Aerodynamic Analysis Models for Vertical-Axis Wind Turbines

International Journal of Rotating Machinery1995, Vol. 2, No. 1, pp. 15-21Reprints available directly from the publisherPhotocopying permitted by license only

(C) 1995 OPA (Overseas Publishers Association)Amsterdam B.V. Published under license byGordon and Breach Science Publishers SA

Printed in Singapore

Aerodynamic Analysis Models for Vertical-AxisWind Turbines

M.T. BRAHIMP, A. ALLET* and I. PARASCHIVOIU*lcole Polytechnique de Montr6al, Department of Mechanical Engineering, C.E 6079, Centre Ville A, Montr6al, EQ.

Canada H3C 3A7

This work details the progress made in the development of aerodynamic models for studying Vertical-Axis Wind Turbines(VAWT s) with particular emphasis on the prediction of aerodynamic loads and rotor performance as well as dynamic stallsimulations. The paper describes current effort and some important findings using streamtube models, 3-D viscous model,stochastic wind model and numerical simulation of the flow around the turbine blades. Comparison of the analytical resultswith available experimental data have shown good agreement.

Key Words: Wind turbines; Aerodynamics; Atmospheric turbulence; Dynamic stall; Navier-Stokes equations.

hrough advanced technology, wind turbine has be-come an important commercial option for large-

scale power production. As a result, there are now morethan 2000 megawatts of wind power in the world, mostof them in the US and Denmark. Modern wind turbinescan basically be classified as either of vertical-axisdesign, such as the Darrieus model, or as horizontal-axisvariety like the traditional farm windpump. In both cases,the turbines rotating blades extract kinetic energy fromthe wind to generate electricity or pump water. Thevertical-axis wind turbine, such as Darrieus model withcurved blades offers an advantageous alternative to thehorizontal axis wind turbine due to its mechanical andstructurally simplicity of harnessing the wind energy.This simplicity, however, does not extend to the rotorsaerodynamic since the blade elements encounter theirown wakes and those generated by other elements andoperate in a dynamic stall regime (Brochier et al., 1986).Added to this is the increasing awareness that atmo-spheric turbulence and fluctuating loads significantlyaffect the turbine output (Turyan et al., 1987). The J.A.Bombardier Aeronautical Chair Group at lcole Poly-

Research AssociateSAeronautical Chair Professor

technique de Montr6al has conducted many research onthe development of computer codes for .studying Dar-rieus rotor aerodynamics (Paraschivoiu, 1988).The objective of the computer programs is to deter-

mine aerodynamic forces and power output of thevertical-axis wind turbine of any geometry at a chosenrotational speed and ambient as well as turbulent wind.Three computer code variants based on the double-multiple streamtube model, stochastic wind and viscousflow field have been developed. The 3DVF viscous flowmodel based on Navier-Stokes equations analyses theDarrieus rotor in a steady incompressible laminar flowfield by solving the Navier-Stokes equations in a cylin-drical coordinates with the finite volume method wherethe conservation of mass and momentum are solved byusing the primitive variables p, u, v, and w (Allet et al.,1992). Since the ambient wind has been considered to beconstant the predicted loads on the blades are identicalfor each rotor revolution. In order to take the fluctuatingnature of the wind into account a 3-D stochastic modelhas been developed and incorporated first into thedouble-multiple streamtube model to analyse the effectof atmospheric turbulence on aerodynamic loads (Bra-himi et al. 1992). Actually more work are underway toincorporate the stochastic wind into the 3DVF viscousmodel. For the dynamic stall simulation the computer

16 M.T. BRAHIMI ET AL.

codes developed use empirical model. Although theempirical dynamic stall models used predict well theaerodynamic loads and rotor performance, they arelimited to the type of airfoil and motion used in theexperiment from which they were derived. Thus, a newcode based on the Navier-Stokes equations and uses thefinite element method for simulating the dynamic stallaround Darrieus wind turbine has been developed (Tchonet al., 1993). Since 3-D simulation would be veryexpensive a 2-D simulation has been adopted. The modeluses a non-inertial stream function-vorticity formulation(q,0) of the 2-D incompressible unsteady Navier-Stokesequations. The computer code was first validated for theflow around a rotating cylinder then, it was applied tosimulate the flow around a NACA 0015 airfoil inDarrieus motion. The present paper presents some devel-opment of the streamtube models, the 3-D viscousmodel, the stochastic wind as well as. the numericalsimulation of dynamic stall.

rotor are calculated by using the principle of two actuatordisks in tandem. Three categories of computer codeshave been developed (Fig. 1): CARDAA which uses twoconstant interference factors in the induced velocitiescalculated by a double iteration, CARDAAV code whichconsiders the variation of the interference factors as afunction of the azimuth and CARDAAX code which takesthe streamtube expansion into account. These codes havebeen used at IREQ, Sandia National Laboratories, DAFIndal Co., IMST Marseilles and elsewhere.

