Prediction of steady aerodynamic performance of

rotors with winglets using simple prescribed wake

methods

Mac Gaunaa∗, Niels N. Sørensen† and Mads Døssing ‡

Risø-DTU, Roskilde, DK-4000, Denmark

The present work includes analysis of the results obtained with a free wake code todetermine two simple, non-iterative prescribed wake models aimed for use in steady loadand design calculations for rotors with winglets. The results show that results from a pre-scribed wake model using only the integral axial load on the rotor and the main geometryof the wing/winglet can produce results in qualitative agreement with the physically morecorrect free wake model. The results indicate, however, that the power production esti-mated from the prescribed wake models using only the integral loads and no local loadingsand/or induced velocities along the blade will be erroneous if the load distribution alongthe wing is not similar to the one the model was calibrated for.

I. Introduction

Many wind turbine sites have restrictions on rotor diameter in one form or another. In those cases, theonly way the power production can be optimized at any specific wind velocity is through maximizing thepower coefficient, CP , of the wind turbine. To that end, winglets can be used, since these can improve thepower production without increasing the projected rotor area. Further, there may be beneficial effects withrespect to noise by using winglets.

For the application of winglets on horizontal-axis wind turbines, previous works has been done by VanBussel,1 Imamura et al.,7 Johansen et.al.,9,10 Gaunaa and Johansen4–6 and Chattot.2 Van Bussel1 developeda momentum theory for winglets on horizontal-axis wind turbines, explaining the positive effect of a wingleton a rotor by the downwind shift of the wake vorticity. This result was, later shown to be incorrect byGaunaa and Johansen.4 The works by Imamura et al.7 and Johansen et.al.9,10 are numerical in nature,as the former work is based on a free-wake vortex-lattice method. The work of Johansen et.al. is basedon Navier-Stokes simulations with the CFD code EllipSys3D, and predicts an increase in CP of 1.74% on amodern MW sized turbine. Gaunaa and Johansens work4–6 included their explanation of the main principlefor power augmentation by the use of winglets: a reduction of tip effects. That work further employed a freewake method to optimize power production for rotors with winglets of different sizes. Using this approachthe power production on a rotor with winglets of length ∆l/R = 0.02 was increased 2.2%a compared to anon-wingletted rotor designed for maximum power output.8 The simulations further showed that downwindwinglets performed better than their upwind counterparts. The work of Chattot2 is based on the Goldsteinapproach, where each blade is treated as a lifting line generating a helicoidal vortex sheet, along prescribedhelices whose pitch is determined to satisfy the wake equilibrium condition. This work included also theeffect of sweep and dihedral of the winglet, and used the NREL blade as the point of reference. It wasfound that the backward swept winglets performed better than the forward swept ones. Along the samelines as for the non-rotating case, the main purpose of adding a winglet to a wind turbine rotor is to reducetip-losses more than the increase in viscous surface drag corresponding to the increased surface area andthereby increase the aerodynamic efficiency of the turbine. The use of winglets on rotors other than wind

∗Senior Scientist, Risø-DTU, Wind Energy Division, P.O Box 49, DK-4000, AIAA Member.†Professor, Risø-DTU, Wind Energy Division, P.O Box 49, DK-4000, Senior AIAA Member.‡PhD student, Risø-DTU, Wind Energy Division, P.O Box 49, DK-4000.aThe power production was evaluated using the CFD code EllipSys3D using the rotor geometry obtained from the free wake

code results.

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AIAA 2011-543

Copyright © 2011 by the American Institute of Aeronautics and Astronautics, Inc. All rights reserved.

energy applications was performed by Muller & Staufenbiel13 and Muller,14 and deals with using wingletson helicopters.

The design codes and load computation codes used in the wind turbine industry are for the vast majoritybased on Blade Element Momentum (BEM) theory, which can not take the effects of nonplanar rotors intoaccount due to the assumption of annularly independent streamtubes. One method that can take all theseeffects into account is for instance Navier-Stokes CFD methods.9,10 The drawback with this type of methodis it’s computational cost, which limits these methods’ use in the design process. Another method whichcan include correctly the effect the winglet has on the aerodynamic response of the rotor is free wake vortexbased methods. These methods are, however, also too computationally costly for use in load calculations.Additionally, the free wake methods are generally inherently numerically relatively unstable, which usuallyrequire a great deal of attention from the user. One viable type of methods that could provide the desiredcombination of accuracy, numerical stability and computational speed could be the prescribed wake models.However, the wake shape depends on tip speed ratio, bound circulation as well as the geometry of the rotor(winglet length/direction). To the knowledge of the authors, there are no previous works where such modelshave been applied on wind turbine rotors with winglets.

