An aerodynamic method for the analysis of isolated horizontal-axis wind turbines

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Titre:

Title

:

An aerodynamic method for the analysis of isolated horizontal-axis

wind turbines

Auteurs:

Authors

: Christian Masson, Idriss Ammara et Ion Paraschivoiu

Date: 1997

Type:

Article de revue / Journal article

Référence:

Citation

:

Masson, C., Ammara, I. & Paraschivoiu, I. (1997). An aerodynamic method for the analysis of isolated horizontal-axis wind turbines. International Journal of Rotating Machinery, 3(1), p. 21-32. doi:10.1155/s1023621x97000031

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Journal Title: International Journal of Rotating Machinery (vol. 3, no 1)

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An

Aerodynamic

Method for the

Analysis of

Isolated

Horizontal-Axis Wind Turbines

CHRISTIAN MASSON*,IDRISSAMMARAandION PARASCHIVOIU

Bombardier AeronauticalChair,colePolytechnique, Montr6al, CANADA,H3C3A7

(Received 4 April1996)

Theaerodynamic analysisof a wind turbinerepresentsaverycomplextask since itinvolves

anunsteadythree-dimensional viscousflow.Inmostexisting performance-analysis methods,

wind turbinesare considered isolated sothat interference effects causedbyother rotors orby

the site topologyareneglected. Studyingtheseeffects in ordertooptimizethearrangement

and thepositioningof Horizontal-Axis Wind Turbines(HAWTs)on awind farm is one of the research activities of the Bombardier Aeronautical Chair. As a preliminary step in the

progress of this project, a method that includes some of the essential ingredients for the

analysisof windfarms has been developedand is presentedin the paper.Inthis proposed method, the flow field around isolated HAWTs is predicted by solving the steady-state, incompressible,two-dimensionalaxisymmetricNavier-Stokesequations.The turbine is

rep-resented by a distribution of momentum sources. The resulting governing equations are solvedusing aControl-Volume Finite ElementMethod(CVFEM). This axisymmetric

im-plementation efficientlyillustrates theapplicabilityandviabilityof theproposed methodol-ogy, by using aformulation that necessitates aminimum ofcomputerresources. The

axi-symmetric methodproduces performance predictions for isolated machines with the same level ofaccuracythan thewell-knownmomentum-strip theory. Itcantherefore be considered

tobe a useful tool forthedesignofHAWTs.Itsmainadvantage, however,is itscapacityto

predicttheflow in the wake which constitutes one of the essential features needed for the

performance predictions ofwind farms of dense clusterarrangements.

Keywords: WindFarm,WindPower,HAWT,NumericalMethod,Navier-StokesEquations

1.

INTRODUCTION

Aftersomeunsuccessful attemptsatconstructing and

operating very-large-scaleisolated windturbines, the

recenttendency is to construct wind farms of medi-um-sizemachines

(500

kW). The strategycurrently

used during the conception of such wind farms con-sists in installing the turbines far from each otherin

orderto minimizethe interferenceeffects. This

prac-ticeresults invery sparse wind farms where the wind

energy potential of a site is inefficiently used. It is

justified by the performance losses associated to the

*Correspondingauthor. Tel.: (514)340-4582.Fax: (514)340-5917. E-mail:christian, masson@meca.polymtl.ca. 21

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22 C. MASSONetal.

wake effects, which are significantin dense

arrange-ments.

However,

arelatively dense but staggered

ar-rangement ofthe turbines is expectedto produce an

increase in the performance ofthe downstream tur-bines with respect to the isolated-turbine situation. This isdueto the beneficial venturieffects thatoccur

between two adjacent turbine wakes. The efficiency ofthe dense staggered cluster is not expected to be

significantly higher than that of the sparse

arrange-ment.

However,

for a given number of turbines, a

dense staggered cluster can be easily conceived that would occupy up to 25% less land than the sparse

arrangement. Notwithstanding the farmefficiency in-crease inducedby the venturi effects, a reduction of 25% of the wind farmarea canrepresentasignificant

economy in operating expenses. Furthermore, some oftheconstructionexpenses, suchasthe grading and

electric infrastructure costs, willbe reduced.

Various aerodynamic methods, appropriate for the conception ofisolated turbines, are available to de-signers (Gohard [1978], Paraschivoiu [1981],

Strick-land et al. [1980], Templin [1974], Wilson [1984]).

