240 300 3600
0.5
1
1.5
2
2.5
3
3.5
4V.A.W.T.sinusoidal motion
b) Incoming velocity variationFig. 2 Comparison of
angle-of-attack variation and incoming velocity variation between
the VAWT and the sinusoidal motion at λ � 2.
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plotted in blue and red contour levels, respectively, and all
vorticitycontour 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 vortexshedding in the wake. The flow reattaches by the time
airfoil reachesα� 5∘ (Fig. 4b). When the angle of attack increases
further, the wakebehind the airfoil starts to oscillate and vortex
shedding commences.Dynamic stall then takes place, and is marked by
the growth, pinch-off, and advection of a leading edge vortex (LEV)
on the suction sideof the airfoil (Figs. 4c–4e). The vortices
generated will propagatedownstream into the wake of the VAWTor
interact with the blades inthe 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
trailing-edge shear layer at this Reynolds number, which resembles
theconvectively unstable Kelvin–Helmholtz instability observed
inplane mixing layers. This TEV couples with an LEV to form a
vortexpair that travels downstream together with the airfoil (Figs.
4g–4i).
10−3
10−2
10−3
10−2
10−1
Fig. 3 The L2 norm of the error of the velocity field in the
streamwisedirection in a single-bladedVAWTwithl � 1.5 rotating at λ
� 2 at t � 1.
Fig. 4 Vorticity field for a (clockwise rotating) VAWT at
various azimuthal angles at λ � 2, Ro � 1.5, and Re � 1500.
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This vortex pair interacts with the airfoil in the downwind half
of acycle (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 nowgenerated on the other
sideof the airfoil and shed into thewake of the VAWT (Figs. 4j–4p).
For amultibladedVAWT,when a blade is traveling in the downwind half
ofa cycle, it interacts with vortices generated upstream from
otherblades or from the wake it generated at an earlier time (Fig.
4o).
B. Comparison of VAWT and EPM
In this section, we compare flows around an airfoil undergoing
theEPM and a single-bladed VAWT at λ � 2, Ro � 1.5, andRe � 1500.
We are interested in the tangential force response of theblade over
a cycle because the power output is proportional to thetangential
force acting on the blade when VAWTs operate at aconstant tip-speed
ratio. The tangential force acting on the blade canbe written as a
linear combination of lift and drag:
Fig. 5 Vorticity field for EPM and VAWT and the Coriolis force
for VAWT at various azimuthal angles.
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CT � CL sin α − CD cos α � CL�sin α −
1
CL∕CDcos α
�(18)
where α is the angle of attack of the blade. From
preliminarysimulations, a three-bladed VAWT with l � 4 will be
free-spinningwith a time-averaged tip-speed ratio λ � 0.95 atRe �
1500, so that inthe flowweare examining the average tangential
force is expected tobenegative. However, as the Reynolds number
increases to the rangewhere commercial VAWTs usually operate,Re
∼O�105 − 106�, dragcoefficient drops dramatically while the change
in the lift coefficient issmall. This leads to a large increase in
the lift-to-drag ratioCL∕CD [28].Therefore, the contribution of
lift to the tangential force dominates athigh Reynolds
numbers.Moreover, the power of a VAWT is generatedmostly in the
upwind half cycle because large vorticity–bladeinteractions cancel
out the driving torque in the downwind half cycle[17]. Therefore,
in this studywewill focus on the lift in the upwind halfof a
cycle.
Figure 5 shows the comparison of VAWTand EPM motion in
thesurging–pitching configuration. Negative and positive vorticity
areplotted in blue and red contour levels, respectively, and all
vorticitycontour plots are using the same contour levels. In the
Coriolis forceplots, black arrows show the direction of velocity,
blue arrows pointthe direction of the Coriolis force, and the color
contour plots themagnitude of the Coriolis force. Since the frame
of reference isrotating clockwise, the Coriolis force deflects the
fluid in theclockwise direction. Figure 6 shows a comparison of the
liftcoefficient against dimensionless time and angle of attack for
a singlerotation and for the average of both lift coefficients over
five cycles.Although there are still the remnants of earlier vortex
shedding in thewakewhen the airfoil just returns to zero angle of
attack (Fig. 5a), theflow reattaches by α � 5° (Fig. 5b), which
leads to a smoothlyincreasing lift coefficient at low angle of
attack. The differences in thelift coefficient between the EPM
andVAWTare small (Fig. 6). As theangle of attack increases, dynamic
stall commences (Figs. 5c–5e),which leads to rapidly increasing
lift. EPM- and VAWT-induced
0 30 60 90 120 150 180
−2
−1
0
1
2
3VAWTEPM
a) The lift coefficients over a cycle against dimensionless
time.
