Aerodynamic Winglet Optimization

Søren Hjort, Jesper Laursen, Peder B. Enevoldsen Siemens Wind Power A/S, Borupvej 16, 7330 Brande, Denmark

shj@siemens.com

pe@siemens.com

laursen.jesper@siemens.com Abstract. During the last couple of years winglets for wind turbine blades have experienced increasing interest. Gaunaa & Johansen [2] showed that electric power can be enhanced by winglets only due to a reduction in tip-loss effects. As a consequence, methods that do not take azimuthal variation of the inducted velocity field into account are not suitable for winglet evaluation. This excludes common BEM codes as viable tools for winglet design, at least with the tip-loss corrections presently available. In response to this need for an applicable engineering code, an efficient free wake vortex line method has been developed for general rotor evaluation. Uniform axial inflow is assumed. The new methods are validated by comparison with BEM calculations, full cfd rotor simulations and field measurements for a Siemens multi-MW turbine. A gradient-based aerodynamic blade optimization is then performed for a representative windspeed and rotor rotational velocity. The target parameter is the mechanical power coefficient. Four testcases are presented: Chord optimization along the entire blade span, then edgewise, flapwise and combined edge/flap-wise optimization of the outermost 2.5 radial meters of the blade. A brief summary of results concludes the paper.

1. Introduction In these years the wind turbine industry seems to have entered a maturing stage, and is no longer to be considered a pioneering technology. As a consequence there has also been a shift in focus. Earlier development has been heavily challenged by the tremendous up-scaling of rotor size during the last 3 decades. Now, as the up-scaling has slowed down, aerodynamic attention is increasingly directed towards model improvement and exotic features involving complex flows not suited for computation by existing engineering aerodynamic design methods. An example is the winglet, well known from the aircraft industry. Is it applicable to wind turbines as well ? Can we evaluate winglets with standard aerodynamic tools ? The answer to the second question is non-affirmative. Today’s workhorse for aerodynamic wind turbine rotor design and evaluation is the Blade-Element-Momentum (BEM) method [1]. For straight slender blades geometrically bound to a plane BEM delivers surprisingly accurate results considering the simplicity of the method. In contrast, for non-straight blades and/or blades not geometrically bounded to a plane, the BEM assumptions become invalid. Examples of blade features not accounted for by BEM include: Large cone angles, large edge- and/or flap-wise pre-deflections, and of course winglets. On an axis for numerical complexity (storage/cpu resources) BEM is located at the very start as the simplest rotor induction method available. A BEM calculated rotor-evaluation takes a couple of milli-seconds on a standard pc. At the other extreme we find full cfd rotor calculations solving the Reynolds-Averaged Navier-Stokes equations (RANS), e.g [8], [9]. Such a calculation typically runs for 24 hrs on a parallel pc-cluster with 10 nodes. In the present paper we aim at a general induction method suitable for slender arbitrary geometry blades, but with a cycle time for a rotor evaluation of up to a few minutes. This limit still allows rotor-design optimizations requiring thousands of rotor evaluations to be performed within one or a few days.