However, efficientandaccuratemethods for the

anal-ysisofadensecluster where the effects of the

three-dimensionalturbulentturbine wakesare includedare not available. The development ofa method that in-cludesthe essential ingredients for the successful

per-formancepredictions of wind turbines in adense

ar-rangementis theauthors’ main objective. In the

pro-posed method, the flow field of the wind farm is

predictedbysolving thesteady-state, incompressible,

three-dimensional Navier-Stokes equations. The tur-bines are represented by distributions ofmomentum sources, a technique introduced by Rajagopalan

[1984]and Rajagopalan andFanucci [1985].This is a

general formulationwhich can be applied, in

princi-ple,tohorizontal-axis andvertical-axis windturbines

andcaninclude the effects ofhubs, towers, and local

topography. The Navier-Stokes equations are solved

using the three-dimensional Control-Volume Finite

Element Method (CVFEM) of Saabas and Baliga

[1994].

The developmentof thismethodis stillin itsearly

stage, and the paper is aimed at presenting the

progressand atdemonstrating theapplicabilityof the

proposed method along with its capacity to analyze

the performances of wind farms. The mathematical

model and numerical method described in the"paper

are atwo-dimensional axisymmetric formulation

ap-plicable to isolated

HAWTs.

This implementation is

used in thepapertodemonstrate theapplicabilityand

viabilityof theproposed methodology atmuchlower

CPUcosts than a fullythree-dimensional

implemen-tation. Several aspects related to the numerical

solu-tionof the mathematical model, suchas(i)the

appro-priate extent of the computational domain, (ii) the

grid spacing needed to obtain accurate performance

predictions, and(iii) the optimum choice of the vari-ouscontrolparameterstoobtainconvergedsolutions,

canalso be studiedmoreefficiently using the axisym-metric formulation. Comparisons between the

perfor-mancepredictions obtainedwiththeproposed

formu-lation and those of the momentum-strip theory are

presented to illustrate the accuracy of the proposed

methodology. Comparisons with experimental data

arealso included.

2.

GOVERNING EQUATIONS AND ROTOR

REPRESENTATION

The flow around an isolated

HAWT

is governed by

the unsteady three-dimensional Navier-Stokes

equa-tions whichhavetobesolved in a.domainwith

mov-ingboundaries. Analytical solutions of such a

prob-lem are hardly possible while numerical solutions

represent a formidable task on today’s computers.

However, a more tractable model producing

mean-ingful results canbe obtained bytime-averaging the

governing equations and byrepresenting the turbine

with momentum sources.

The derivationofthe,gov.erning equations

andmo-mentum sources is based on time-averaging tech-niques and the blade-element theory. The interested readerisreferred totheworks ofRajagopalan [1984]

and Rajagopalan and Fanucci [1985] for a detailed

derivation of the source terms ofvertical-axis wind

turbines.Inthecaseof horizontal-axis wind turbines, ofinteresthere, the derivationisvery similarand the

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reader is referred to Fig. 1 for the definition of the

various parameters involved in the evaluation ofthe

momentum sources.

In steady-state, laminar, two-dimensional

axisym-metricflow aroundanisolated

HAWT,

the

mathemat-ical model consists ofa set offour differential

equa-tions: a continuity equation and three momentum

equations. The four dependent variables are

P

(hav-ing the components u, v, and w in the x, r, and 0

directions) and p. The fluiddensityis represented by

p, anditsdynamic viscosityby ta.Theturbine is

com-posed of B blades having a coning angle y and a

chord c that can vary along the blade. The turbine rotational speedis f.

Theflow around anHAWTcan berepresented by

the following generalformulation:

V.(I3V+)--

_V.FV+

-ff

(Sp)qb

q--

(ST)

₊

(1)

Theappropriate governingequationscanbeobtained

from

Eq.

(1) by defining the dependent variable,

thediffusioncoefficient,

F,

andthevolumetric source

terms,

_(Sp)+

and

_(ST)+,

accordingto Table

I,

where

(ST)

K

(UC

D

+ WCL)

cos

(2)

(ST)

-K

(UC

D

WCL)

sin3’

(3)

(ST)

K

(UC

L

WCD)

(4)

BVrelcAg

K -p

(5)

2Vcv

Wre

/U

2_q- _W2

₍₆₎

U ucos/- v sin/ (7)

w

rf- w (8,)

Thesource terms

_{(ST)u, (ST)v, (ST)}

arethe x-,r-,and

0-component of the mutual time-averaged force,

ex-ertedby the fluid andthe blades on one another, per

unit volume ofthe fluid. They will be referredto as

the momentum-source terms.