Therefore, the aim of the present work is to use a steady free wake code which have earlier given results invery good agreement for both standard rotor cases8 and rotors with winglets4–6 to simulate a number of rotorgeometries with different bound circulations/loadings, and from these results to formulate and subsequentlytest the bounds of application of very simple prescribed wake models.

II. Free Wake Modelling

II.A. Description of free wake method

In the Free-Wake Lifting Line method, the wings are represented by concentrated line vortices, from whichshed vorticity emanates into the wake. The method does not take into consideration the actual local geometryof the wing cross-section (airfoil shape), but models only the effect of the circulations that the airfoilsgenerate. The inviscid lift forces from the fluid on the wings are evaluated from the Kutta-JoukowskiTheorem, L = ρVrel × Γ, using the relative velocity of the flow with respect to the wings, including also thecontributions from the free wakes of the wings. Since viscous effects are not naturally a part of a lifting linealgorithm, they are taken into account separately. The local drag forces act in the direction of the relativeflow direction, and the magnitudes are obtained from the lift forces using 2D lift-to-drag ratios of the airfoilsections. An algorithm for evaluating the induced forces from the velocities that the nonplanar boundvorticity induces on itself is included in the model. In order to avoid the lifting line singularity on itself, thisextra induced velocity is computed using a vortex-lattice type grid. The self-induced velocities are obtainedusing an iterative scheme using a weighted difference between 3D and 2D self-induced velocities. From theconverged self-induced velocity, the induced forces are computed using the lifting line. The self-inducedforces in the case of winglets straight up- or down-wind produces negligible changes in power production.A non-negligible effect on power production from this term is only seen in cases where the winglets havesweep, i.e. where the winglets are tilted forward or backward. Since the winglets considered in the presentwork have no sweep, the self-induced effect will not be discussed further. In order to determine the shapeof the free wake, a steady-state free wake method was adopted. Due to the inherent unstable nature of freewake methods for wind turbine applications, some care must be taken to obtain converging solutions. Sincethe freestream velocity is constant, and the turbine is assumed not to be operating in yawed conditions,only the vortices from one wing need be updated; the other ones are obtained from symmetry conditions.In order to ensure adequate resolution of the wake, the position of the wake is determined in specific planesparallel to the rotor plane, with narrow spacing near the rotor plane and increased spacing further down thewake (∆Z=0.003R at the rotor disc and ∆Z=0.009R at Z=3R using 450 cross-sections). In the first part ofthe wake (up to 3R), the wake is updated freely, and the wake velocities are evaluated at the intersectionbetween the linear vortex elements. In the second part of the wake, the vortex strings keep constant radialdistance to the rotational axis, and the azimuthal positions are obtained from extrapolation of the values atthe end of the free wake zone. The last zone is a semi-infinite vortex tube to model the effect of the far wake.The positions of the free wake are updated for one cross-section at the time, and the differences in radial andazimuthal positions are convected to all downstream coordinates of the wake after updating all positions atthat specific axial position. In order to avoid stability problems with the free wake method, the cross-sectionsare not updated in the typically parabolic marching fashion, but according to a scheme that ensures that

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the update cross-section position varies as much as possible in space while still covering all cross-sections inthe free wake domain during a single global iteration. This scheme is an adoption of the integer sequenceA049773 in the on-line encyclopedia of integer sequences.18 Furthermore, relaxation is employed to facilitatea more stable numerical behavior. The relaxation coefficient used for the present computations was set to0.6. The wake vorticity is modeled by rectilinear vortices with a viscous Rankine vortex core: 0.01R at rotordisc going toward 0.05R exponentially with a half-distance of 2R. The results shown in this work were allobtained with the bound vorticity along the main wings discretized in 40 elements, with finer resolutiontowards root and tip where gradients are steeper. The winglet part of the wings was discretized using anadditional 10 elements. Investigations of the discretization have shown that the present setup producesresults that change only marginally by further increasing resolution. The integral forces and dimensionlessnumbers are obtained from integration of the total distributed forces.