For the upstream half-cycle of the rotor, the relativevelocity and the local angle of attack as a function of tipspeed ration "X" are given by:

W2 V2 (X sinO) 2 + cos2Ocos2 (1)

cosOcosc arcsin

V (X snn0i cos Zo cos2 (2)

MOMENTUM MODELSSeveral aerodynamic prediction models currently existfor studying Darrieus wind turbines and a complete stateof the art review including the appropriate references isgiven by Strickland (1986) and Paraschivoiu (1988).Generally, the main objective of all aerodynamic modelsis to evaluate the induced velocity field of the turbinesince knowledge of this velocity field allows all theforces on the blade and the power generated by theturbine to be determined.The first approach to analyze the flow field around

vertical-axis wind turbine was developed by Templin(1974) who considered the rotor as an actuator diskenclosed in a simple streamtube where the inducedvelocity through the swept volume of the turbine isassumed to be constant. An extension of this method tothe multiple-streamtube model was then developed byStrickland (1975) who considered the swept volume ofthe turbine as a series of adjacent streamtubes. Otheraerodynamic methods for modeling the wind turbine arebased on the vortex theory (Strickland, et al., 1980).Basically two types of the vortex model have been used:the fixed-wake and the free-wake models. Although thesemodels have the advantage to predict the aerodynamicloads and performance more exactly than the momentummodels, they require a considerable amount of computertime. Paraschivoiu (1981) developed an analytical model(DMS) that considers a multiple-streamtube system di-vided into two parts where the upwind and downwindcomponents of the induced velocities at each level of the

The normal and tangential forces coefficients are evalu-ated for each streamtube as a function of the bladeposition using the blade airfoil sectional force coeffi-cients:

CN CLcoso + Czsino (3)

CT CLsina CDCOSa (4)

where the blade airfoil section lift and drag coefficients,CL and CD, are obtained by interpolating the availabletest data using both the local Reynolds number (Reb Wc/v) and the local angle of attack.

flight palh

CAROAAX j.o.

FIGURE DMS model; CARDAA, CARDAAV, and CARDAAXcomputer codes.

AERODYNAMIC ANALYSIS 17

STOCHASTIC WIND MODELThe earlier aerodynamic models for studying VerticalAxis Wind Turbine (VAWT) are based on a constantincident wind conditions and are thus capable of predict-ing only periodic variations in the loads (Veers, 1984),(Malcolm, 1987), (Marchand et al., 1987), (Strickland,1987) and (Homicz, 1988). As a result, the predictedloads on the blade are identical for each rotor revolutionand we have no information about the effect of turbu-lence on the rotor. Indeed the atmospheric tuibulence asseen by rotating wind turbines has become an importantfactor for studying stochastic aerodynamic loads andturbulent flow effects have been identified as one of themajor source of rotor blade fatigue life.

Continuing the development of the DMS model, a newcode (CARDAAS) has been developed to predict loads onVAWT by taking the fluctuating nature of the wind intoaccount. The velocity field of the wind is assumed to bea linear superposition of a steady or mean componentand a fluctuating component. The main objective of thewind model is to simulate the turbulent velocity fluctua-tions. It includes both the streamwise and lateral com-ponent of the turbulent velocity. The one dimensionalvariations of this turbulent wind are introduced bycreating time series of the wind velocity at a fixed pointupwind the rotor and assuming that the wind speed isconstant in a plane perpendicular to the mean winddirection.The turbulent wind speed downstream of the fixed

point is obtained by calculating a time delay in the timeseries. The decrease in the streamwise velocity as theflow passes through the rotor is taken into account byassuming a linear variation in the streamwise direction.The method used for the 3-D wind model is to simulatewind speed time series at several points in the planeperpendicular to the mean wind direction (Fig. 2). For

Z y

::::::: ::. R:

rotational plane

FIGURE 2 Schematic of 3-D wind simulation.

each point the time series is generated to represent thevariation about the mean velocity in the longitudinal andvertical directions. The relative velocity and the localangle of attack are:

W2= [Or (V + uf)sinO vfcosO] 2 +[(V q- uf)cosO vf sirtO]2cos 2 (5)

arcsin [((V + uf)cosO vfsinO)cos]/W (6)where f represents the turbine rotational speed, r thelocal rotor radius, V the induced velocity for eachstreamtube as a function of the azimuthal angle 0, uf andvf the fluctuating velocities and the meridional angle.The fluctuation velocities due to the turbulent wind are

represented by a Fourier time-series (Brahimi, 1992):

Np/2

Vf cr+ ., [Af sin2rrrl "r+) + Bf cos(2rrrlf +)]j=l

(7)+

where V f represents the normalized fluctuation veloci-+

ties, Vf Vf/ Vi( Vf =u Vy), and o

"+ represents thenormalized turbulence intensity, cr cr/Vi. The un-known Fourier coefficients A + and B+. are given in terms

J Jof the non dimensional spectral power density, , thedimensionless frequency band Aq and a random phaseangle Oj by:

Af (26PjA +) 1/2 sin(Oj) (8)Bf (2q)jArl+) 1,2 COS(Oj) (9)

The fluctuation velocities are performed by using a FastFourier Transform, then the aerodynamic loads areevaluated for each streamtube as function of the bladepositions using the total flow velocities.