CL

and

CD

are the lift

anddragcoefficients oftheblade-defining airfoiland

TABLE SpecificFormsof theGeneralEquation

+

F (Sp)b (ST)+

continuity equation 0 0 0 x-momentumequation u _! --Op OX _(ST) r-momentumequation v _a -Op Or (ST)

0-momentum equation w p 0 _(ST)w

are, ingeneral,functions of theangleof attackoand the local Reynolds number Re In the proposed

model, these coefficientscanbe taken fromeither

ex-perimentaldataornumerical resultsobtained overthe

appropriate two-dimensional airfoil.

Inthe above_{equations, Vcv}isthe control volumeto

which the conservation principles have beenapplied,

and Af corresponds tothe blades length thatcrosses

thiscontrol volume.Itis tobe noted thatAfis zeroin

the region of the flow which is not crossed by the blades. Consequently, the momentum-source terms are non-zeroonlyin theregion of the flow crossedby

the turbineblades.

The mathematical model described in this section

has been obtainedby considering laminarflow. This

assumptionis difficulttojustifyphysically, sinceit is

well known that thewake ofawind turbine is turbu-lent. Nevertheless, theproposedmathematical model still includes someof the essential elements for

accu-rateperformancepredictions ofwindfarms. The

ma-jor features of theproposedmodelare (i)its capacity

to model the details of the wake behindtheturbines

and (ii) its implicit introduction of the interferences

between the rotors. The interferences are due to nu-merous effects such asthepressurevariationand the

blockage phenomenon.

In this early stage of development, the

laminar-flow assumption is used in order to emphasize the

various aspects relatedtothe windturbinemodelling without introducing uncertainties inherentto the

tur-bulence models. It is demonstrated in the RESULTS

Section thataccurateperformance predictionscan be

obtained using the laminar-flow assumption in the

caseofisolatedHAWTs since theirperformancesare

not highly influenced by the details oftheir wakes.

However,

when a turbine is located behind another, the turbulent wake has to be consideredin order to

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24 C.MASSONetat. /

--

Blade FIGURE

Lift/

Dra

SECTION

A-A’

RotorGeometricParameters.

U

obtain accurateperformancepredictions of the

down-windturbine.

3.NUMERICAL

METHOD

The proposed numericalmethod is aCVFEM based

on the primitive-variables, co-located, equal-order

formulation of Masson et al. [1994]. Detailed de-scriptions ofthis CVFEM, pertaining to the

simula-tionof particulatetwo-phaseflows and internal

three-dimensional turbulent flows, are available in the works ofMasson [1993] and Saabas [1991]. There-fore, for sake of conciseness, only the aspects

rele-vant tothe successful simulation of theflow arounda windturbine arepresented inthis section.

3.1 Interpolation of in Convective

Term

In the derivation of algebraic approximations to

sur-face integrals of the convective fluxes, two different

interpolation schemes for

+

werepresented bySaabas

[1991]: the FLow Oriented upwind scheme (FLO);

anda

MAss

Weighted upwind scheme (MAW).

The FLO scheme is based on the earlier work of Baliga and Patankar [1980,1988]. The interpolation

function used in this scheme

responds

appropriately

to an element-based Peclet number and to the direc-tion of theelement-averaged velocity vector. In

pla-nartwo-dimensionalproblems that involve

acute-an-gled triangular elements and relatively low element Peclet numbers, Saabas and Baliga [1994] have

proventhattheFLO schemecan be quite successful.

If high values of the element Peclet number are

en-countered, however,theFLOschemecanleadto

neg-ative coefficients in the algebraic discretised

equa-tions (Saabas and Baliga[1994]) andthis difficultyis

compoundedwhenobtuse-angledtriangular elements

areused. The donor-cellscheme of Prakash [1987]is oneway of ensuring positive coefficients: in this

ap-proach,the value ofa scalarconvected outofa

con-trol volume, across its surface, is set equal to the valueof the scalaratthe node within the control

vol-ume. This approach guarantees positive coefficients, but takes littleaccount ofthe influenceof the direc-tionof theflow. Thusit ispronetoconsiderable false diffusion(Prakash 1987]).