Our new method should be able to handle the examples mentioned where BEM is not applicable. Specifically for winglets it has been shown [2], [3], that the optimum rotor power coefficient, Cp, can be enhanced by winglets, but only due to a reduction in tip-loss. Hence, it was proven that under the actuator disc assumption, maximum power only depends on the plane-projected area swept by the blades, whereas maximum power is insensitive to winglets bended up- or down-wind as long as the swept area is kept constant. This means that our search for an engineering method for arbitrary geometry blades must go further, in terms of complexity, than the cfd actuator disc solver, since this method is based on the assumption that the blade forces can be smeared out tangentially, transforming the rotor into a disc. The reader is referred to [4] for an introduction to the actuator disc method. Beyond what we consider computationally efficient reside the iterative free wake vortex-line methods [2], [5] and actuator line method [4]. The remaining part is organized as follows: Our new model is presented in chapter 2 and validated in chapter 3. Chapter 4 presents a series of testcases dealing primarily with winglet evaluation and optimization. A short summary concludes the paper. 2. The model Assuming familiarity with the concepts and methods mentioned in the introduction, our model can be described briefly as a non-iterative free wake vortex-line method. To the authors’ knowledge this method is new. It consists of a few main components, each of which being of ancient origin. The novel part that allows the free-wake vortex-line method to be non-iterative ensures low computational complexity: The shed vorticity is convected downstream along streamlines found from a cfd actuator disc solution. Although not perfect, the actuator disc solution offers a very qualified estimate for the traces of the shed vorticity used in the vortex-line method. The components of our new induction method are presented below. Mesher The computational domain for the cfd actuator disc solver is 2D due to tangential periodicity. It is discretized using a curvi-linear structured quad mesher. Essentially the method is of “advancing front” type. It starts from the disc line which has been extended to the radial farfield, and works its way advancing upstream to the inlet. Then the second block is generated starting once again at the rotor-disc line, but advancing downstream all the way to the outlet. The 2 blocks are then connected. An example is shown in fig.1. CFD actuator disc solver Just like BEM the actuator disc method uses profile coefficients to find the forces between the blades and the fluid, but the flowfield is now calculated by solving the Navier-Stokes equations. The forces from the blades are smeared out tangentially and applied to the system of equations as RHS volume forcing on the momentum equations. The forcing is applied for those of the mesh elements which are located on the disc line. The steady incompressible NS-equations in cylindrical co-ordinates with tangential periodicity on differential form are: Continuity:

0)(=

∂∂

+∂

∂xv

rr

rv xr (1a)

r denotes radius or radial dimension, x is the rotor axial dimension and θ the azimuthal (tangential) dimension.

Figure 1: Left: Computational domain. Left boundary: Inflow, Right boundary: Outflow, Lower boundary: Rotational axis of symmetry. Upper boundary: Outflow condition ensured due to radial farfield inclination. Right upper: Close-up. Right lower: Close-up in vicinity of winglet tip. Note how the mesh is strictly fitted to the disc line.

−2 0 2 4

44

45

46

47

48

Axial distance [m]

256 x 224 domain (close−up)

−20 −10 0 10 20

30

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Axial distance [m]

256 x 224 domain (close−up)

−400 −300 −200 −100 0 100 200 300 400−100

0

100

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300

400

500

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Axial distance [m]

radi

al d

ista

nce

[m]

256 x 224 domain

Momentum:

xxx

rx

xx

r fxv

rvr

rxp

xvv

rvv =⎟

⎟⎠

⎞⎜⎜⎝

⎛

∂

∂+⎟

⎠

⎞⎜⎝

⎛∂∂

∂∂

−∂∂

+∂∂

+∂∂

2

211

ρµ

ρ (1b)

rrr

rr

xr

r fxv

rrv

rrp

rv

xvv

rvv =⎟

⎟⎠

⎞⎜⎜⎝

⎛

∂

∂+⎟

⎠

⎞⎜⎝

⎛∂

∂∂∂

−∂∂

+−∂∂

+∂∂

2

211

2 )(ρµ

ρθ (1c)

θθθ

ρµθθθ f

xv

rrv

rrvv

xvv

rvv r

rxr =⎟

⎟⎠

⎞⎜⎜⎝

⎛

∂

∂+⎟

⎠

⎞⎜⎝

⎛∂

∂∂∂

−+∂∂

+∂∂

2

21 )( (1d)

The term contains the mass-specific external forcing, where the actuator disc’s forcing on the fluid is applied. The lift force along the disc in the x-r 2D domain is recalculated at each iteration. Given the instantaneous flowfield, the velocity vector relative to the rotating blade is projected onto the profile plane at each point along the disc line. The incidence angle is computed and lift/drag coefficients are then looked up from the polar tables. For arbitrary geometry blades the span-wise velocity component might be large. This is accounted for by applying the drag force based on the size and direction of the non-projected velocity vector.

f

The system of equations (1a-d) is cast in a curvilinear finite volume formulation, and solved using the SIMPLE algorithm by Patankar [6]. To speed up convergence multigrid acceleration is applied. For details, see [7]. Applying V-cycles on 3 grid levels substantially reduces the number of iterations needed for full convergence, typically to around 300.