0 5 10 15 20 25 30
−2
−1
0
1
2
3VAWTEPM
b) The lift coefficients over a cycle against angle of
attack.
0 30 60 90 120 150 180
−2
−1
0
1
2
3VAWTEPM
c)The average lift coefficients over five cycles.
Fig. 6 Comparing CL;VAWT and CL;EPM at λ � 2, Ro � 1.5, and Re �
1500.
0 5 10 15 20 25 30−2
−1
0
1
2
3VAWTEPMSPMSSPM
0 5 10 15 20 25 30−2
−1
0
1
2
3VAWTEPMSPMSSPM
0 2 4 6 8 10 12 14
−0.5
0
0.5
1VAWTEPMSPMSSPM
0 2 4 6 8 10 12 14
−0.5
0
0.5
1VAWTEPMSPMSSPM
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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flows are quite similar, with just a small phase difference when
theairfoils pitch up. They result in comparable lift throughout
theupstroke phase.When the downstroke phase starts (Figs. 5e and
5f), the
development of a TEV leads to a decrease in lift. The
aforementionedBloor instability in the shear layer at the trailing
edge produces high-frequency fluctuations in the lift coefficient.
For EPM, the TEV shedsinto the wake and a secondary vortex [29]
appears as the angle ofattack decreases further (Fig. 5g), which
results in a sudden increasein the lift coefficient. On the other
hand, for VAWT, as described inSec. III.A, this TEV coupleswith the
LEVand forms a vortex pair thattravels together with the airfoil
(Figs. 5g and 5h). This generates highpressure on the suction side
and further decreases the lift. This vortexpair is “captured” by
the rotating airfoil. By analogy with flowobserved in insect flight
by Dickinson et al. [30], we refer to thisphenomenon as the
wake-capturing of a vortex pair in VAWT. Thewake-capturing occurs
at a slightly different phase in each cycle andleads to a
significant decrease in the average lift in the downstrokephase. In
general, the lift of an airfoil undergoing the EPM is
overestimated in the downstroke phase. Moreover, when this
vortexpair travels downstream, it interacts with the airfoil in the
downwindhalf of a cycle. This leads to a lift coefficient with
large fluctuationsand 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
theclockwise direction. The magnitude of the Coriolis force acting
onthe background fluid decreases as the azimuthal angle
increases.Therefore, the Coriolis force acting on the fluid around
vorticesbecomes relatively important in the downstroke phase. A
strongerCoriolis force is exerted on the fluid around the vortex
pair, whichdeflects the fluid in such a way that the vortex pair
travels with theairfoil (Figs. 5f–5h).
C. Comparison with an Airfoil Undergoing a Sinusoidal Motion
Flows around an airfoil undergoing SPM and SSPM introduced
inSec. II.A are compared with one undergoing EPM and in a VAWT.
Acomparison of the lift response at λ � 2 and 4, Ro � 1.5, andRe �
1000 is shown in Fig. 7.
0 30 60 90 120 150 180−3
−2
−1
0
1
2
3
0 30 60 90 120 150 180−3
−2
−1
0
1
2
3
0 30 60 90 120 150 180−3
−2
−1
0
1
2
3
Ro=0.75, VAWTRo=0.75, EPMRo=1.00, VAWTRo=1.00, EPMRo=1.25,
VAWTRo=1.25, EPM
0 30 60 90 120 150 180−3
−2
−1
0
1
2
3
0 30 60 90 120 150 180−3
−2
−1
0
1
2
3
0 30 60 90 120 150 180−3
−2
−1
0
1
2
3
0 30 60 90 120 150 180−3
−2
−1
0
1
2
3
0 30 60 90 120 150 180−3
−2
−1
0
1
2
3
0 30 60 90 120 150 180−3
−2
−1
0
1
2
3
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.
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At lower tip-speed ratio, λ � 2, in the upstroke phase, we can
seethat only CL;EPM is close to CL;VAWT at low angle of attack.
CL;SPMand CL;SSPM overestimate the lift due to the overestimation
of thepitch rate. In the downstroke phase, none of CL;EPM, CL;SPM,
andCL;SSPMmatches the behavior ofCL;VAWT because of the strong
effecton lift of the wake-capturing that occurs in the flows. At
higher tip-speed ratio, λ � 4,CL;SPM andCL;SSPM still overestimate
the lift at thebeginning of the upstroke phase. Nevertheless, as
the angle of attackincreases, and after vortex shedding starts,
differences between thefour lift coefficients are relatively small.