TheMAW schemeis based on the

positive-coeffi-cient schemes of Schneider and Raw [1986]. It en-sures, atthe element level, that the extent to which

thedependentvariableat anode exteriorto acontrol

volume contributes to the convective outflow is less

than orequalto its contribution tothe inflow by

con-vection. Thus, it is a sufficient condition to ensure

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terms add positively to the discretised equation. It

should be noted that the MAW scheme takes better

account ofthe influence of the direction of the flow than thedonor-cell scheme ofPrakash [1987],so itis

less prone to false diffusion. Details ofthe

formula-tionoftheMAWschemearepresentedinthe workof Saabas 1991].

In problems with acute-angledtriangular elements

and relatively low element Peclet numbers, the FLO

scheme is more accurate than the MAW scheme.As

was mentioned earlier, however, when high element Peclet numbers are involved, especially in

conjunc-tion with obtuse-angled elements, the FLO scheme

produces negativecoefficientsinthe discretised

equa-tions. Negative coefficients in the’ discretized

equa-tions canleadto convergence difficulties when

itera-tive solutionalgorithms, suchasSIMPLEorits

vari-ants (Patankar [1980]) and CELS (Galpin et al.

[1985]),thatusesegregatedorcoupled equation

line-by-line iterative algorithms to solve the linearized

setsofdiscretisedequations, areused.

Inthenumerical solutionof the flow aroundawind

turbine, a fine grid has to be used in the immediate

vicinityof therotorinordertocapture thelarge

vari-ations of the flow properties. The grid size is then

increased as the distance to the turbine increases.

This coarseningis appliedin orderto keepa

reason-able number ofgrid points while ensuring the appli-cation of the boundary conditions to be far enough

from therotor. Theuseof suchcoarsegrids resultsin

large Pecletnumbers sincethese are directly

propor-tional to the grid size.

Convergence

difficulties were

encountered during the simulations presented in this

paperwhentheFLOschemewasused.Therefore,the

MAW

scheme has been usedtoproduceall the results

presented in this paper and its application is

recom-mended for the successful solution of the flow around

a wind turbine.

related velocities (u

m,

vm),

as afunction ofa

pseudo-velocity and a pressuregradientterm:

(9)

where t/, and 9are thepseudo-velocities, andd", and

d arecalled thepressure-gradientcoefficients.These expressions come directly from the discretised mo-mentum equations. However, instead of using the

control-volume-averaged pressure gradient, the

ele-mental pressure gradient is used to compute the mass-flow related velocity in each element. This

pre-ventstheoccurrenceof spuriouspressureoscillations

in theproposed CVFEM.

3.3

Momentum-Source Term

Linearization The momentum-source term,

(ST)

₊

is expressed in

the followinggeneralform (Patankar [1980]):

(ST)

₊

(ST)

C q-

(ST)p+

(lO)

In

each triangular element, the values of

(ST)c

and

(ST)p are stored at the vertices, and are assumed to

prevailoverthecorresponding portions of the control

volumes within that element. Whilethe volume

inte-gration of themomentum-source term is straightfor-ward,itsproperlinearizationis crucialtoensure

con-vergence of the overall algorithm, especially in the

contextofthe segregatediterative solutionalgorithm

used in this work. The linearization consists in the specification ofappropriate expressions for

(ST)c

and

(ST)p"

The momentum-source term canbe linearized

ex-plicitly ineach iteration:

3.2

Mass

FlowRate Interpolation

Themassflowrate iscalculated usingaspecial treat-mentborrowed from the works of Prakash and

Patan-kar [1985] and Saabas and Baliga [1994]. This

spe-cial treatment consists in expressing the mass-flow

(ST)

C

(ST)+

(ST)

p 0 (11) where the superscript * means that the source term

hasbeenevaluated using the flowpropertiesobtained at the previous iteration. Implementation ofthis

lin-earization in the segregated iterative algorithm used in this work resultedin severeconvergence problems

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26 C.MASSONetal.

for highly loaded wind turbines.

In

such conditions, the value of the momentum-source terms become

large, and the resulting momentum equations are

source-dominated. This results in very slow

conver-gence rates, and, sometimes the segregated iterative

algorithm even diverges. In an effort to improve the robustnessof the iterative solutionalgorithm, the fol-lowing treatmentis proposed:

(ST);

(ST)

C 0

(ST)

p

_+,

(12)

This linearization has proven to be much less prone

to convergence problems than the explicit

lineariza-tionexpressed by

Eq.