A validation of the main part of the code in a non-winglet setting is found in Johansen,8 where comparisonof a priori results obtained with the current code and an actuator disc code were made with results from thefull 3D CFD code EllipSys3D11,12,15 on an aerodynamically optimal rotor. The agreement between CFDand free wake method results was excellent. please refer to8 for the details. For a comparison of the resultsfrom the free wake method with EllipSys3D for rotors including winglets also very a very good agreementwas observed. Please refer to5 for the details on the comparison between CFD and the free wake results inthe wingletted case.

II.B. Load case computations

A matrix of load cases and rotor geometries were simulated using the free wake method. The geometriesconsidered are three bladed rotors with winglets at a right angle to the main part of the rotor, with wingletlengths ranging from −10% to +10% (positive in the downwind direction) in increments of 2% of the rotorradius. The rotor/winglet combinations under consideration in the current work have no sweep angle, sothey point directly in the upwind or downwind direction. For a tip speed ratio of 8, the 11 different rotorgeometries were loaded in five different ways: high loading, medium loading, low loading, low root/high tiploading, high root/low tip loading. An example of the loading shapes are shown in Figure 1 below The

0 0.2 0.4 0.6 0.8 10

0.02

0.04

0.06

0.08

0.1

0.12

0.14

l/ltot

[−]

Γ B

HighMediumLowLow−HighHigh−Low

Figure 1. The different bound circulation loading shapes used in the present work. Here shown for tip speed ratio 8.

functions used for expressing the bound circulations are given in Equations (1) to (4)

ΓB(l) = Ftip(l)Froot(l)Γlin(l) (1)

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Ftip(l) = 2/π cos−1(exp(−CTip(1− l/ltot))) (2)

Froot(l) = (2/π cos−1(exp(−CRoot(l/ltot))))1.5 (3)

Γlin(l) = l/ltotΓTip + (1− l/ltot)ΓAxis (4)

The constants used to determine the bound circulations for the highly loaded cases in the present calculationsare

CTip = 2.671λ (5)

CRoot = 3.125λ (6)

ΓTip = 0.9864/λ (7)

ΓRoot = 0.9504/λ (8)

where λ is the tip speed ratio of the rotor. For the medium loaded cases, ΓTip and ΓRoot are multiplied with0.75 respectively. For the low load cases the scale factor 0.5 is used.

For the computations shown here the viscous drag forces are determined assuming a lift to drag ratioCl/Cd = 110 corresponding to the RisøB1-15 airfoil at α = 80, which operates at Cl = 1.4.

In addition to the 55 main computational cases outlined above, investigations for different tip speed ratiosand tip loading constants (Equation 5) have been run to be able to assess the generality of the obtainedprescribed wake models.

III. Prescribed Wake Models

The present work contain the results of two different prescribed wake models.

1. Simple helical wake, screw surface with constant pitch angle

2. A prescribed wake model based on the integral loads on the rotor and rotor geometry

The models are outlined below.

III.A. Prescribed wake model A: Simple helical wake, screw surface with constant pitch angle

One of the simplest prescribed wake models imaginable is a simple screw surface with a constant pitchangle and constant radius. The trajectories of the trailed wake vortex filaments can in this type of mode bedescribed by a single constant when described in the rotor fixed polar coordinate system: dΘ/dz, where Θis the angle around the rotor rotational axis, z. If we as a crude approximation think of this wake shapecoming from a case without tangential induction, we see that we can use the definition of the tip speed ratio,λ = ΩR/V∞, and the concept of axial induction to write

dΘ

dz= − Ω

V∞(1− a)= − λ

R(1− a)(9)

From 1D momentum theory results we suspect that the axial induction factor should roughly scale asa = 1/2− 1/2

√1− CT with the thrust coefficient. Therefore we set

a = KA(CT )

(1

2− 1

2

√1− CT

)(10)

The parabolic function KA(CT ) is tuned from the free wake ’Low’, ’Medium’ and ’High’ loading cases forthe non-wingletted case at a tip speed ratio of 8, to match the power production for these three cases:

KA(CT ) = 0.1420C2T − 0.2433CT + 1.1694 (11)

The performance of the model is compared to the free wake results later. First we describe a model thatmimics more of the features found in the free wake solutions.

