Aerodynamic analysis models for vertical-axis wind turbines

(1)

Titre:

Title

: Aerodynamic analysis models for vertical-axis wind turbines

Auteurs:

Authors

: M. T. Brahimi, A. Allet et Ion Paraschivoiu

Date: 1995

Type:

Article de revue / Journal article

Référence:

Citation

:

Brahimi, M. T., Allet, A. & Paraschivoiu, I. (1995). Aerodynamic analysis models for vertical-axis wind turbines. International Journal of Rotating Machinery, 2(1),

p. 15-21. doi:10.1155/s1023621x95000169

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Aerodynamic Analysis Models for

Vertical-Axis

Wind

Turbines

M.T.

B

RAHIMP, A.

ALLET*

and

I. PARASCHIVOIU*

lcole

PolytechniquedeMontr6al,Departmentof MechanicalEngineering, C.E6079, CentreVilleA, Montr6al, EQ.

CanadaH3C3A7

Thisworkdetailstheprogress madeinthedevelopmentof aerodynamic models for studying Vertical-AxisWind Turbines

(VAWT’s)withparticular emphasisonthe prediction of aerodynamic loads androtorperformanceaswellasdynamic stall

simulations.Thepaperdescribes currenteffort andsomeimportant findings using streamtube models, 3-Dviscousmodel, stochasticwindmodel andnumerical simulationof theflow around theturbineblades. Comparison of the analytical results

with availableexperimental data have showngoodagreement.

KeyWords: Wind turbines;Aerodynamics;Atmospheric turbulence; Dynamic stall;Navier-Stokesequations.

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

large-scalepowerproduction.As aresult, thereare now more

than 2000megawattsofwindpower intheworld, most

of them inthe US and Denmark. Modernwindturbines

can basically be classified as either of vertical-axis

design, suchastheDarrieusmodel,or ashorizontal-axis

variety like the traditional farmwindpump.

In

both cases,

the turbine’s rotating blades extract kineticenergyfrom

the wind to generate electricity or pump water. The

vertical-axis windturbine, suchasDarrieus model with

curved blades offers an advantageous alternative to the

horizontal axis wind turbine due to its mechanical and

structurally simplicity of harnessing the wind energy.

This simplicity, however, does notextend to the

rotor’s

aerodynamic since the blade elements encounter their

own wakes and those generated by other elements and

operate inadynamicstallregime (Brochieretal., 1986).

Added to this is the increasing awareness that

atmo-spheric turbulence and fluctuating loads significantly

affect the turbine output(Turyan etal., 1987). The

J.A.

Bombardier Aeronautical Chair

Group

at

lcole

Poly-ResearchAssociate SAeronauticalChairProfessor

technique de Montr6al has conductedmanyresearch 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 and power output of the

vertical-axis wind turbine ofany geometry at achosen

rotationalspeed and ambient as well as turbulentwind.

Three computer code variants based on the

double-multiple streamtube model, stochastic windandviscous

flowfieldhavebeendeveloped.The 3DVFviscousflow

model 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 where

the conservationof mass and momentum are solvedby

using the primitive variablesp, u, v, andw(Allet etal.,

1992).Sincethe ambientwindhas beenconsidered tobe

constant thepredicted loads onthe blades are identical foreachrotor revolution.Inordertotake the fluctuating

natureof the wind into account a 3-D stochastic model

has been developed and incorporated first into the

double-multiple streamtube model to analyse the effect

of atmospheric turbulence on aerodynamic loads

(Bra-himi etal. 1992). Actually more workare underway to

incorporate the stochastic wind into the 3DVF viscous

(3)