(11).

3.4Boundary Conditions

The computational domain consists ofa simple

cyl-inder that includes the wind turbine. Boundary con-ditionshavetobeprescribedonthethreefacesofthis

cylindrical domain.

Inlet Boundary: The inlet boundary is a r-0 plane

located upstream of the wind turbine. In this plane,

the three velocity components are given by the known freestream wind speed while the pressure is

calculated from the discretisedcontinuity equations.

3.5 Overall SolutionAlgorithm

The discretisedequationsforma set ofcoupled non-linear algebraic equations. Inthis work, the iterative

variable adjustment procedure proposed by Saabas and Baliga [1994]isusedtosolve theproposed math-ematical model. Thisprocedureis akinto SIMPLER

(Patankar

[1980])

without the pressure correction

equation. In order to facilitate implementation and

testing of the proposed method, structured grids are

used. Thus, a line Gauss-Seidel algorithm based on

thetridiagonal matrixalgorithmareusedtosolvethe

discretisedequationsfor p, u, v, andw.

The momentum-source linearization proposed in

this work can lead to large differences between the

values of the two pressure-gradient coefficients. In

the case of an horizontal-axis wind turbine

com-pletely containedwithina r-0 plane, for example, d"

is typically much smaller than d

.

This large

differ-ence in the values of the pressure-gradient

coeffi-cientsleadstodifficultiesinensuring the overallmass conservation since the streamwise pressure gradient

has a negligible effect in the discretised pressure

equations. This difficulty is alleviated by prescribing

a relatively large number of iterations in the line

Gauss-Seidel algorithm used for the solution of the discretisedpressure equations.

4.

RESULTS

Outlet Boundary: The outlet boundary is a r-0

planelocated downstream of the wind turbine.Inthis

plane, the pressure is assumed to be uniform and

given while the three velocity components are

com-puted from the discretised momentum equations

ob-tained using the outflow treatment of Patankar

[980].

Top

Boundary: This is acurved surfacelocatedat a

radial distance far from the wind turbineblade tip.In

this plane, the velocityis set to its freestreamvalue.

Pressureis calculated from the discretisedcontinuity

equations.

The results presented in this section are aimed at

demonstrating the capacity of the proposed

method-ology to accurately predict the performances of

iso-lated

HAWTs,

and to analyse wind farms. To this

effect, the details of the computed flow field in the

vicinity of an isolated

HAWT

are shown, and

com-parisons between the predictions of the proposed

method, those of the well-known momentum-strip

theory,and experimental data whenavailable are

pre-sented. Furthermore, the following aspects related to

thesuccessful numerical solutionof the mathematical model arestudied: (i) the determination of the

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obtain acomputational-domain independent solution,

and(ii)the sensitivityoftheperformancepredictions

with respect to thegrid size.

The simulationspresentedin this sectionhave been realizedfor two

HAWTs:

(i) theNASA/DOEMod-0 100-kW Experimental

HAWT

(Puthoff and Sirocky

[1974]) operating at arotational speedof 40rpmand

ablade pitch angle of 3

,

and (ii) theINTA Experi-mental rotor(Hernandez and

Crespo

[1987]) operat-ing at a rotational speed of 1500 rpm and a blade pitch angleof 9.5

4.1

Extent

of the ComputationalDomain

A

detailed study of the behaviour of power

predic-tions with respect to the size of the computational

domainhas been undertakeninordertodeterminethe

minimum extentof the computationaldomainneeded

to produce relevant performance predictions for

HAWTs.

Inthecase of thesimulation of the

axisym-metric/swirling model, the size of the computational

domain is characterized by three length parameters,

namely

Axue, AXDN,

and

_RCD.

These length

parame-ters arepresentedin Fig. 2 whichillustrates the grid

topology. Itis to be noted that the grids used in the

axisymmetric/swirling simulations were much finer

than theone shown in Fig. 2.

Figs. 3-5 show the results of the behaviourof the

performance prediction with respect to the extent of

the computational domain.These graphs present the

variation of the difference between the performance

prediction at a finite value of one of the length

pa-rameters (denoted by either

P(Axup

),

P(AXDN),

P(RcD))

and the power predicted for a very large

value of the correspondinglengthparameter(denoted byp(c)), asafunctionofaspecificlengthparameter.