Vortex-line method For slender blades the lifting line vortex method enables a simple computation of the whole 3-dimensional flowfield. Mathemati-cally this method rests on the principles laid out by potential flow theory and in particular Helmholtz’ theorems:

1. Fluid elements initially free of vorticity remain free of vorticity.

2. Fluid elements lying on a vortex line at some instant continue to lie on that vortex line. More simply, vortex lines move with the fluid.

3. The strength of a vortex tube does not vary with time.

Bound vorticity is computed along the span of the blade via the Kutta-Joukowski Theorem:

Γ

Γ×⋅⋅∆= relL VsF ρ (2)

Figure 2: Fundamental subset of vorticity lines: Bound vorticity from one blade (magenta), shed root vorticity

0 50 100 150 200−50050

−100

−50

0

50

100

Tip− and root−vortex from 1 blade

(blue) and shed tip vorticity (blue)

LF is the known lift force vector from the converged actuator disc solution. is the length of the discretized blade segment, and

s∆

relV is the velocity vector relative to the rotating blade. After computation of bound vorticity distribution, the shed vorticity distribution is computed from the requirement that all vortex lines are continuous and extend infinitely far downstream. Using the actuator disc solution, the shed vorticity can now be traced as it propagates downstream. A large number of shed vortex helices are traced from each blade, a fundamental subset of which is depicted on fig.2. All vortex lines are discretized into filament segments. The induced velocity is calculated from Biot Savart’s law. Specifically for a linear vortex filament we have [10] :

)())((

4)(

212121

2121rrrrrrrrrrxv P ⋅+

×+⋅

Γ=Γ π

(3)

Px is the point in Cartesian space where induction from the filament is to be evaluated. 1r and

2r are position vectors from the vortex filament line start and end positions, 1x and 2x , to the evaluation point Px respectively. Equation (3) is singular for Px approaching the filament line. This is remedied by introducing a viscous cut-off radius:

2212121

2121

)())((

4)(

cP

rrrrrrrrrrrxv+⋅+

×+⋅

Γ=Γ π

(4)

cr is in the order R⋅01.0 . As the shed vortices propagate downstream the need for discretization

diminishes. In response to this, adjacent shed vortices will be merged into one another in a systematic way until approximately 2.5 rotor diameters downstream. Here we adopt a vortex tube formulation and express the residual induction along the blade using semi-analytic closed-form solutions involving elliptic integrals.

Optimization routine The first three steps of our method (mesher, actuator disc cfd solver, vortex-line method) are accomplished in 2-3 minutes on a single-processor laptop pc. This is a significant leap forward in terms of execution speed for this kind of calculations. The disc solver is the most computationally expensive component consuming approximately 2/3 of the calculation time, and it scales almost linearly with problem size due to multigrid acceleration. The size of the discretized domain is annotated on fig.1. Due to the reduced computation time, it is feasible to release the blade layout distributions, e.g. chord, edge-/flap-wise deflection etc, and feed them into an optimization routine. The present optimization package is of line-search/quasi-newton type. It scales fairly linearly in complexity with the number of Degrees of Freedom (DoF) and handles unconstrained as well as bounded optimization problems.

3. Validation The first part of the validation relates to the actuator disc cfd solver. It is compared to a BEM calculation without Prandt’s tip-loss correction. The turbine is an SWT-2.3-93: Siemens Wind Turbine, 2.3 MW, 93 m rotor diameter. The compared quantity is the rotor integrated mechanical power.

Wind speed [m/s]

Rotor rot. Velocity

[rpm]

Xblade BEM code

[kW]

Act. Disc new method

[kW]

Tolerance

6 10.0 424 423 -0.2 % 8 13.5 1009 1006 -0.3 % 10 16.0 1939 1945 0.3 %

Table 1

The energy yield calculated by both methods correlate very well. The second part of the validation deals with the vortex-line method.

