Analysis of a Stall Model for the Aerodynamic Calculation of an

Horizontal Axis Wind Turbine NREL/NWTC with the Lifting

Line Theory

Joao Mateus Rodrigues Caldeirajoaomcaldeira@tecnico.ulisboa.pt

Instituto Superior Tecnico, Lisboa, Portugal

May 2014

Abstract

The purpose of this work is the analysis of the aerodynamic performance of a Horizontal Axis WindTurbine in steady state. The calculation is based on the Lifting Line Theory with the inclusion of theeffect of aerodynamic drag on the induced velocities. This model allowed the calculations to be carriedout for stalled flow conditions. The model is applied to the NREL/NWTC turbine and the resultsshow a good agreement with the experimental power coefficient and the results of the Blade ElementMomentum Theory in a wide range of tip-speed ratios.Keywords: Lifting Line Theory, Stall Models, NREL Turbine

1. Introduction

Several codes for the aerodynamic analysis in awind turbine have been developed and the liftingline theory is one of the options to estimate theperformance of this kind of rotors. The analysis hasfocus on the calculation of the power and thrust forsteady state flow, across the turbine and the resultsshow a good comparison with experimental data.However, the model evaluates the viscous effectsusing the lift and drag steady state data, whichconstrain the calculations to a no stall situation.The purpose of this work is to include a stall modelon a regular lifting line theory implementation andthe results are validated by comparison with exper-imental data and data from other codes.

The NREL (National Renewable Energy Labo-ratory) developed a wind turbine which was sub-stantially studied by the scientific community. Itsresults are available for comparison and it has beenadopted in this work for the validation.

The lifting line method requires the lift and dragdata, which can be obtain from theoretical and ex-perimental. Also different wake models can be usedin the lifting line. In this work a helicoidal vortexwake model and an aligned wake model are adopted.

This work is organized as follows: In section 2 themathematical model and the numerical solution aredescribe. In section 3 results are presented. The ef-fects of the lift and drag data and the wake modelon the calculations are shown. Finally, the compar-ison with experimental data and the results of other

methods is observed. In section 4 the conclusionsare drawn.

2. Mathematical ModelConsider a rotor of a horizontal axis turbine withradius R, Z blades placed symmetrically around acylindrical hub of radius rH . The turbine rotateswith a constant angular speed of ω and the fluid hasa uniform velocity of U in the direction of the rotoraxis. The fluid is also considered as incompressiblewith a density of ρ.

In a reference frame rotating with the turbinerotor the flow is considered as steady. The coordi-nate system (x, y, z) with the x axis aligned withthe rotating axis of the turbine, the y axis coin-cides with the reference line of one of the bladesand the z axis completing the right-hand side sys-tem where ( ~ex, ~ey, ~ez) are the unit vectors. It isalso important to define the cylindrical coordinatesystem (x, r, θ), the unit vectors ( ~ex, ~er, ~eθ) are alsodefined. The Figure 1 illustrate the coordinate sys-tems explained.

Each turbine blade is represented by a radial linevortex and a radial line source with a variable inten-sity Γ(r) and σ(r), to model the effects of aerody-namic lift and drag forces, respectively. The liftinglines of each blade are define as:

rH < r < R θk =2π(k − 1)

Z, (1)

where k = 1, 2, ..., Z is the index which representseach lifting line.

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Figure 1: Coordinate Systems

In the lifting line model, the vortex sheet gener-ated at each blade corresponds to a surface alignedwith the local flow. Figure 2 illustrates the liftinglines with the vortex wake model for a two bladeturbine rotor.