Thelengthparameters have beennondimensionalized

with respectto the rotor diameterD. The grids used were ofuniformtype.withequalgridspacinginther

andxdirections.Thisanalysis has been undertakenat

themostcriticaltip speedratioTSRwhich isbelieved

tobe the TSR corresponding to themaximumpower

coefficient

Cp.

Thepower coefficient is definedby

P

ce

3)

2

where P is the mechanical power,

U

is the freestream wind velocity, and R is the rotor radius.

The operational condition at which the maximum

powercoefficient occurs isthe mostcritical situation

since it corresponds to the regime where a greater

portion of the energy available in the flow is

ex-[::CD

r

T

Il_

AXup

_--lJ-"

_AXDN

---1

(9)

28 C. MASSONetal. I0"0

I

8.0 .o 4.0

NASA/DOEMod-0HAWT D.=40rpm

U.=8m/s

BladePitchAngle of 3

FIGURE3 Variationof the Performance PredictionwithRespect

toAXup.

tractedbytherotor. Forablade pitchangleof 3 and

a rotational speed of 40 rpm, the freestream wind

speed at which the maximum

Cp

is reached for the

NASA/DOE Mod-0 rotor is near 8 rrds. This study

hasrevealed that the power predictions ofanisolated rotor aresignificantly influencedbythelengthofthe computational domain upstream of the rotor (i.e.

AXup)

and the positionoftheconstant-radius

bound-ary (i.e.

RcD).

Based ontheresultspresentedin Figs.

3 and5,

AXup/D

7.5 and

_RcD/D

4.0 seems tobe

large enoughto produce performance predictions

in-dependentoftheextentofthedomain.Forthe

down-stream extent of the computational domain, a much

0.50

NASA/DOE Mod-0HAWT

D,=40rpm

U.=8m/s

I--2.0 4.0 8.0

Axo

D

FIGURE 4 Variationofthe PerformancePredictionwithRespect

to

AXDN.

0.0

1.500 3.000

NASA/DOEMod-0HAWT D.=40rpm

U.=8m/s

1, 4.500 6.000

RcD/D

FIGURE5 Variationofthe PerformancePredictionwithRespect

toRCD.

smaller value can be used. Fig. 4 suggests that

AXDN/D

4.5 is more than sufficient. These values oftheextentofthe computational domain have been

usedtoproducethe resultspresentedin the remainder

of this section.

4.2 GridDependence Study

Fig. 6 presents the difference between the

perfor-manceprediction obtained atagiven number of grid points

N,

P(N), and the grid-independentpower

pre-diction, p(c). This grid dependence study has been

3.0

NASA/DOEMod-0HAWT

.Q=40rpm

2.s

U.=8m/s

2.0

i/

Uniform grid

1.0 0.5

0,0

2.5E4 5.0E4 7.5E4

N

FIGURE6 Variationof the Performance Prediction withRespect

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realized on uniform grids with equal gridspacing in

the randx directions in order to facilitatethe

deter-mination of the grid-independent power prediction. Fig. 6 shows that the grid-independent solution is

reachednearN 25 000. Thisvery largenumber of grid points needed to obtain the grid-independent

powerprediction is due to the use ofuniformgrids.

However,

for practical calculations, nonuniform grids should be used in order to minimize the number of gridpoints needed. Using auniform and finegridin

the vicinity of therotoralongwith anexpanding grid

in the rest of the computational domain, it has been

possible to reduce the number of points needed to

obtain asolutionclosetothegrid-independentoneto

3 000(see Fig.

6).

Cp 0.40 0.35 0.30 0.25 0.20 0.15 0.10 0.05 0.00 5.0

,,,

.=40

_!pro

Blade PitchAngle of 3 Proposed Method Momentum-StripTheory

7.5 10.0 12.5

TSR

FIGURE 8 Power Coefficient Predictions for the NASA/DOE Mod-0HAWT.

4.3 Comparisonswith the

Momentum

Strip

Theory andExperimental

Data

Figs. 7 and 8 show comparisons between the

perfor-mance predictions of the NASA/DOE Mod-0 100-kW Experimental

HAWT

produced by the

pro-posedmethodologyand the results of the

momentum-striptheory. The agreement between thetwomethods

is good. This was to be expected since the factor

limiting theaccuracyofthe resultsistheuseof static

two-dimensional lift and drag coefficients, on which

both methods are based. As stated before, the main

motivation in developing the proposed method re-sides in its inherentmodelling of therotor/rotor,

ro-tor/ground,androtor/towerinteractionsandits

capac-itytoproducethedetails ofthe flowfield around the

turbines.