THREE-DIMENSIONAL VISCOUS MODELSince the DMS codes do not take the viscous effects intoaccount a computer code named 3DVF (Allet, 1993) hasbeen developed. This code analyses the Darrieus rotor(Fig. 3) in a steady incompressible laminar flow field bysolving the Navier-Stokes equations in a cylindricalcoordinates using the finite volume method. The conser-vation of mass and momentum are solved using theprimitive variables p, u, v, and w. The effect of thespinning blades is simulated by distributing source termsin the ring of control volumes that lie in the path of theturbine blades.

18 M.T. BRAHIMI ET AL.

By using Fig. 3 the relative velocity V ret is calculatedby the following equation:

Vret Vn en + Vo eo ( Vabs e.) en+ (-Or+ Vabseo) e (10)

The only unknown parameters in the above equation (Eq.(10)) is the absolute velocity, V abs, which is computed byusing the Navier-Stokes equations. The discretizationmethod used to solve the governing equations is based onthe finite volume method (FVM) proposed by Patankar(1980). For velocity-pressure coupled flows, a staggeredgrid system is known to give more realistic solutions andis adopted in the present study. The calculating methodbased on the control volume approach used here is thewidely known "SIMPLER" algorithm. Details of thegoverning equations with the numerical procedure aregiven by Allet (1993). The motion of the Darrieus bladesare time averaged and introduced through the sourceterms into the momentum equations. The source termsare valid for all the computational cells that lie in the pathof the turbine.

A0Sr NcOVret

-w COsCDUCOS CLV + CDwsin (11)

AOS NCpVre (CDv + CLV CDYIr) (12)

AOSz NcpVre

-w sinS(CDwsin8 CLV + CDUCSS) (13)All forces and power computations are integrated for thegrid points that lie in the path of the blades.

DYNAMIC STALL SIMULATIONDynamic stall is an unsteady flow phenomenon whichrefers to the stalling behavior of an airfoil when the angle

FIGURE 3 Angles, velocity vectors and forces for Sandia 17-mVAWT.

of attack is changing rapidly with time. It is characterizedby dynamic delay of stall to angles significantly beyondthe static stall angle and by massive recirculating regionsmoving downstream over the airfoil surface.

In the case of Darrieus wind turbine, when theoperational speed approaches its maximum, all the bladesections exceed the static stall angle, the angle of attackchanges rapidly and the whole blade operates underdynamic stall conditions. This increases the unsteadyblade loads and structural fatigue (Ham, 1967, Philippeet al., 1973, Gormont, 1973, McCroskey et al., 1976 andMcCrosky, 1981). Semi-empirical dynamic stall modelshave already been included in our computer codes basedon DMS models as well as on the 3-D viscous model.These are namely the Gormont model (Gormont, 1973),MIT model and Indicial model (Paraschivoiu et al.,1988, Proulx et al., 1989). Although the dynamic stallmodels predict well the aerodynamic loads and perfor-mance on Darrieus wind turbine, they are limited to thetype of airfoil and motion used in the experiment fromwhich they were derived.A new code for simulating the dynamic stall around

Darrieus wind turbine has been developed, it is called"TKFLOW" (Tchon et al., 1993). This code is based onthe Navier-Stokes equations and uses the finite elementmethod. Since 3-D simulation would be very expensive a2-D simulation has been adopted. The model uses anon-inertial stream function-vorticity formulation (q*,o)of the 2-D incompressible unsteady Navier-Stokes equa-tions. The computer code was first validated for the flowaround a rotating cylinder (Tchon et al., 1990). Then, itwas applied to simulate the flow around a NACA 0015airfoil in Darrieus motion.The vorticity transport equation and the stream func-

tion compatibility are given by:

---

"[(.0g- V])e.O + 2S.,] (14)Ot

0x2 0x 0x 0x0xThe stream function compatibility equation is:

2 + 0 (15)The effective viscosity is given by v v + v where vrepresents the eddy viscosity. The computational meshused in the TKFLOW is an hybrid one composed of astructured region of highly stretched quadrilateral ele-ments in the vicinity of solid boundaries and an unstruc-tured region of triangular elements elsewhere (Fig. 4).