In the downstroke phase,behaviors of CL;EPM, CL;SPM, and CL;SSPM
are close to that ofCL;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 aVAWT in the upstroke phase for the subscale Reynolds
numbers
considered in this study. However, it overestimates the
liftcoefficients in the downstroke phase due to its inability to
predict thewake-capturing phenomenon.
D. Effect of Tip-Speed Ratio, Rossby Number, and Reynolds
Number
In this section, the effect of tip-speed ratio, Rossby number,
andReynolds number on the flow in a VAWT is investigated
tounderstand when wake-capturing will occur. We compare
thesimulations of a rotating wing with a wing undergoing EPM.
Weexamine the flows at tip-speed ratios λ � 2, 3, and 4, and
Reynoldsnumbers Re � 500, 1000, and 1500. The corresponding
liftcoefficients with Rossby numbers Ro � 0.75, 1.00, and 1.25
areshown in Fig. 8.As the tip-speed ratio increases, the amplitudes
of angle-of-attack
variation and the corresponding lift decrease. Because the
maximum
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 thecorresponding simulations (color scale) at
λ � 2 and Ro � 1.
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angle of attack is slightly above the static stall angle of a
NACA 0018airfoil predicted byMorris and Rusak [5], the lift
coefficients of EPMare close to that of VAWT at λ � 4 for all
Rossby numbers andReynolds numbers examined. Therefore, EPM motion
is a goodapproximation of VAWT at larger tip-speed ratios due to
the lowangle of attack. However, at lower tip-speed ratios, CL;EPM
remainsclose to CL;VAWT only in the upstroke phase. In the
downstrokephase, the discrepancy in lift coefficients due to the
wake-capturingeffect becomes larger as Rossby number decreases and
Reynoldsnumber increases. As the VAWT rotates faster, on the one
hand, thewake-capturing effect is strengthened due to the
intensifying Coriolisforce, which corresponds to decreasing Rossby
number; on the otherhand, it is attenuated because of the
decreasing amplitude of theangle-of-attack variation due to the
increasing tip-speed ratio.Therefore, the growth of the discrepancy
depends subtly on theincrease of the rotating speed of the VAWT.To
probe the existence of wake-capturing at higher Reynolds
numbers, the vorticity field inVAWT (Re � 1500) for a single
periodis 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
ingray are the phase-averaged vorticity field taken from
theexperiments. In Figs. 9a–9f, their phase-averaged field was
filteredto plot only the LEVgenerated around θ� 72° and the plot
representsa composite of overlaid fields from various azimuthal
angles.Similarly, in Figs. 9g–9i, the contours in the gray scale
show thefiltered, phase-averaged TEV evolution. To make
qualitativecomparisons, our vorticity fields at the corresponding
azimuthalangles are overlaid in the color scale on top of the
results from theexperiments. Our blue contours correspond to
negative vorticity,which should be compared with the LEV-filtered
PIV data inFigs. 9a–9f, while the red contours correspond to
positive vorticitythat should be compared with the TEV-filtered PIV
data inFigs. 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 acycle. The disagreement in Figs. 9f and 9i may be due to
strongvortex–blade and vortex–vortex interactions in the downwind
half ofa cycle. An LEV is generated around θ� 72∘ and
wake-capturingoccurs around θ� 90°, which forms a vortex pair
traveling with theblade (Figs. 9a–9c). The vortex pair then
detaches around θ� 133°and propagates downstream (Figs. 9d–9f). The
location of the vortexpair composed of the phase-averaged LEVand
TEVagrees with thecurrent simulation, especially at θ� 158° (Figs.
9e and 9h). Thequalitative agreement in the upwind half of a cycle
suggests thatwake-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
notonly by the Coriolis effect but also because the angle of attack
andincoming velocity vary simultaneously. It is interesting to
understandwhether the Coriolis effect has strong coupling with the
angle-of-
attack or incoming velocity variations. Therefore, we
independentlyexamine airfoils undergoing the decoupled pitching and
surgingmotion 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
rotatingairfoil undergoing only the surgingmotion of a VAWT is
achieved bypitching the airfoil around the leading edge
simultaneously as itrotates so that the angle of attack is
fixedwith respect to the incomingvelocity. For an airfoil surging
at an angle of attack of 15°, liftcoefficients are shown in Fig.