Figure 2: Lifting Line and Vortex Wake

Let the vortex sheet Sk be relative to the liftingline number k and ~γ is the free vortex vector onthe same vortex sheet. The Biot-Savart Law allowsto calculate the induced velocity on a point by thelifting line k:

(2)

~vk(x, r, θ) =1

4π

∫ R

rH

~S1 × ~erkS3

1

Γ(r′)dr′

− 1

4π

∫∫Sk

~S × ~γ(x′, r′, θ′)

S3dA′

where:~erk = cos(θk)~ey + sin(θk)~ez, ~S1 = (x, y − y′k, z −z′k) is the radial vector from a point over the blade(0, y′k, z

′k) until a point over the lifting line (x, y, z).

S1 is the modulus of the vector ~S1, | ~S1|, ~S = (x −x′, y − y′k, z − z′k) is the radial vector from a pointover the free vortex line (x′, y′k, z

′k) until a point over

the vortex sheet (x, y, z). S is the modulus of the

vector ~S, |~S|The drag force is modelled by a radial source

line, coincident with the lifting line with intensity

of σ(r) = dmdr where m is the strength of a point

source. The induced velocity in each point by thesource line is given by:

~vk(x, y, z) = − 1

4π

∫ R

rH

σ(r′)

(~S1

S31

)dr′ (3)

where ~S1 = (x, y − y′k, z − z′k)

In the particular case when ~S1 is equal to 0 theinduced velocity calculated by Equation 2 goes toinfinity. An alternative formulation is to consider asmall rectangular area with a width equal to δ andcalculate the flow which pass through it [Sparen-berg]. The small area works as a section of a vortextube and the induced velocity modulus is:

vk =σ(r)

2δ(4)

The parameter δ has a major influence, once itis possible to control the flux on the tube throughit. Assuming that the total flux inside the tube isequal to the one which crosses the disk annulus atradius r. The mass flow rate is:

2πrρVx (5)

and the flux emitted by the small area of width δis:

ZρV δ, (6)

where V is the velocity in the blade section. Equity(6) and (7):

δ =2πr

Z

VxV

(7)

Since the wind turbine is composed by Z blades,the contribution of the Z blades to the induced ve-locity on a point is the sum of the induced velocityby each blade.

~v(x, y, z) =

Z∑k=1

~vk(x, y, z) (8)

Introducing the velocity triangle and forces ona blade section, the induced velocities representedare the combination of the induced velocities by thevortex and by the sources.

va = vΓa + vσa (9a)

vt = vΓt − vσt (9b)

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Figure 3: Velocity Triangle and Forces on a BladeSection

The forces on the lifting line can be computedfrom the Kutta-Joukowski Law:

dD = −ρV σdr (10a)

dL = −ρV Γdr (10b)

The thrust and the torque resulting from the liftand drag forces, are given by, see Figure 3:

dT = ρ|V |Γ(r) cosβidr + ρ|V |σ(r) sinβidr (11a)

dQ = ρ|V |Γ(r)r sinβidr − ρ|V |σ(r)r cosβidr(11b)

From the velocity triangle in Figure 3 the inducedpitch angle is:

tanβi =U − vaωr + vt

(12)

and the drag lift rate

ε =dD

dL=σ

Γ(13)

Integrating along the radius and summing on thenumber of blades:

T = ρZ

∫ R

rH

(ωr + vt)Γ(r)(1 + ε tanβi)dr (14a)

Q = ρZ

∫ R

rH

(U − va)Γ(r)(1− ε cotβi)rdr (14b)

The dimensionless coefficients which characterizethe thrust and the power are defined below, andwhen combined with the Equations 14 give the fi-nal form of CT and CP . All the quantities are di-mensionless using the rotor radius R as a referencelength and the speed U as reference velocity.