, ie,! il 50lhrevolutlon I....... ,00th ,ution ml0 .o....e .......i RDAAV mo i ,,, iiii

..

,

...............,.,...90 -60 -30 30 60 90 120 150 180 210 240 270

Azimuthal Angle, deg,

FIGURE 4 Computational mesh around NACA 0015 airfoil in FIGURE 6 Angle of attack vs azimuthal angle at TSR=2.33 inDarricus motion, turbulent wind.

RESULTS AND DISCUSSIONThe prediction of the performance coefficient vs tipspeed ratio (TSR) for Sandia 17-m wind turbine usingthe DMS model is given by Fig. 5. Comparison withvortex model (VDART3) (Strickland et al., 1980) andexperimental data (Paraschivoiu, 1988) shows that theprediction by CARDAA code is well improved usingCARDAAV and CARDAAX codes. Figs. 6 and 7 showthe effect of atmospheric turbulence on the angle ofattack and on the aerodynamic torque distribution. Un-like the periodic distribution predicted by CARDAAV,when turbulence is included the angle of attack distribu-tion varies from one revolution to another. Furthermore,the ensemble-averaged aerodynamic torque distributions(Fig. 7) do not coincide with the periodic distribution.CARDAAS (1-D and 3-D) results predict well theexperimental data given by Akins et al., (1987).The simulation of the dynamic stall hysteresis loop

using 3DVF code with indicial model is given by Fig. 8.

0.3

0.2

Tip speed ratio, TSR

Compared to CARDAAV results, predictions in theupwind and downwind regions of the turbine are wellcompared to experimental data (Akins, 1989). The per-formance predictions at different tip speed ratio (Fig. 9)is also well predicted using 3DVF code.

16

14

-90 -75

,i.-"" ""..,, RPM=508,.";;": ;"! ":".:

i/ ,-" ".,,:,," "xl

:/: :.t" x.

-60 -45 -30 -15 15 30 45 60 75 90Azimuthal angle, deg.

FIGURE 7 Aerodynamic torque at TSR-4.61 and turb.=(27%, 25%).

-O,5

,1

-1,5

0 -24 -18 -12 -6 12 18 24 30

Angle of attack, deg.

FIGURE 5 Performance coefficient vs tip speed ratio at RPM=50.6. FIGURE 8 Normal force coefficient vs angle of attack at TSR=2.49.

2O M.T. BRAHIMI ET AL.

0.5

Sandi.- 17mPM=,z: I /

0.2

/i_ : v /1/10

Tip-speed ratio TSR

FIGURE 9 Power coefficient vs tip-speed ratio for Sandia 17-m.

being used in many places and are judged to be satisfac-tory because of its inexpensive approaches it is veryimportant to include atmospheric turbulence into accountespecially for large wind turbines. In the case of windturbine interferences such as wind farms the 3DVFseems to be more suitable since it can compute the flowvelocity everywhere in the rotational plane as well as inits vicinity. The dynamic stall simulation using Navier-Stokes equations presents a good tool to predict theunsteady flow for Darrieus motion and more effort areunderway to introduce a turbulence model in the presentcode.

Acknowledgements

In the above aerodynamic codes the model used fordynamic stall is based on semi-empirical methods.. Thesimulation of the flow around an airfoil in Darrieusmotion using Navier-Stokes solver is given by Fig. 10 interm of streamline evolution. Results using TKFLOWcode can predict the region where the dynamic stall mayoccur.

CONCLUSIONAerodynamic loads and performance of Darrieus windturbine depend on the flowfield of the wind through thesurface swept by the blades. The computer codes devel-oped in this study represent a good tool for calculatingthe aerodynamic loads and performance of the vertical-axis wind turbines. Although the DMS models are still

FIGURE 10 Computed streamlines around NACA 0015 airfoil inDarrieus motion.

This work was prepared in the context of J.-Armand BombardierAeronautical Chair. The authors gratefully acknowledge the supportprovided by the Chair.

NomenclatureAf, Bf Fourrier coefficientsc blade chord, mCo blade airfoil section drag coefficientCa blade airfoil section lift coefficientCN blade airfoil section normal-force coefficientCT blade airfoil section tangential-force coefficientN number of bladesr local rotor radius, mRe local Reynolds numberSr.z,o.o source termsT S R tip speed ratiou, v components of the absolute velocity, m/suf, vf fluctuations velocities, m/sV,,. absolute velocity, m/sV:; local ambient wind, velocity m/sW relative inflow velocity, m/sX tip speed ratioc angle of attack, deg.0 azimuthal angle, deg.5 meridional angle, deg.v cinematic viscosity, m-/sv effective viscosity, m-/sv, eddy viscosity, m/sI] turbine rotational speed, s-9 density, kg/m"r dimensionless timerl reduced frequency

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