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III.B. Prescribed wake model B: Based on the integral loads on the rotor and rotor geometry

Whereas the previous model was sought as simple as possible to be able to investigate the possibilitiesof using the simplest possible wake geometry, the present model reflects more features found in the freewake solutions: wake expansion, the outermost vortex filaments being convected faster than the inner ones,modeling tangential induction, varying axial transport velocity through the wake. In order to keep themodel in as simple a form possible, the model still only uses the integral quantity CT and winglet size andorientation in the formulation. The model for the main part of the wake basically consist of two parts: thefirst one expressing the radial expansion of the wake, and the second part expresses the pitch angle of thetrailed vortex filaments throughout the wake. The first part of the model is given by Equations (12) to (15)

f1(CT ) = 1.0− 0.715−20.5 CT+25.9 (12)

f2(∆Z) = 1.073(1.0− exp(−0.91∆Z))0.88 (13)

(ri/R)FW (CT ) = f1

√0.5(1/

√(1− CT ) + 1) (14)

(ri/R)(∆Z,CT ) = ((ri/R)FW − (ri/R)Rotor)f2 + (ri/R)Rotor (15)

The function f1 expresses the deviation of the free wake far field radial expansion from what would beexpected from 1D momentum theory, given the thrust coefficient. The other function, f2, expresses theradial expansion as a function of downstream distance, and is fitted the results from the free wake results.

The other part of the model, which deal with the pitch angle of the trailed vortex filaments throughoutthe wake, is easiest described by treating the axial transport velocity and the tangential induction separately,and then combining the the two to finally get the pitch angle of the vortex filaments. The axial transportvelocity correspond to the radial expansion in Equation (15), and can be expressed as in Equations (16)to (19), in terms of an local axial induction factor defined as commonly done in 1D momentum and BEMtheory: Vax = V∞(1− aloc)

a(CT ) = 0.5− 0.5√1− CT (16)

abasis(CT ,∆Z) = 1− a(1/(f2(f1

√(1/

√1− CT + 1)/2− 1) + 1))2 (17)

Further, in order to take into account the faster convection of the tip vortices, all vortices trailed frompositions outside of r/R = 0.9 have a reduced induction factor: the axial induction factor for these vorticesare reduced to a value lower than the ”basis” value all along the filaments. So for the ”inner” vortex filamentsthe local axial induction is given by

aloc,inner(CT ,∆Z) = abasis(CT ,∆Z) (18)

Correspondingly the ”outer” vortex filaments the local axial induction is given by

aloc,outer(CT ,∆Z) = KB(lwl)abasis(CT ,∆Z) (19)

As indicated, the value of the constant KB depends only on the winglet length, and has been tuned to matchthe power production for the ’high’ loaded case for λ = 8. The values corresponding to the different wingletlengths are given in the table below: In fact, the vortices trailed closest to the innermost portion of the

Table 1. Values for KB as function of nondimensional winglet length lwl/R.

lwl/R -0.10 -0.08 -0.06 -0.04 -0.02 0.00 0.02 0.04 0.06 0.08 0.10

KB 0.419 0.426 0.441 0.458 0.478 0.511 0.469 0.450 0.438 0.435 0.423

blade are also convected faster downstream, corresponding also to what happens in the tip section. This is,however, not included in the present model, because the influence from including this effect in the model hasa negligible effect on both the power production and the loads.

The tangential induction is modeled using the result from vortex tube modeling (see the works by Øyeand coworkers in16,17), where it can be shown that the tangential induction in the wake can be shown tohave the magnitude exactly corresponding to the influence of a root vortex having the same strength as thebound vortex strength from all wings at the same position on the wings where the vorticity filament was

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trailed from. Combining this result with the expressions for the axial convection velocity above, we get forthe derivative of the azimuth coordinate of the wake filaments with respect to downwind coordinate

dΘ

dz(CT ,∆Z, λ,NB ,ΓB,i) = −(

λ

R+

NBΓB,i

2πV∞(ri/R)2)/(1− aloc) (20)

Note, that the coordinate system in which the wake positions are described follows the rotor. Therefore,the first term in Equation (20) correspond to the relative tangential velocity due to the rotation of therotor, while the second term correspond to the actual induced tangential velocity in the wake as seen froma ”ground based” observer.