16 M.T. BRAHIMI ET AL.

codes developed use empirical model. Although the

empirical dynamic stall models used predict well the

aerodynamic loads and rotor performance, they are

limited to the type of airfoil and motion used in the

experimentfromwhich they were derived. Thus, anew

codebasedontheNavier-Stokesequations anduses the

finite element method for simulating the dynamic stall

around Darrieuswind turbinehas beendeveloped (Tchon

et al., 1993). Since 3-D simulation would be very

expensive a 2-Dsimulationhas beenadopted.Themodel

uses anon-inertial streamfunction-vorticityformulation

(q,0)of the2-D incompressibleunsteadyNavier-Stokes

equations. The computer codewas first validatedfor the

flow around a rotating cylinder then, it was applied to

simulate the flow around a NACA 0015 airfoil in

Darrieusmotion.The presentpaperpresentssome devel-opment of the streamtube models, the 3-D viscous

model, the stochastic wind as well as. the numerical

simulation of dynamic stall.

rotor arecalculatedbyusing theprincipleoftwo actuator

disks in tandem. Three categories of computer codes

have been developed(Fig. 1): CARDAAwhich uses two

constant interference factors in the induced velocities

calculatedby adouble iteration, CARDAAVcodewhich

considers the variation of the interference factors as a

functionof theazimuthandCARDAAXcodewhichtakes

the streamtube expansioninto account.These codes have

been usedat

IREQ,

Sandia NationalLaboratories,

DAF

Indal Co.,IMST Marseillesand elsewhere.

Forthe upstream half-cycle of the rotor, the relative

velocityand the local angleof attackas afunction oftip

speed ration

"X"

are givenby:

W2 _V2

_(X

_sinO)

2

₊

cos2Ocos2

₍₁₎

cosOcos

c arcsin

V

(X

snn0i

cos

Zo

cos2

(2)

MOMENTUM MODELS

Several aerodynamic prediction models currently exist

for studyingDarrieuswind turbinesand acompletestate

of the artreview including the appropriate referencesis

given by Strickland ₍₁₉₈₆₎ and Paraschivoiu (1988).

Generally,themainobjective of allaerodynamicmodels

is to evaluate the induced velocity field ofthe turbine since knowledge of this velocity field allows all the

forces on the blade and the power generated by the

turbine 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 disk

enclosed in a simple streamtube where the induced

velocity through the swept volume of the turbine is

assumed tobe constant.Anextensionof this method to

the multiple-streamtube model was then developed by

Strickland

(1975)

who considered the swept volume of the turbine as a series of adjacent streamtubes. Other

aerodynamic methods formodelingthe wind turbineare

based on the vortex theory (Strickland, et al., 1980).

Basicallytwotypes of thevortexmodelhave been used:

the fixed-wake and the free-wake models.Althoughthese

models have the advantage to predict the aerodynamic

loads andperformancemoreexactlythan themomentum

models, theyrequireaconsiderableamountof computer

time. Paraschivoiu(1981) developedananalytical model

(DMS) that considers a multiple-streamtube system

di-vided into two parts where the upwind and downwind

components of theinduced velocities ateach level of the

The normaland 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)

C_T

_CLsina

_CDCOSa

(4)

where the bladeairfoil sectionlift anddragcoefficients,

CL

and

_CD,

are obtained by interpolating the available

testdatausingboth the localReynoldsnumber(Reb W

c/v) and thelocal angle of attack.

flightpalh

CAROAAX j.o.

FIGURE DMS model; CARDAA, CARDAAV, and CARDAAX

(4)

STOCHASTIC

WIND

MODEL

The earlier aerodynamic models for studying Vertical

Axis Wind Turbine (VAWT) are based on a constant

incident wind conditions and are thuscapableof

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 predicted

loadsonthe bladeareidenticalfor eachrotorrevolution

and we have no information about the effect of

turbu-lenceon therotor. Indeed theatmospherictui’bulence as

seenbyrotating windturbineshas become an important

factor for studying stochastic aerodynamic loads and

turbulentflow effects have been identified as one of the

majorsource ofrotorblade fatiguelife.

