aerodynamic analysis models for vertical-axis brahimi. allet. montreal canada

8/6/2019 Aerodynamic Analysis Models for Vertical-Axis Brahimi. Allet. Montreal Canada

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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 an d Breach Science Publishers SA

Printed in Singapore

Aerodynamic Analysis Models for Vertical-AxisWind Turbines

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

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 an d some important findings using streamtube models, 3-D viscous model,stochastic wind model an d 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 turbinesca n basically be classified as either of vertical-axis

design, such as the Darrieus model, or as horizontal-axis

variety like the traditional farm windpump. In both cases,the turbines rotating blades extract kinetic energy from

the 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 du e to its mechanical and

structurally simplicityof

harnessingthe wind

energy.This simplicity, however, does no t extend to the rotorsaerodynamic since the blade elements encounter their

ow n wakes an d those generated by other elements an d

operate in a dynamic stall regime (Brochier et al., 1986).Added to this is the increasing awareness that atmo-spheric turbulence an d fluctuating loads significantlyaffect the turbine output (Turyan et al., 1987). The J.A.Bombardier Aeronautical Chair Group at lcole Poly-

Research Associate

SAeronautical Chair Professor

technique de Montr6al has conducted many research on

the development of computer codes for .studying Dar-rieus rotor aerodynamics (Paraschivoiu, 1988).

The objective of the computer programs is to deter-

mine aerodynamic forces an d power output of the

vertical-axis wind turbine of any geometry at a chosen

rotational speed an d ambient as well as turbulent wind.

Three computer code variants based on the double-

multiple streamtube model, stochastic wind and viscous

flow field have been developed. The 3DVF viscous flowmodel based on Navier-Stokes equations analyses the

Darrieus rotor in a steady incompressible laminar flow

field by solving the Navier-Stokes equations in a cylin-drical coordinates with the finite volume method wherethe conservation of mass an d momentum are solved by

using the primitive variables p, u, v, an d w (Allet et al.,1992). Since the ambient wind has been considered to be

constant the predicted loads on the blades are identical

for each rotor revolution. In order to take the fluctuatingnature of the wind into account a 3-D stochastic model

ha s been developed an d incorporated first into the

double-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


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16 M.T. BRAHIMI ET AL.

codes developed us e empirical model. Although theempirical dynamic stall models used predict well the

aerodynamic loads an d rotor performance, they arelimited to the type of airfoil an d mot io n u se d in theexperiment from which they were derived. Thus, a newcode based on the Navier-Stokes equations an d uses the

finite 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 to

s imu la te the flow around aNACA

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 codes

have been developed (Fig. 1): CARDAAwhich uses twoconstant interference factors in the induced velocities

calculated by a double iteration, CARDAAVcode whichconsiders the variation of the interference factors as a

function of the azimuth an d CARDAAXcode which takesthe streamtube expansion into account. These codes have

been 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)cosOcos

c arcsinV (X snn0i co s Zocos2

(2)

MOMENTUM MODELS

Several aerodynamic prediction models currentlyexistfor studying Darrieus wind turbines an d a complete stateof the art review including the appropriate references isgiven by Strickland (1986) an d 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 an d the power generated by theturbine to be determined.

The first approach to analyze the flow field aroundvertical-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 by

Strickland (1975) who considered the swept volume ofthe turbine as a series of adjacent streamtubes. Other

aerodynamic 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 an d the free-wake models. Although thesemodel s h av e the advantage to predict the aerodynamicloads an d performance more exactly than the momentummodels, they require a considerable amount of computertime. Paraschivoiu (1981) developed an analyticalmodel(DMS) that considers a multiple-streamtube system di -vided into two parts where the upwind an d 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 blade

position using the blade airfoil sectional force coeffi-cients:

CN CLcoso+Czsino

(3)

CT CLsina CDCOSa (4)

where the blade airfoil section lift an d drag coefficients,

CL an d CD,are obtained by interpolating the availabletest data using both the local Reynolds number (Re b W

c/v) an d the local angle of attack.

flight palh

CAROAAX j.o.

FIGURE DMS model; CARDAA, CARDAAV, and CARDAAX

computer codes.


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AERODYNAMICANALYSIS 17

STOCHASTIC WIND MODEL

The earlier aerodynamic models for studying VerticalAxis Wind Turbine (VAWT) are based on a c on st an tincident 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 revolutionan d 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 an dturbulent 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 fluctuatingnature of the wind intoaccount. The velocity field of the wind is assumed to bea linear superposition of a steady or mean componentan d a fluctuating component. The main objective of thewind model is to simulate the turbulent velocity fluctua-tions. It includes both the streamwise an d 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 an d assuming that the wind speed is

constant in a plane perpendicular to the mean winddirection.