Once the radial expansion and trailed vortex filament pitch angles are determined from Equations (15)and (20), the wake vortex system can be constructed, and the induced velocities, etc. can be determinedfrom them. Note also that when this method is implemented in a tool where the geometry of the rotor isgiven, the determination of a wake corresponding to the loading situation (thrust coefficient) will requireiteration, since the bound circulation, and from these the thrust coefficient, will depend on the inducedvelocities, which in turn depend on the thrust coefficient. However, the numerical solution of this should befairly straightforward.

IV. Results

In this section we first show some overview results from the free wake code computations. Hereafter, weshow results that highlight the behavior of the prescribed wake models compared to the free wake model.

IV.A. Free wake code: results overview

Figure 2 below show the main integral output of the simulated cases with tip speed ratio 8. It is seen that theresults are in agreement with earlier findings: the downwind winglets are more beneficial than the upwindones in terms of power production, and that the increase in power is more rapid closer to having no winglet.That is: an addition of a 2% winglet to an unwingletted rotor can increase the power production more thanwhen increasing winglet length from 8% to 10%. Furthermore, the beneficial effect of the winglets are higherfor higher loading. This also corresponds well with the earlier conclusion by Gaunaa et.al., which stated thatthe beneficial effect of a winglet was an effective reduction of ”tip effects”. Tip related losses increases withloading, so the cases with higher loading, which have higher tip losses, can potentially benefit more from areduction of these.

IV.B. Comparison between free wake model and prescribed wake model A: constant pitchangle

Since the quantities determining the performance in the lifting line approach of determining the forces arethe induced velocities, we will show comparisons of induced velocities as well as of the resulting distributedloads here. Notice, that the axial induced velocity influence the power contribution forces on the main partof the rotor. However, on the winglet part, what determines the part of the forces which go to/from powerproduction is the radially induced velocities. Figure 3 show specific comparisons between prescribed wakemodel A and free wake results. It is seen that the helical wake model overestimates the axial induction (z)and thereby underestimates the power production on the inner part of the rotor. On the outer part thistrend is reversed. Also, it is seen that the radial induction (x) is overestimated on the winglet, which resultsin too low production on the winglet part of the rotor. All trends in the inductions, however, are correctlycaptured using this model. It is also seen that the model does not capture the power production correctlyon the ’low-high’ loaded and winglet cases. This is not surprising, since the model is not tuned for thesecases. Further, it is noted that the error on the cases with low loading is low. The same trends are furtherobserved in Figure 4, where an overview of the integral quantities are seen for this prescribed wake model.It is noted, that while the error on CT is negligible (this is general for these cases since the CT value stemsmainly from the bound vorticity and the rotational eigenmotion, which by fat overshadows the tangentialinduction contribution), the error on CP is substantial for all cases where a part of the rotor is highly loadedif it is not exactly the case for which the model is tuned (no winglet lwl = 0, ’high’, ’medium’ and ’low’loaded cases). On the lower graph it is seen that the predicted increase in production due to the addition ofan winglet is quite substantial in both the ’high’ and the ’low-high’ cases, where errors above 100% are seen.

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−0.1 −0.05 0 0.05 0.1

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

lwl/R [−]

CT [−

]

HighMediumLowLow−HighHigh−Low

−0.1 −0.05 0 0.05 0.10.25

0.3

0.35

0.4

0.45

0.5

0.55

lwl/R [−]

CP [−

]

HighMediumLowLow−HighHigh−Low

Figure 2. Overview of free wake integral results for tip speed ratio λ = 8. Upper: CT versus winglet length. Lower: CP versuswinglet length. The legend indicates the loading type according to figure 1.