Continuingthedevelopmentofthe DMS model,a new

code

(CARDAAS)

has beendevelopedtopredictloadson

VAWT by takingthefluctuating natureof the wind into

account. Thevelocityfieldof thewindis assumedtobe

a linear superposition of a steady or mean component

and a fluctuating component. The main objectiveofthe

windmodelis to simulatethe turbulent velocity

fluctua-tions. It includes both the streamwise and lateral

com-ponent of the turbulent velocity. The one dimensional

variations of this turbulent wind are introduced by

creatingtime series of the windvelocityata fixed point

upwind the rotor and assuming that the wind speed is

constant in a plane perpendicular to the mean wind direction.

The turbulent wind speed downstream of the fixed

pointis obtainedbycalculating a timedelayin the time

series. The decrease in the streamwise velocity as the

flow passes through the rotor is taken into account by

assuming a linearvariation in the streamwise direction.

Themethod used for the 3-D windmodel is to simulate

wind speed time series at several points in the plane

perpendicular to the mean wind direction (Fig. 2). For

Z y

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

rotationalplane

FIGURE2 Schematicof3-D windsimulation.

each pointthe time series is generated to represent the

variationaboutthe meanvelocityin thelongitudinaland

vertical directions. The relative velocity and the local

angleof 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 the

local rotor radius,

V

the induced velocity for each

streamtube as a functionofthe azimuthalangle0,

_uf

and

vf

the fluctuating velocities and themeridional angle.

The fluctuation velocities duetothe turbulent wind are

represented by aFourier 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, V

f

Vf/

Vi(

Vf

=u

Vy),

and o"+ represents the

normalized turbulence intensity, cr

_cr/Vi.

The

un-knownFouriercoefficients

A

+and

B

+.

aregiven interms

J J

of the non dimensional spectral power density,

,

the

dimensionless frequency band

A’q

andarandom

phase

angle Oj by:

Af

(26PjA+)

1/2

sin(Oj)

(8)

Bf

(2q)jArl

+)

1,2

COS(Oj)

(9)

The fluctuation velocities areperformed byusingaFast

Fourier Transform, then the aerodynamic loads are

evaluated for each streamtube as function of the blade

positions using the total flow velocities.

THREE-DIMENSIONAL VISCOUS

MODEL

SincetheDMScodes donottake the viscous effectsinto account acomputercodenamed3DVF(Allet, 1993)has

been developed. This code analyses the Darrieus rotor

(Fig. 3)ina steadyincompressible laminar flowfieldby

solving the Navier-Stokes equations in a cylindrical

coordinatesusing thefinitevolume method. The

conser-vation of mass and momentum are solved using the

primitive variables p, u, v, and w. The effect of the

spinningblades is simulatedby distributingsourceterms

in theringof control volumes that lie in the pathofthe

(5)

18 M.T. BRAHIMI ET AL.

By

usingFig. 3 the relativevelocity Vretiscalculated

by thefollowing equation:

Vret Vn

en

+

_V’o

_eo

(

_Vabs

e

.)

en

+

_{(-Or+ Vab}

seo)e

(10)

Theonlyunknown parameters in the aboveequation(Eq.

(10))isthe absolutevelocity, Vabs,whichiscomputedby

using the Navier-Stokes equations. The discretization

method usedtosolvethegoverning equationsisbased on

the finite volume method (FVM) proposed by Patankar

(1980). For velocity-pressure coupledflows, astaggered

gridsystemisknowntogivemorerealistic solutionsand

is adopted inthe present study. Thecalculating method

based on the control volume approach used here is the

widely known "SIMPLER" algorithm. Details of the

governing equations with the numerical procedure are

givenbyAllet

(1993).

Themotionofthe Darrieus blades

are time averaged and introduced through the source

terms into the momentum equations. The source terms

arevalidfor allthecomputationalcellsthatlieinthepath

of the turbine.