Wind speed [m/s]

Rotor rot. Velocity

[rpm]

Field measurement

[kW]

ANSYS-CFX / Ellipsys

[kW]

Vortex line New method

[kW] 6 10.0 400 396/392 413 8 13.5 986 950/945 982 10 16.0 1894 1853/1853 1897

Table 2

The comparisons in table 2 involve field measurements from the SWT-2.3-93 at Høvsøre National Test Site in Jutland (Denmark), a commercial and an academic research cfd code and of course the non-iterative free wake vortex-line method. Overall nice agreement. Both cfd codes seem to underestimate the power by a few percent. This discrepancy relates to the applied type of boundary layer laminar-turbulent transition. The above cfd results are for fully turbulent cases. If run with laminar-turbulent transition enabled, the CFX results would have been approximately 7% higher [8]. So much for the rotor integrated power output. If we draw closer attention to the radial blade distributions, an important trend is to be noticed, see fig.3. The actuator disc method gives lower induction and higher power on the inboard part of the blade on expense of higher induction and lower power on the outboard part of the blade. The same holds for the vortex-line method, although here, the inherent inclusion of tiploss obscures comparison on equal

terms. The slight redistribution, which is quite notable in testcase 1 because of the power-optimal high loading, is a confirmation of similar results obtained by Risø [11].

10 15 20 25 30 35 40 450

0.1

0.2

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0.4

0.5

rotor radius [m]

Axi

al in

duct

ion

Radial distribution comparisons

BEM without tiploss corr.Act.disc without tiploss corr.Vortex−line (inherent tiploss)

10 20 30 40

0

5

10

15

20

25

30

35

Rotor radius [m]

Mec

h.po

wer

[kW

/m]

Radial distribution comparisons

BEM without tiploss corr.Act.disc without tiploss corr.Vortex−line (inherent tiploss)

Figure 3: Comparison of relative axial induction (left) and mechanical power per radial blade meter for the three induction methods. The blade and operational point is the optimized result from testcase 1, see below.

4. Testcases The four testcases to be presented all share the same settings unless otherwise stated. The inflow is purely axial and uniform 10 m/s. The rotor is non-coned and non-tilted with a 46.2 m radius. Each of the three blades has fixed lift-coefficient 1.0 all along the span. The corresponding drag-coefficient is 0.01, such that the glidenumber is exactly 100. The rotor rotational velocity is 16.0 rpm. The blade layout radial distributions, chord, edge-/flap-wise deflection all consist of 33 points distributed with increasing resolution towards the tip. For each testcase all or a subset of the points for one or multiple distributions are released as Degrees of Freedom (DoFs) for the optimization routine. The optimization target parameter is the mechanical power coefficient, Cp. Testcase 1: Straight blade chord optimization We use classic BEM without tiploss correction to find the Cp-optimal distri-bution of chord along the blade span. All 33 points are released. A maximum chord constraint of 3.5 m is imposed. Fig.4 shows the celebrated Betz’ limit (black) as the maximum obtainable Cp derived from the BEM method, 16/27 constantly along the span. The dark blue line is the BEM-calculated Cp for the BEM-optimized chord distribution with no drag. Notice how the Cp approaches Betz’ limit asymptotically towards the blade tip as the tip-speed ratio increases. The green line is for the same chord distribution, but Cp is now calculated with the actuator disc method. Notice the afore-mentioned redistribution of power to-

Figure 4: A selection of Cp-distributions

10 15 20 25 30 35 40 450

0.1

0.2

0.3

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rotor radius [m]

mec

h. c

p

Act.Disc. : Cp = 0.5169Act.Disc. no drag: Cp = 0.5610BEM no drag: Cp = 0.5581Betz limit: Cp = 0.5926

maintained constant throughout the remainintestcases.