IV.C. Comparison between free wake model and prescribed wake model B

Figure 5 show specific comparisons between prescribed wake model A and free wake results. It is seen thatthis model better captures the distribution of the quantities over the span of the rotor than the simplermodel for the cases for which they are tuned. As the other model it does have some problems with the casewith a different loading along the blade (’low-high’) than what it was tuned for. Even though the comparisonbetween the prescribed and the free wake looks fairly good, and even the integral loads for the ’low-high’loading case looks decent, Figure 6, investigation of other loading cases reveal how sensitive to tuning thistype of model is. Figure7 show that the change in power production predicted with prescribed wake modelB can be much different than what the free wake code predicts. A contributing factor to this result maybe due to the combination of numerical noise on the free wake results and the relatively small change inpredicted power production for this change in circulation. The last comparison, Figure8 show the effect ofchanging the tip speed ratio away from the basis value 8. Again, it is noted that if the model is used out ofthe region it is tuned for the results may be misleading.

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0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1−0.5

−0.4

−0.3

−0.2

−0.1

0

0.1

0.2

0.3

0.4

0.5

blade surface length/R [−]

Indu

ced

velo

citie

s/V

∞ [−

]

ViX

ViY

ViZ

ViXHelical

ViYHelical

ViZHelical

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10

0.2

0.4

0.6

0.8

1

blade surface length/R [−]

Cp

and

Ct [

−]

CP=0.51242

CT=0.87347

CPHelical=0.51241

CTHelical=0.87341

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1−0.5

−0.4

−0.3

−0.2

−0.1

0

0.1

0.2

0.3

0.4

0.5

blade surface length/R [−]

Indu

ced

velo

citie

s/V

∞ [−

]

ViXViYViZViXHelicalViYHelicalViZHelical

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10

0.2

0.4

0.6

0.8

1

blade surface length/R [−]

Cp

and

Ct [

−]

CP=0.34822

CT=0.43289

CPHelical=0.34822

CTHelical=0.43292

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1−0.5

−0.4

−0.3

−0.2

−0.1

0

0.1

0.2

0.3

0.4

0.5

blade surface length/R [−]

Indu

ced

velo

citie

s/V

∞ [−

]

ViXViYViZViXHelicalViYHelicalViZHelical

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10

0.2

0.4

0.6

0.8

1

blade surface length/R [−]

Cp

and

Ct [

−]

CP=0.47832

CT=0.71258

CPHelical=0.4817

CTHelical=0.71264

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1−0.5

−0.4

−0.3

−0.2

−0.1

0

0.1

0.2

0.3

0.4

0.5

blade surface length/R [−]

Indu

ced

velo

citie

s/V

∞ [−

]

ViXViYViZViXHelicalViYHelicalViZHelical

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1−0.4

−0.2

0

0.2

0.4

0.6

0.8

1

blade surface length/R [−]

Cp

and

Ct [

−]

CP=0.53554

CT=0.91937

CPHelical=0.51869

CTHelical=0.91931

Figure 3. Comparison between free wake and prescribed wake model A. Left figures induced velocities, right figures show localCt and Cp. Both versus position along the blade. Note that the crosses in the tangential induction correspond to the vortex tubeapproximation where the induced velocity correspond to a half 2D vortex at the root position with the vortex strength correspondingto the bound vorticity on all three blades at that radial position.

V. Conclusion

A free wake method was used to determine the wake shape for rotors with up to 10% radius length upand downstream for different loadings and tip speed ratios. From these results, two different prescribed wakemethod have been derived. The results show that results from a prescribed wake model using only the integralaxial load on the rotor and the main geometry of the wing/winglet can produce results in good qualitativeagreement with the physically more correct free wake model. The results indicate, however, that the power

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−0.1 −0.05 0 0.05 0.1−1.5

−1

−0.5

0

0.5

1

1.5x 10

−4

lwl/R [−]

(CT

,hel

ical−

CT)/

CT [−

]

HighMediumLowLow−HighHigh−Low

−0.1 −0.05 0 0.05 0.1−0.05

−0.04

−0.03

−0.02

−0.01

0

0.01

lwl/R [−]

(CP

,hel

ical−

CP)/

CP [−

]

HighMediumLowLow−HighHigh−Low

−0.1 −0.05 0 0.05 0.1−1.8

−1.6

−1.4

−1.2

−1

−0.8

−0.6

−0.4

−0.2

0

lwl/R [−]

(∆C

P,h

elic

al−

∆CP

,FW

)/∆C

P,F

W [−

]

HighMediumLowLow−HighHigh−Low

Figure 4. Comparison between free wake and prescribed wake model A. Upper left: Error on CT . Upper right: Error on CP .Lower: Relative error in power production difference compared relative to the corresponding case with no winglet.

production estimated from these prescribed wake models may be erroneous if the load distribution alongthe wing is not similar to the one the model was calibrated for. Further work is needed to clarify whethera prescribed wake model using local characteristics (loading, induction) along the rotors can overcome theshortcomings of the simpler models illuminated in this work.