The turbulent wind speed downstream of the fixedpoint 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 an d

vertical directions. The relative velocity and the localangle of attack are:

W2= [Or (V + uf)sinO vfcosO]2 +[(Vq- uf)cosO vfsirtO]2cos2 (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 fluctuatingvelocities an d the meridional angle.

The fluctuation velocities due to the turbulent wind arerepresented by a Fourier time-series (Brahimi, 1992):

Np/2

Vf cr+ ., [Afsin2rrrl"r+) + Bfcos(2rrrlf+)]j=l

(7)

+where V

frepresents the normalized fluctuation veloci-

+ties, V

f Vf/Vi(Vf =u Vy),and o"+ represents the

normalized turbulence intensity, cr cr/Vi. The un -

known Fourier coefficients A+

and B+.are given in termsJ Jof the non dimensional spectral power density,, thedimensionless frequency band Aq an d 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 blade

positions using the total flow velocities.

THREE-DIMENSIONAL VISCOUS MODEL

Since the DMS codes do no t take the viscous effects intoaccount a computer code named 3DVF (Allet, 1993) ha sbeen 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 the

primitive variables p, u, v, an d w. The effect of thespinning blades is simulated by distributingsource termsin the ring of control volumes that lie in the path of theturbine blades.


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18 M.T. BRAHIMI ET AL.

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

Vret Vnen + Voeo ( Vabse .) 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 an d

is adopted in the present study. The calculating methodbased on the control volume approach used here is the

widely known "SIMPLER" algorithm. Details of thegoverning equations with the numerical procedure are

given by Allet (1993).The motion of the Darrieus bladesare time averaged and introduced through the sourceterms into the momentum equations. The source termsare val id for all the computational cells that lie in the pathof the turbine.

A0Sr NcOVret-wCOsCDUCOS CLV+ CDwsin (11)

AOS NCpVre (CDv + CLV CDYIr) (12)

AOSz NcpVre -wsinS(CDwsin8 CLV+ CDUCSS) (13)All forces an d power computations are integrated for thegrid points that lie in the path of the blades.

DYNAMIC STALL SIMULATION

Dynamic 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 characterized

by 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 the

operational speed approaches its maximum, all the bladesections exceed the static stall angle, the angle of attackchanges rapidly and the whole blade operates under

dynamic stall conditions. This increases the unsteadyblade loads an d structural fatigue (Ham, 1967, Philippeet al., 1973, Gormont, 1973, McCroskey et al., 1976 an dMcCrosky, 1981). Semi-empirical dynamic stall modelshave already been included in ou r 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 an d perfor-mance on Darrieus wind turbine, they are limited to the

type of airfoil an d motion used in the experiment fromwhich they were derived.

A new code for simulating the dynamic stall aroundDarrieus wind turbine ha s been developed, it is called

"TKFLOW" (Tchon et al., 1993). This code is based onthe Navier-Stokes equations an d 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 0015

airfoil in Darrieus motion.The vorticity transport equation and the s tr eam func-

tion compatibility are given by:

---" [(.0g- V])e.O+ 2S.,] (14)Ot0x20x 0x 0x0x

The stream function compatibility equation is:

2 + 0 (15)

The effective viscosity is given by v v + v where v

represents 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 an d an unstruc-tured region of triangular elements elsewhere (Fig. 4).


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AERODYNAMIC ANALYSIS 19

, i e , ! il50lhrevolutlon

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 in

Darricus motion, turbulent wind.

RESULTS AND DISCUSSION

The 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 an d 7 showthe effect of atmospheric turbulence on the angle ofattack an d 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 on e revolution to another. Furthermore,

the ensemble-averaged aerodynamic torque distributions

(Fig. 7) do not coincide with the periodic distribution.CARDAAS (1-D an d 3-D) results predict well theexperimental data given by Akins et al., (1987).

The simulation of the dynamic stall hysteresis loopusing 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 an d downwind regions of the turbine are well

compared 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 90

Azimuthal 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 coefficientvs angle of attack at TSR=2.49.


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2O M.T. BRAHIMI ET AL.

0.5

Sandi.- 17m

PM=,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 an d are judged to be satisfac-tory because of its inexpensive approaches it is very

important 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 an d 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.

CONCLUSION

Aerodynamic loads an d performance of Darrieus wind

turbine 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 an d 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.

Nomenclature

Af ,Bf Fourrier coefficientsc blade chord, m

Co 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, m

Re local Reynolds numberSr.z,o.o source termsT S R tip speed ratiou, v components of the absolute velocity, m/s

uf, 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-/s

v effective viscosity, m-/sv, eddy viscosity, m/sI] turbine rotational speed, s-9 density, kg/m"r dimensionless time

rl reduced frequency

References

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AERODYNAMICANALYSIS 21

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