Figs. 9 and 10 present similar comparisons for the

case of the

INTA

Experimental

HAWT.

In this case,

theagreementisverygoodbetween thetwomethods.

Fig. 10 also showssomeexperimental

data.

The

per-formance predictions produced by the two methods

are in relatively good agreement with the measured

performances.

400

P (kW)

300

NASA/DOEMod-0HAWT

D,=40rpm

Proposed Method Momentum-StripTheory

7.5 10.0 12.s

U

(m/s) 15.0

FIGURE 7 Power Predictions for the NASA/DOE Mod-0

HAWT. P (kW) 2.0 1.0 INTAHAWT O.= 1500rpm

Blade

_{Pitc___h}

_{Angl_____e}

of

_{9.5___}

Proposed Method

Fm

10.0 15.0 20.0 25.0 30.0

U

(m/s)

(11)

30 C.MASSONetal.

Cp

INTA HAWT f=1500rpm

""-"

-’’-a- BladePtchAngle of9.5

\

Proposed Method

Momentum-StripTheory

Experiments

TSR

FIGURE 10 PowerCoefficientPredictionsfor theINTA HAWT.

4.4 Predicted Flow Field

Using thenonuniform3 000-point grid, the flowfield

around the NASA/DOE Mod-0 HAWT has been

computed. The resulting pressure and velocityfields

in the vicinity of the rotor are presented in Figs. 11

and 12. These results illustrate the capacity of the

proposed methodology to produce the details of the

flowbehind a wind turbine, whichconstitutes one of the essential features for the successful analysis of

windfarms.

5.

CONCLUSION

Theideaof using theventurieffects inwindfarms is a new concept. More fundamental research is

re-quiredtodetermine its applicability.Forinstance, the

sensitivityof the cluster efficiencyto thewind

orien-tationis a crucialaspect relatedtothe viability ofthis

concept. The development of a method appropriate for the analysis of dense cluster arrangements

corre-spondstothe first step inthisfeasibilitystudy.Sucha

method can allow the determination of the optimum cluster arrangement, which is expected to be highly

relatedto a specific site (wind speedand orientation variations,topography, etc.). Furthermore, thedigital

versionof such a method representsapowerful tool for the designers of wind farm projects.

The methodpresentedin thispaper includessome

of the important ingredients needed for the successful analysis of dense cluster arrangements. The

imple-mentationofthismethod has beenrealizedunder the assumption of axisymmetric swirling flow. This

im-plementation has allowed to efficiently illustrate the

applicability andviability of the proposed

methodol-ogy byusing a formulationthat.necessitates a mini-mumof computer resources. Additional features have

tobeincludedin thismethodbeforeitsapplicationto

FIGURE Computed PressureFieldfortheNASA/DOEMod-0HAWT.

p-p_ (Pa) 7.0 5,0 3.0 1.0 -:3.0 -7.0 -11.0

(12)

(m)

15

NASA/DOEMod-O HAW

____-__.--____-, _> ).=40 _rpm

,Blade

PitchAngleof 3

_/

20

x (m)

FIGURE 12 ComputedVelocityFieldfor theNASA/DOEMod-0HAWT.

theperformancepredictions of wind farms. Thefully

three-dimensional formulation has to be imple-mented. Furthermore, an appropriate turbulence model should be selected and implemented in order

to obtain accurate performance predictions for

tur-bines located behind others. Nevertheless, the

axi-symmetric formulationproduces performance

predic-tions for isolated HAWTswith the same level of

ac-curacy than the well-known momentum-strip theory.

Itcanbe consideredtobeauseful tool for thedesign

of horizontal-axis windturbines.

Acknowledgement

Thisworkissupported bythe "Ministrede

l"Inergie

et des Ressources du Qudbec" through the

"Pro-gramme d’aide au d6veloppement des technologies

del’6nergie".The authors wouldlike toexpresstheir

gratitude to R.

E

Legault and J. E. Wade, from Kenetech Windpower, for the numerous helpful dis-cussions they havehad withthem. The results of the

momentum-striptheoryhavebeen obtainedusing the

computer codedeveloped by TayebBrahimiandDan

Chocron, of the Bombardier AeronauticalChair.

References

Baliga, B.R.,Patankar,S.V., 1980.A NewFinite-Element

Formu-lation for Convection-Diffusion Problems, Numerical Heat Transfer,Vol. 3, pp. 393-409.