VI. Further work & Outlook

1. Investigate the performance of prescribed wake models that employ also the local loading along thewing (for example HAWTDAWG3 initially developed at Glasgow University)

2. Add a winglet of ”sensible” shape to an existing wing (for instance the NREL/Upwind rotor), andperform the final test of the prescribed wake models by comparing the predicted performance withCFD results

3. Investigate the effect of sweep angles with the free wake tool, and compare such results with CFD

Acknowledgments

The authors gratefully acknowledge that this work is partly funded by the Danish Energy Council in theproject EFP vingetipper.

References

1Van Bussel, G.J.W. A momentum theory for winglets on horizontal axis wind turbine rotors and some comparisonwith experiments Fourth IEA Symposium on the Aerodynamics of Wind Turbines, November 1990, Rome, Italy, Proc. edt.K.McAnulty, ETSU-N-118, Harwell, Didcot, U.K., January 1991

2Chattot, J.J., Effects of Blade Tip Modifications on Wind Turbine Performance Using Vortex Model, AIAA 2008-1315,46th AIAA Aerospace Sciences Meeting and Exhibit, 7-10 January 2008, Reno, Nevada.

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0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1−0.5

−0.4

−0.3

−0.2

−0.1

0

0.1

0.2

0.3

0.4

0.5

blade surface length/R [−]

Indu

ced

velo

citie

s/V

∞ [−

]

ViX

ViY

ViZ

ViXModel

ViYModel

ViZModel

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10

0.2

0.4

0.6

0.8

1

blade surface length/R [−]

Cp

and

Ct [

−]

CP=0.51242

CT=0.87347

CPModel=0.51252

CTModel=0.87351

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1−0.5

−0.4

−0.3

−0.2

−0.1

0

0.1

0.2

0.3

0.4

0.5

blade surface length/R [−]

Indu

ced

velo

citie

s/V

∞ [−

]

ViXViYViZViXModelViYModelViZModel

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10

0.2

0.4

0.6

0.8

1

blade surface length/R [−]

Cp

and

Ct [

−]

CP=0.47832

CT=0.71258

CPModel=0.48975

CTModel=0.71273

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1−0.5

−0.4

−0.3

−0.2

−0.1

0

0.1

0.2

0.3

0.4

0.5

blade surface length/R [−]

Indu

ced

velo

citie

s/V

∞ [−

]

ViXViYViZViXModelViYModelViZModel

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1−0.4

−0.2

0

0.2

0.4

0.6

0.8

1

blade surface length/R [−]

Cp

and

Ct [

−]

CP=0.53554

CT=0.91937

CPModel=0.53601

CTModel=0.91936

Figure 5. Comparison between free wake and prescribed wake model B. Left figures induced velocities, right figures show local Ct

and Cp. Both versus position along the blade.

−0.1 −0.05 0 0.05 0.10

0.005

0.01

0.015

0.02

0.025

0.03

0.035

lwl/R [−]

(CP

,Mod

el−

CP)/

CP [−

]

HighMediumLowLow−HighHigh−Low

−0.1 −0.05 0 0.05 0.1−0.05

0

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0.4

lwl/R [−]

(∆C

P,M

odel−

∆CP

,FW

)/∆C

P,F

W [−

]

HighMediumLowLow−HighHigh−Low

Figure 6. Comparison between free wake and prescribed wake model B. Left: Error on CP . Right: Relative error in powerproduction difference compared relative to the corresponding case with no winglet.

3Currin, H.D., Coton, F.N. & Wood, B. Dynamic Prescribed Vortex Wake Model for AERODYN/FAST, Journal of SolarEnergy Engineering, Vol. 130. 7 p. August 2008.