CT =T

12ρU

2πR2(15a)

CP =ωQ

12ρU

2πR2(15b)

CT =2Z

π

∫ 1

r∗H

(rλ+v∗t )Γ∗(r)(1+ε tan(βi))dr (16a)

CP =2Zλ

π

∫ 1

r∗H

(1− v∗a)Γ∗(r)(1− ε cot(βi))rdr

(16b)

where λ = ωRU , r∗H = rH

R , v∗a = vaU , v∗t = vt

U e

Γ∗ = ΓUR

Introducing the lift and drag coefficients:

CL =dLdr

12ρV

2c(17)

CD =dDdr

12ρV

2c(18)

where c is the section chord. We may relate thelift and drag forces to the local properties of theblade section. For a blade airfoil the coefficientsare functions of the angle of attack and Reynoldsnumber. The angle of attack is:

α = βi − ψ (19)

where ψ is the pitch angle of the blade section.

Re =V c

ν(20)

where ν is the kinematic viscosity.From Kutta-Joukowski Law the dimensionless

circulation are related to the section lift and dragcoefficients by:

CL =2Γ

V c(21)

CD =2σ

V c(22)

In this work a numerical approach is followed, sothe lifting lines need to be discretized in M elementsalong the radius. A cosine distribution is used forthe elements corner points because it allows a goodconvergence in the classical rotor problems.

ri =1

2(1 + rH)− 1

2(1− rH) cos

(i− 1)π

M(23)

On each element ri < r < ri+1, i = 1, ...,M theintensity of the vortex and the sources are assumedto be constant Γ(r) = Γi and σ(r) = σi.

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The induced velocities are calculated at the con-trol points given by:

ri =1

2(1 + rH)− 1

2(1− rH) cos

(i− 1/2)π

M(24)

and are written as:

vΓa,ti =

M+1∑j=1

vΓa,tγij

γj =

M∑j=1

vΓa,tijΓj (25)

vσa,ti =

M∑j=1

vσa,tijσj =

M∑j=1

vσa,tijεiΓj (26)

According to the discretization, the axial forceand the power coefficients are given by:

CT =2Z

π

M∑i=1

(riλ+vti)Γi(1 + εi tan(βi)i)(ri+1− ri)

(27a)

CP =2Zλ

π

M∑i=1

(1−vai)Γi(1−εi cot(βi)i)ri(ri+1−ri)

(27b)

Resolution of the analysis problem is based onEquation 12 which is discretized from reads

(tanβi)i =1− vΓ

ai − vσai

λr + vΓti − v

σti

(28)

Substituting the induced velocities from 2 and 2:

(tanβi)i =1−

∑Mj=1 v

ΓaijΓj −

∑Mj=1 v

σaijεiΓj

λr +∑Mj=1 v

ΓtijΓj −

∑Mj=1 v

σtijεiΓj

(29)and, finally, reordering the terms in order to thecirculation Γj to get a system of M independentequations:

(30)

M∑j =1

vΓaijΓj +

M∑j =1

vσaijεiΓj + tanβi

M∑j=1

vΓtijΓj +

M∑j=1

vσtijεiΓj

= 1− tanβiλri

i=1,...,M

3. ResultsIn order to make the calculation, the tip speed ratiowas changed in each simulation and all the studiedvalues are given in Table 1. In this paper only theresults for TSR = 5.42 and TSR = 1.52 So, the

higher TSR models a situation where there is noflow separation on the blade, on the other hand,the lower TSR models a totally separated flow and,in the between, at least one of the studied TSR’swill model a situation where the separation occursin the mid part of the blade.

U(m/s) TSR7 5.4210 3.8013 2.9215 2.5320 1.9025 1.52

Table 1: Wind Velocity and Tip Speed Ratio

To study the lift and drag data influence, threedifferent types of data were used, one based on theinviscid theory, one based on experiments made inDelft and the last one is a compilation of all theexperiments and simulations made by the NREL.

In table 2 there are the thrust and power coeffi-cients obtain by the lifting line method using thesethree different lift and drag types of data. Theresults are compared with the experimental valuesmeasured in a wind tunnel from NASA Ames.