A0

Sr NcOVret

-w

COsCDUCOS

CLV’

+ CDwsin

(11)

AO

S

NCpVre

(CD

v

+

CLV

CDYIr)

(12)

AO

Sz

NcpVre

-w

sinS(CDwsin8

_CLV’

+

CDUCSS)

(13)

All forces andpower computationsareintegratedfor the

grid points that lie in the path of the blades.

DYNAMIC STALL

SIMULATION

Dynamic stall is an unsteady flow phenomenon which

referstothestallingbehaviorofanairfoilwhentheangle

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

VAWT.

ofattack ischanging rapidlywithtime.Itischaracterized

by dynamic delayof stallto angles significantly beyond

the staticstallangleandbymassiverecirculating regions

moving downstream over the airfoil surface.

In

the case of Darrieus wind turbine, when the

operational speed approachesitsmaximum, all the blade

sectionsexceed the static stallangle, theangleof attack

changes rapidly and the whole blade operates under

dynamic stall conditions. This increases the unsteady

blade loads and structural fatigue (Ham, 1967, Philippe

etal., 1973, Gormont, 1973,McCroskeyetal., 1976 and

McCrosky,

1981).

Semi-empirical dynamic stall models

havealreadybeen included inourcomputer codes based

on DMS models as well as on the 3-D viscous model.

Thesearenamely theGormontmodel(Gormont, 1973),

MIT

model and Indicial model (Paraschivoiu et al.,

1988, Proulx et al.,

1989).

Although the dynamic stall

models predict well the aerodynamic loads and

perfor-manceonDarrieus windturbine, theyarelimited tothe

type of airfoil and motion used in the experimentfrom

which theywere derived.

A

new code for simulating the dynamic stall around

Darrieus wind turbine has been developed, it is called

"TKFLOW" (Tchon etal., 1993). Thiscode isbased on

the Navier-Stokesequations and uses the finite element

method. Since3-D simulation would bevery expensivea

2-D simulation has been adopted. The model uses a

non-inertial streamfunction-vorticity formulation(q*,o)

of the 2-DincompressibleunsteadyNavier-Stokes

equa-tions.The computer code wasfirstvalidated for the flow

aroundarotating cylinder

(Tchon

etal., 1990). Then, it

was appliedto simulate the flow arounda NACA0015

airfoil in Darrieusmotion.

Thevorticity transportequation and the stream

func-tioncompatibility are givenby:

---

_Ot

"

[(.0

g-

V])e.O

+

2S.,]

(14)

0x2

_0x

0x 0x0x

The streamfunctioncompatibility equation is:

2 +

0

(15)

The effective viscosity is given byv v+ v where v

represents the eddy viscosity. The computational mesh

used in the TKFLOWis an hybrid one composed of a

structured region of highly stretched quadrilateral

ele-mentsinthevicinityofsolidboundaries andan

(6)

,

...

ie,!

_il

50lhrevolutlon

I

,00th ,ution

m

l0

o

.o....e ...i RDAAV

m

i

,,,

iiii

..

,

...,.,..

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

AzimuthalAngle,deg,

FIGURE 4 Computational mesh around NACA 0015 airfoil in FIGURE 6 Angle of attack vs azimuthal angle at TSR=2.33 in

Darricusmotion, turbulentwind.

RESULTS

AND

DISCUSSION

The prediction of the performance coefficient vs tip

speed ratio

(TSR)

for Sandia 17-m wind turbine using

the DMS model is given by Fig. 5. Comparison with

vortex model

(VDART3)

(Strickland et al.,

1980)

and

experimental data (Paraschivoiu,

1988)

shows that the

prediction by

CARDAA

code is well improved using

CARDAAVand

CARDAAX

codes.Figs. 6 and 7 show

the effectof atmospheric turbulence on the angle of

attack and on the aerodynamic torque distribution.

