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.

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http://dx.doi.org/10.2514/1.J054199http://crossmark.crossref.org/dialog/?doi=10.2514%2F1.J054199&domain=pdf&date_stamp=2015-09-15

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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