Type of DataThrust Power

Coefficient CoefficientInviscid Theory 0.5557 0.4187DELFT Data 0.5100 0.3660NREL Data 0.5288 0.3662Experimental - 0.3574

Table 2: Thrust and Power Coefficients for differentLift and Drag Data and TSR = 5.42

The results show that the Inviscid Theory is notthe better option once it neglected all the viscouseffects. Between the other two options there is not agreat difference because they both model effectivelythe turbine behaviour.

The accuracy of the aerodynamic coefficients isimportant for the success of the results obtained.A good correlation with the reality is essential toget a better aerodynamic analysis.

The wake model is the other external data whichcan influence the lifting line theory implementation.Two models are in study, a aligned wake with thelocal velocity of the flow with radial expansion andinduction factors method.

In this case, two values of TSR are studied,TSR = 5, 42 and TSR = 1, 52 to show the wakeinfluence in two different cases, when the flow isseparated and when it is not.

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Wake ModelThrust Power

Coefficient CoefficientAligned Wake 0.5288 0.3662Induction Factors

0.4183 0.2403MethodExperimental - 0.3574

Table 3: Influence of the Wake Model for TSR =5, 42

Wake ModelThrust Power

Coefficient CoefficientAligned Wake 0.1140 0.0105Induced Factors

0.0991 0.0098MethodExperimental - 0.0144

Table 4: Influence of the Wake Model for TSR =1, 52

For a lower TSR the results for both modelsare close to the experimental ones. However, fora higher TSR the aligned wake produces a muchbetter result than the Induction Factors Method.This shows the great influence of the wake modelused and an advantage of the aligned wake over theInduction Factors Method, that has a good perfor-mance for a low TSR, because of the rising of theinduced pitch angle which causes a approximationof both models, but for a higher TSR the resultsremain far away from the expected.

A wake aligned with the flow velocity is the bestoption for modelling the flow around the wind tur-bine, between the studied options.

The final goal of the Lifting Line Implementationis the comparison with other methods. One of themost developed method for the study of wind tur-bines is the Blade Element Method. Both methodallow a wide range of the TSR, so the compari-son is made in two different situations, when theflow is separated (TSR = 1, 52) and when it is not(TSR = 5, 42). Besides the thrust and power co-efficients, the radial variation of other aerodynamicparameters is shown. The parameters are the angleof attack, the lift and drag coefficients, the Reynoldsnumber and the inducted velocities by the vorticesand by the sources.

The first parameter shown is the angle of attack,which is always connected to the lift and drag co-efficients through the lift and drag data analysedbefore. It is presented the radial variation of theangle of attack for the two TSR values, in order tomake a comparison between the behaviour of thisparameter on both situations.

The results for TSR = 5.42 and TSR = 1.52 areshown in parallel. For all the variables it is possibleto make a comparison between the two methods andas well as between the two values of TSR.

Figure 4: Angle of Attack for TSR = 5, 42

Figure 5: Angle of Attack for TSR = 1, 52

It is verified a clear increasing of the angle of at-tack with the decrease of the TSR. The lift anddrag coefficients are calculated through the lift anddrag data and the angle of attack. So, these threeparameters are not related inside the lifting line rou-tine, proving the importance of external data accu-racy. For a high TSR, it is expected a low dragcoefficient and a lift coefficient without a stall asfor a low TSR the drag should increase due theflow separation.

It is shown that near the hub and the tip thetwo methods in study diverge. The Lifting LineMethod presents decay due the assumption that thecirculation is 0 on the tips of the blade.

Now, it is presented the radial variation of the liftand drag coefficients.

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Figure 6: Lift Coefficient for TSR = 5, 42

Figure 7: Lift Coefficient for TSR = 1, 52

Figure 8: Drag Coefficient for TSR = 5, 42

Figure 9: Drag Coefficient for TSR = 1, 52

As before, the Lifting Line Method and the BladeElement Momentum have a difference near the tipand the hub. The coefficients are related throughthe lift and drag data so the behaviour on the angleof attack in propagated.