Figure 5

Testcase 2: Edgewise winglet optimization

wards the root, leading to local spanwise values for Cp on the inboard part of the blade exceeding Betz. The cyan line of fig.4 is similar to the green,

for Cd = 0.01. The BEM-optimized chord on is depicted on fig.5 and will be

g

but nowdistributi

This time the outermost 16 points (2.5 radial meters) are released for the edge-wise deflectiodistribution. The vortex-line method optimized geometry is shown in fig.6. The rotor rotationirection is clock-wise, so the edgewise winglet is down-winded w.r.t. the re

n al

lative velocity rojected onto the rotor (edgewise) plane. The obtained Cp-gain is 1.1 percent. This number

should be compared to the Cp difference between the actuator disc efficiency (0.5168) and the vortex-line calculated efficiency (0.4950) which is 4.3 percent and thus a quantification of the tiploss, which according to [2] should be the theoretical maximum for the obtainable Cp-gain from applying winglets.

Figure 6

dp

41 42 43 44 45 46 47

−3

−2

−1

0

1

2

Edge−wise winglet geometry

10 20 30 400

1

2

3

4BEM − optimized chord distribution

Testcase 3: Flapwise winglet optimization Same procedure as for the preceding testcaseedgewise. Again the vortex-line method is flapwise deflection of 2.5 m is imposed. Fig.7and the pressure iso-lines in the surroundinggain is 2.6 percent, or 60% of what is theoretic

, but now for the flapwise distribution and not the used, and in the optimization settings a maximum shows the flapwise geometry along with the mesh vicinity of the downwind winglet. The obtained Cp-

ally obtainable [2].

et optimization

Figure 7 Testcase 4: Combined edge-/flap-wise wingl

Figure 8

−0.5 0 0.5 1 1.5 2 2.5 3

44.5

45

45.5

46

46.5

47

Velocity magnitude iso−lines

−0.5 0 0.5 1 1.5 2 2.5 3

44.5

45

45.5

46

46.5

47

Static pressure iso−lines

No meflap-wise deflection distribution. A 2.5 m flapin the optimization routine. Fig.8 shows theNote the reduction of the downwind curvatufrom the preceding testcase (fig.7). The Cp-sum of the individual Cp-gains from the isola 5. Conclusions In response to a need for more complex blade evaluation, a computationally fast non-iterative free-wake vortex line method suitable for a itrary geometry slender turbine blades has been developed, validated and applied specifically for winglet optimization. Although for a straight slen r blade seems convincing, further validation is need ore complex blade geometries, .g. the optimized winglets. Comparison with cfd is

1. Durand W F, Aerodynamic Theory, Vol IV, Div. L: Airplane Propellers by H. Glauert, 1935. 2. Gaun ic Efficiency of Wind Tu s 75 (2007)

012006 3. Øye S, etween thrust

and indu um on the Aero

4. Mikkelsen R, is, Technical Uni

5. Van Garrel A, ule. ECN-C-03-0

6. Patankar ss and momentu al of Heat and Mass Tran

7. Ferziger J ition, Springer, 2002

8. Laurse ance of a Multi- Megawatt Wi 2007 9. Johansen ades using CFD10. Phillips W F & Sny Prandtl’s Classic Lifting-Line Theory, Journal of Aircraft, Vol.37, No.4, pp.662-670, 2000.

1. Madsen H Aa, Mikkelsen R, Øye S, Bak C, Johansen J, A Detailed investigation of the Blade Element Momentum (BEM) model based on analytical and numerical results and proposal for modifications of the BEM model. Journal of Physics: Conference Series 75 (2007) 012016

ters) are released as DoFs for both the edge- and wise maximum deflection is imposed as upper limit final optimized geometry in the flapwise direction. re radius compared to the purely flapwise winglet

gain for the combined optimization is more than the ted edge- and flap-wise winglet optimizations.

w the outermost 16 points (2.5 radial

rb

the validation deed for m e

a logical next step. Discussing the specific optimized winglet geometries is beyond the scope of this paper, as is any load related issue. The industrially relevant blade design seeks to optimize the energy yield per load, and not only the energy output. Still, pursuing maximum Cp remains an issue of academic interest.

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1