Un-like the periodic distribution predicted by

CARDAAV,

whenturbulence is included theangleof attack

distribu-tionvaries fromone revolutionto another.Furthermore,

theensemble-averaged aerodynamic torquedistributions

(Fig. 7) do not coincide with the periodic distribution.

CARDAAS (1-D

and

3-D)

results predict well the

experimental datagiven byAkins etal.,

(1987).

The simulation of the dynamic stall hysteresis loop

using 3DVF code with indicial model isgiven by Fig. 8.

0.3

0.2

Tipspeedratio,TSR

Compared

to

CARDAAV

results, predictions in the

upwind and downwind regions of the turbine are well

comparedto

experimental

data (Akins, 1989). The

per-formancepredictionsat different tipspeedratio (Fig. 9)

is also wellpredicted using 3DVF code.

16 14 -90 -75 ,i.-"" ’""’..,, RPM=508 ,.";;"’: ’;"!":".: i/’,-" ".,, :,," "xl :/: :’. t" ’x. -60 -45 -30 -15 15 30 45 60 75 90

Azimuthalangle, deg.

FIGURE7 Aerodynamic torqueatTSR-4.61 andturb.=(27%, 25%).

-O,5 ,1 -1,5

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

Angleof attack,deg.

(7)

2O M.T.BRAHIMI ETAL. 0.5 Sandi.- 17m PM=,z: _I _/ 0.2

/i_

:

v _/’’1 / 10 Tip-speedratioTSR

FIGURE9 Powercoefficientvstip-speedratioforSandia 17-m.

beingused inmany placesandarejudgedtobe

satisfac-tory because of its inexpensive approaches it is very

importanttoincludeatmosphericturbulenceinto account

especially for large wind turbines.

In

the case ofwind

turbine interferences such as wind farms the 3DVF

seems tobe more suitable since itcan compute the flow

velocity everywhereinthe rotationalplaneaswell asin

its vicinity. The dynamic stall simulation using

Navier-Stokes equations presents a good tool to predict the

unsteady flow for Darrieus motion and more effort are

underwaytointroduceaturbulence model inthepresent

code.

Acknowledgements

In

the above aerodynamic codes the model used for

dynamic stall is based on semi-empirical methods.. The

simulation of the flow around an airfoil in Darrieus

motionusingNavier-Stokessolverisgivenby Fig. 10 in

term of streamline evolution. Results using TKFLOW

codecanpredict the region where thedynamicstallmay

occur.

CONCLUSION

Aerodynamic loads and performance of Darrieus wind

turbine dependon the flowfield ofthe windthrough the

surface sweptby the blades. Thecomputercodes

devel-oped in this studyrepresent a goodtool for calculating

the aerodynamic loads andperformance of the

vertical-axis wind turbines. Although the DMS models are still

FIGURE 10 Computed streamlines around NACA 0015 airfoil in

Darrieus motion.

This work was prepared in the context of J.-Armand Bombardier

Aeronautical Chair. The authors gratefullyacknowledge the support providedbytheChair.

Nomenclature

Af,

_Bf

Fourrier coefficients

c blade chord,m

Co

blade airfoilsectiondragcoefficient

Ca

bladeairfoil sectionliftcoefficient

CN

blade airfoilsectionnormal-force coefficient

CT

blade airfoilsectiontangential-forcecoefficient

N numberof blades

r localrotorradius,m

Re

localReynoldsnumber

Sr.z,o.o source terms

TSR tipspeedratio

u,v components of the absolutevelocity,m/s

uf,vf fluctuationsvelocities, m/s

V,,. absolute velocity,m/s

V:; local ambient wind, velocitym/s

W relativeinflowvelocity,m/s

X tipspeedratio

c angleofattack, deg.

0 azimuthalangle,deg.

5 meridionalangle, deg.

v cinematicviscosity,m-/s

v

effectiveviscosity,m-/s

v, eddyviscosity,m/s

I] turbine rotationalspeed,

s-9 density,kg/m

"r dimensionless time

rl reducedfrequency

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