10a. We can see that dynamic stall isrelatively stable and no
wake-capturing phenomenon is observed.Moreover, from the analysis
by Choi et al. [7], when the reducedfrequency is low enough, the
flow can be approximated as quasi-steady, which results in both
lift coefficients for VAWT and EPMfluctuating about a slowly
increasing mean value. For the case ofα � 30°, lift coefficients
are shown in Fig. 10b. The flow is wellseparated so that there is
no stationary vortex shedding.Moreover, nowake-capturing phenomenon
is observed in the flow.
2. Airfoil Undergoing Only Pitching Motion
We consider a pitching motion in a freestream velocity Wavg �λU∞
at λ � 2, Ro � 1.5, and Re � 1500. A rotating airfoilundergoing
only the pitching motion in a VAWT is achieved byrotating an
airfoil in a VAWT without the external freestream andsimultaneously
pitching it around the leading edge with the exactangle-of-attack
variation. The corresponding lift coefficients areshown in Fig. 11.
We can see dynamic stall in both lift coefficients asangle of
attack increases. However, there is no
wake-capturing.Thewake-capturing effect is therefore only present
when pitching,
surging, and the Coriolis force are all present.
0 30 60 90 120 150 1800
0.5
1
1.5VAWTEquiv. motion
0 30 60 90 120 150 1800
0.5
1
1.5
2
2.5
3
3.5VAWTEquiv. motion
b) = 30°αa) = 15°αFig. 10 Comparing lift responses of airfoils
undergoing only surging motion at λ � 2, Ro � 1.5, and Re �
1500.
0 30 60 90 120 150 180−1
−0.5
0
0.5
1
1.5
2
2.5
3VAWTEquiv. motion
Fig. 11 Comparing lift responses of airfoils undergoing only
pitching
motion at λ � 2, Ro � 1.5, and Re � 1500.
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IV. Conclusions
In simulating the flow around a single-bladed vertical axis
windturbine (VAWT), an interesting wake-capturing phenomenon
thatoccurs during the pitch-down portion of the upstream,
lift-generatingportion of the VAWT cycle was observed. This
phenomenon leads toa substantial decrease in lift coefficient due
to the presence of a vortexpair traveling together with the
rotating airfoil. Our results show thatthis flow feature persists
and grows stronger as tip-speed ratio andRossby number are reduced
and Reynolds number is increased.Therefore, the growth of this
features depends subtly on the increaseof the rotating speed of the
VAWT, which, on the one hand, isstrengthened due to the
intensifying Coriolis force. On the otherhand, it is attenuated
because of the decreasing amplitude of theangle-of-attack
variation. Moreover, although our study is restrictedto 2-D flow at
relatively low Reynolds numbers, the qualitativeagreement of the
leading edge vortex and trailing edge vortexevolutions with
Ferreira et al. [17] experiment suggests that thisfeature may
persist in real applications. The corresponding decreasein
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
pitchand surgewere also examined.Except at the beginning of the
pitch-upmotion, all of the simplified motions are good
approximations toVAWT motion at sufficiently high tip-speed ratios
because thecorresponding maximum angle of attack is close to or
lower than thestall angle of the blade. However, at low tip-speed
ratios, while theequivalent planar motion captures the pitch-up
part of the cycle, allthe motions show significant differences in
forces during the pitch-down motion. The results show that the
equivalent motion is a goodapproximation to a rotating airfoil in a
VAWT in the upstroke phasewhere the Coriolis force has a relatively
small effect on vortices.However, it overestimates the average lift
coefficient in thedownstroke phase by eliminating the
aforementioned wake-capturing.The flow by decomposing the planar
motion into surging- and
pitching-only motions was further investigated. Wake-capturing
wasobserved onlywhen the combination of surging, pitching, and
rotationare present, which suggests that this feature is associated
with anunique combination of angle-of-attack variation,
instantaneousvelocity variation, and the Coriolis effect.
Acknowledgments
This project is sponsored by the Caltech Field Laboratory
forOptimized Wind Energy with John Dabiri as principal
investigatorunder the support of the Gordon and Betty Moore
Foundation. Wewould like to thank John Dabiri, Beverley McKeon,
Reeve Dunne,and Daniel Araya for their helpful comments on our
work. Theparametric study in this work used the Extreme Science
andEngineering Discovery Environment (XSEDE), which is supportedby
National Science Foundation grant number ACI-1053575.
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