As expected, the drag coefficient raise when theTSR goes down, due the flow separation and sub-sequent increase of the drag forces. For the LiftingLine Method the induced velocities have two parts,the one from the vortices and the one from thesources. Both are presented in the same graph andcan be compared to each other. For TSR = 5.42the velocities induced by the sources are very low,because it is a no separated flow. Due to flow sep-aration the velocities increase and assume valuesnear the ones from the velocity induced by the vor-tices. The induced velocity by the vortices assumethe same evolution for both TSR.

Figure 10: Axial Induced Velocity for TSR = 5, 42

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Figure 11: Axial Induced Velocity for TSR = 1, 52

Figure 12: Tangential Induced Velocity for TSR =5, 42

Figure 13: Tangential Induced Velocity for TSR =1, 52

As in the previous comparisons, the thrust andpower coefficients are presented. They provide a im-portant information to the method because it is pos-sible to check the correlation between the method

implemented and the real data. Like in the resultspresented before, the lifting line and the blade el-ement method are compared as well. In these twofinal graphs, the results from the panel method arealso shown.

Figure 14: Power Coefficient for Different Methods

Figure 15: Thrust Coefficient for Different Methods

For the modelation of the stall model, the panelmethod is not a good option because it neglectedthe viscous effects. So, the viscous effects are provenas essential for a stall model. Both Lifting Line andBlade Element Momentum have a good correlationwith the experimental results.

This last results prove the success of the routineon the implementation of a stall model in the LiftingLine Method.

4. Conclusions

The first conclusion to take about this work is thatthe lifting line method is well implemented and theinclusion of a stall model to it, allows to model the

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flow separation over a wind turbine and use the lift-ing line theory in a wider range of angles of attack.

It was evaluated the influence of the lift and dragdata. We reached the conclusion that the lifting linemodel has a dependence on this data. The influenceof the wake model was studied as well. Two modelwere computed and the one with a wake alignedwith its velocity has a better correlation. However,the one calculated from the induction factors arealso able to model at lower TSR′s.

The Lifting Line Method was compared with theBlade Element Momentum and the Panel Methodfor the stall modelling on a wind turbine. The re-sults for the Panel Method have a bad correlationwith the experimental ones because this methoddoes not include de viscous forces. A stall modelis essential to compute the performance in a windturbine.

Both Lifting Line Method and Blade ElementMomentum are able to model the stall, providinga good agreement with experimental results.

The inclusion of the stall model in the LiftingLine Method Implementation allows to model thestall on the NREL/NWTC turbine, reaching thepurpose of this paper.

AcknowledgementsThe author would like to thank to Professor Falcaode Campos for all the time he spend helping andtutoring during the development of the project.

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performance by numerical lifting-surface theory.Relatorio Tecnico, 1978.(8) Joao Leite Machado: Projecto hidrodinamicode turbinas de corrente marıtima de eixo horizontalcom o modelo da linha sustentadora. Tese deMestrado, Instituto Superior Tecnico, 2010.(9) William B Morgan e JW Wrench: Some com-putational aspects of propeller design. Methods inComputational Physics, 4:301331, 1965.(10) Joao Filipe Tiago Potra: Projecto e analiseaerodinamica do rotor de pequenas turbinaseolicas. Tese de Mestrado, Instituto SuperiorTecnico, 2006.(11) David A Simms, S Schreck, M Hand e LJ Fin-gersh: NREL unsteady aerodynamics experimentin the NASA-Ames wind tunnel: a comparison ofpredictions to measurements. National RenewableEnergy Laboratory Colorado, USA, 2001.(12) Niels N Srensen e JA Michelsen:NREL/NWTC aerodynamics code blind com-parison. Em 2. Science Panel meeting, NREL,2000.(13) JA Sparenberg: Hydrodynamic Propulsionand Its Optimization:(Analytic Theory), volume27. Springer, 1994.

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