Baliga, B.R., Patankar, S.V., 1988. Elliptic Systems: Finite Ele-mentMethodII,inMinkowycz,W.J.etal. (Eds.),Handbookof NumericalHeat Transfer,Wiley,NewYork,Chap. 11, pp. 421-461.

Galpin,RE, VanDoormaal,J.R,Raithby,G.D., 1985.Solutionof the IncompressibleMassandMomentumEquationsby Applica-tionofaCoupledEquationLineSolver,Int.J._forNum.Meth.in Fluids, Vol.5,pp.615-625.

Gohard,J.D.,1978.Free Wake Analysis_ofWind Turbine

Aerody-namics, TR 184-14, Massachusetts Institute of Technology,

Cambridge.

Hernandez,J.,Crespo, A., 1987. Aerodynamic Calculationof the Performance ofHorizontal Axis Wind Turbinesand Comparison withExperimentalResults,Wind Engineering,Vol. 11,pp. 177-187.

Masson, C., 1993. Numerical Prediction _of Gas-Solid Particle FlowsoveraWideRangeofConcentration inIrregular

Geom-etries, Ph.D. Thesis,Dept. ofMech.Eng., McGill University, Montreal, Canada.

Masson, C., Saabas, H.J. and Baliga, B.R., 1994. Co-Located

Equal-OrderControl-Volume Finite-ElementMethod for

Two-DimensionalAxisymmetricIncompressibleFluidFlow,

Interna-tionalJournalforNumericalMethods in Fluids, Vol. 18, pp.

1-26.

Paraschivoiu, I., 1981. Double-Multiple Streamtube Model for DarrieusWindTurbines, SecondDOEWind Turbines

Dy-namics Workshop,NASA CP-2185, pp. 19-25.

Patankar, S.V., 1980. NumericalHeat TransferandFluid Flow, McGraw-Hill, New-York.

(13)

32 C. MASSONetal.

Prakash,C.,Patankar,S.V., 1985AControlVolume-Based Finite-Element Method for Solving theNavier-StokesEquations Using

Equal-Order Velocity-Pressure Interpolation, Numerical Heat Transfer,Vol. 8, pp. 259-280.

Prakash, C., 1987.Examinationof the Upwind (Donor-Cell)

For-mulation inControlVolumeFinite-ElementMethodsforFluid Flow andHeatTransfer,NumericalHeat Transfer,Vol. 11,pp.

401-416.

Puthoff,R.L.,Sirocky,RJ., 1974 Preliminary Design_ofalO0-kW Wind TurbineGenerator,NASA TMX-71585.

Rajagopalan,R.G.,1984.InviscidUpwindFinite_DifferenceModel forTwo-DimensionalVertical Axis WindTurbines, Ph.D. Thesis,

WestVirginia University,Morgantown.

Rajagopalan, R.G., Fanucci, J.B., 1985. FiniteDifference Model for theVertical Axis WindTurbines, Journal_ofPropulsion and

Power,Vol. 1, pp. 432-436.

Saabas,H.J., 1991.A Control VolumeFiniteElement Method_for Three-Dimensional,Incompressible, ViscousFluidFlow, Ph.D. Thesis,Dept. of Mech.Eng.,McGillUniversity, Montreal,

Can-ada.

Saabas,H.J.and Baliga,B.R.,1994. Co-LocatedEqual-Order Con-trol-Volume Finite-Element Method forMultidimensional,

In-compressible, Fluid FlowmPart I, Numerical Heat Transfer,

Vol.26B, pp. 381-407.

Schneider, G.E., Raw, M.J., 1986. ASkewed Positive Influence Coefficient Upwinding Procedure for Control Volume Based Fi-nite Element Convection Diffusion Computation, Numerical

Heat Transfer,Vol.9, pp. 1-26.

Strickland, J.H.,Webster, B.T., and Nguyen, T., 1980. A Vortex

ModeloftheDarrieusTurbine:AnAnalytical andExperimental Study, SAND79-7058,SandiaNationalLaboratories,

Albuquer-que.

Templin, R.S., 1974. Aerodynamic _Performance Theory_for the NRCVertical-Axis WindTurbine, LTR-160, NationalResearch CouncilofCanada, Ottawa.

Wilson,R.E.,Walker, S.N., 1984.PerformanceAnalysisof

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