4Gaunaa, M & Johansen, J. Determination of the maximum aerodynamic efficiency of wind turbine rotors with winglets,Journal of Physics 5, 2007 and oral presentation on 2nd EWEA,EAWE The Science of making Torque from Wind conference

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10 15 20 25 30−5

−4

−3

−2

−1

0

1

2

3x 10

−3

TipConst [−]

(CP

,Mod

el−

CP)/

CP [−

]

10 15 20 25 300

0.5

1

1.5

2

2.5

3

3.5

4

4.5

TipConst [−]

(∆C

P,M

odel−

∆CP

,FW

)/∆C

P,F

W [−

]

Figure 7. Comparison between free wake and prescribed wake model B for different tip loading shapes (The reference case value is20). No winglet. Left: Error on CP . Right: Relative error in power production difference compared relative to the reference case.

4 5 6 7 8 9 10−1

0

1

2

3

4

5

6

7

8x 10

−4

λ [−]

(CT

,mod

el−

CT)/

CT [−

]

4 5 6 7 8 9 10−0.01

−0.005

0

0.005

0.01

0.015

0.02

0.025

λ [−]

(CP

,mod

el−

CP)/

CP [−

]

Figure 8. Comparison between free wake and prescribed wake model B for different tip tip speed ratios (The reference case isλ = 8.). No winglet. Left: Error on CT . Right: Error on CP .

at DTU, 28-31 August 20075Gaunaa, M & Johansen, J. Can CP be Increased by the Use of Winglets? or A Theoretical and Numerical Investigation

of the Maximum Aerodynamic Efficiency of Wind Turbine Rotors with Winglets, 46th AIAA Aerospace Sciences Meeting andExhibit, 7-10 January 2008, Reno, Nevada.

6Gaunaa, M & Johansen, J. Estimation of possible increase in Cp by use of Winglets, In: Bak, C. (ed.), Ris NationalLaboratory (DK). Wind Energy Department. Research in aeroelasticity EFP-2006. Ris-R-1611(EN) p. 47-62

7Imamura, H. et al Numerical Analysis of the Horizontal Axis Wind Turbine with Winglets, JSME International Journal,Series B vol. 41 nr. 1, Feb 1998 JSME Tokyo Japan pp 170-176

8Johansen, J., Madsen, H, Aa., Gaunaa, M. and Bak C. Design of a wind turbine rotor for maximum aerodynamic efficiencyWind Energy, Vol. 12, 2009, p 261-273.

9Johansen, J.; Sørensen, N.N.; Aagaard Madsen, H.; Wen Zhong Shen; Okulov, V. Advanced rotor aerodynamics - in-cluding tip and root aerodynamics. In: Bak, C. (ed.), Ris National Laboratory (DK). Wind Energy Department. Research inaeroelasticity EFP-2005. Ris-R-1559(EN) p. 11-17

10Johansen, J. and Sørensen, N.N. Numerical Analysis of Winglets on Wind Turbine Blades using CFD, EWEC 2007Conference proceedings, Madrid, Spain

11Michelsen J.A. Basis3D - a Platform for Development of Multiblock PDE Solvers. Technical Report AFM 92-05, TechnicalUniversity of Denmark, 1992

12Michelsen J.A. Block structured Multigrid solution of 2D and 3D elliptic PDE’s. Technical Report AFM 94-06, TechnicalUniversity of Denmark, 1994

13Muller R.H.G & Staufenbiel R The Influence of Winglets on Rotor Aerodynamics Vertica Vol. 11, No. 4, p.601-618. 198714Muller R.H.G Winglets on Rotor Blades in Forward Flight a Theoretical and Experimental Investigation Vertica Vol.

14, No. 1, p.31-46, 199015Sørensen N.N. General Purpose Flow Solver Applied to Flow over Hills. Ris-R-827-(EN), Ris National Laboratory,

Roskilde, Denmark, June 199516Øye, S. A simple vortex model. In Proc. of the third IEA Symposium on the Aerodynamics of Wind Turbines, ETSU,

Harwell, 1990, p. 4.1-4.15.17Madsen, H.Aa., Mikkelsen, R., Johansen, J., Bak, C., Øye, S. & Sørensen, N.N Inboard rotor/blade aerodynamics and its

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influence on blade design. Chapter in Risøtechnical report: Research in aeroelasticity EFP-2005. Ed. Bak, C. Risø-R-1559(EN),Roskilde, Denmark, May 2006. p.19-39.

18The On-Line Encyclopedia of Integer Sequences. http://www.research.att.com/ njas/sequences/A049773

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