Tong W. Wind Power Generation and Wind Turbine Design

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Aerodynamics and Aeroelastics of Wind Turbines 95

0.3

0.35

0.4

0.45

0.5

0.55

0.6

5 5.5 6 6.5 7 7.5 8

TSR

Blade performance B = 3

Swirl-Glauert

Tip-Wilson

L/D 200

L/D 100

L/D 40

Figure 6: Comparison of swirl drag and proﬁ le-drag losses against measured

values of actual turbines.

despite the fact that the otherwise neglected pressure decrease in the near wake

gives higher c

in the inboard section, Betz’ limit is valid globally.

When considering drag losses, one has to imagine that a rotor blade can be regarded

as an aerodynamic device experiencing two forces drag (in ﬂ ow direction) and lift

(orthogonal to that). The lift force and part of the drag force (per unit span) are due to

the pressure around the airfoil. Figure 7 illustrates this. The c

is deﬁ ned as

(2)

( 14 )

where c is the chord of the airfoil.

Here:

(2) · 1

vcr

( 15)

and

(2) · 1

vcr

(16 )

are deﬁ ned. It can be shown that lift gives rise to no loss as it is perpendicular to

the ﬂ ow, which is not the case for drag. A measure of efﬁ ciency for proﬁ les is

96 Wind Power Generation and Wind Turbine Design

deﬁ ned as the lift-to-drag-ratio L / D. Usually this number is around 100. In total

Figs. 10 and 11 are obtained. Compared to Fig. 3 the quantitative inﬂ uence of drag

and ﬁ nite bladenumber on c

are presented. In addition Fig. 6 shows the improve-

ment of state-of-the art commercial wind turbines of one manufacturer over a span

of 20 years. Clearly one can see that high-performance airfoils have to be used to

reach for a

max

c in the order of 0.52. It is clear that B = 1 is very special, and one

might assume that no such turbines were manufactured. This in fact was not the

case. In the late 1980s the German company MBB manufactured the so-called

Monopteros (see Fig. 9 ), a single-blade turbine. To demonstrate the big differences

data from Rohrbach et al. [ 46 ] is included.

Figure 7: Pressure coefﬁ cient around an airfoil.

Figure 8 : Lift and drag around an airfoil; v is a sample inﬂ ow velocity and is not

to be confused with a relative velocity.

Aerodynamics and Aeroelastics of Wind Turbines 97

0.3

0.35

0.4

0.45

0.5

0.55

0.6

0 5 10 15 20

TSR

L/D-Variation, B=3

Glauert Ideal

L/D = Inf

L/D = 160

L/D = 80

L/D = 40

L/D = 20

Figure 10: c

via TSR: B = 3, L / D varies.

Figure 9 : Blade elements of a rotor.

Sørensen and Okulov [ 39 , 40 ] recently formulated a vortex theory for these

types of rotors (see Fig. 11 ).

To sum up: at the present time a limit of c

= 0.52 seems to have been reached

by modern turbines, which is only possible if specially designed airfoils are used.

A c

of 0.45 (typical for turbines in the early 1990s) is obtained using old proﬁ les

from the aerospace industry (see Figs 3 and 6 ).

98 Wind Power Generation and Wind Turbine Design

2.1 Blade element theories

Blade element theory (see Fig. 9 ) divides the rotor into several ﬁ nite length sec-

tions of Δ r (or dr mentioned above). Then forces applied on these annuli are com-

pared to those from airfoil theory. Implicitly it is assumed that there is no mutual

interference of the sections. With reference to Fig. 9 and denoting with dm˙ the

mass ﬂ ow through the disk and with v

the tangential velocity, a thrust

0.1

0.2

0.3

0.4

0.5

0.6

0 5 10 15 20

TSR

B Variation; D/L = 0

Glauert Ideal

Wilson B=4

Wilson B=3

Wilson B=2

Wilson B=1

RohrbachWorobel B=1

Okulov B=1

Figure 11: c

via TSR: D = 0, bladenumber varies.

Figure 12: Single bladed wind turbine Monopteros by MBB, Photo: Schaffarczyk.

Aerodynamics and Aeroelastics of Wind Turbines 99

(),dT dm v v=−



( 17)

4(1 )2

dT a a r dr v

p=+

( 18)

and torque

dQ dm v r=



( 19)

2,2dQ r dr v rprw=

(20 )

4(1).dQ r v a apr=Ω−

′

( 21 )

increment is obtained.

Now comparison with the forces resulting from the airfoil sections is performed.

From Fig. 8 the force coefﬁ cient in the inﬂ ow direction is

NL D

·() ·().CCcos Csinaa=+

(22 )

Here the ﬂ ow angle ϕ can be computed from the velocity triangle .

(1 )

() .

(1 )

tan

−

′

( 23)

The angle of attack and the ﬂ ow angle is related via the twist angle ϑ :

.ajJ=−

( 24 )

The square of the velocity is

22 2

((1 ) ) ( (1 )) .wav aw=− + +

′

( 25 )

All relevant data is then calculated using an interaction scheme. It is to be noted that

in the inner part of a blade, values of greater than 0.5 and even 0.52 are observed.

Then the simple momentum theoretical value c

( a ) = 4(1 – a ) · a is no longer valid.

An empirical extension for 0.5 < a < 1.0 must be used. The deviation starts at a ≈

0.3 and gives c

( a = 1) of approximately 2.

Many engineering codes, such as the PROP Code of Walker and co-workers [ 18 ]

follow this approach. These codes rely heavily on measured aerodynamic data for

airfoils. In the early days of wind energy, airfoils from ordinary airplanes were used.

Since then, special airfoils for wind turbines have been developed, mainly at Stutt-

gart (FX-Series), Delft (DU proﬁ les) and Risø. Today, not only power optimization

but also load reduction has to be included into proﬁ le and blade design. Another

important issue is a phenomenon called stall delay within rotating boundary layers.

Since its ﬁ rst observation by Himmelskamp [ 30 ] in 1945, it has become evident that

much behavior cannot be explained without the phenomenon, which is also impor-

tant for swept ﬂ ow. For 3D boundary layers the ECN model [ 45 ]:

⎛⎞

⎜⎟

⎝⎠

( 26 )

with f deﬁ ned as C

L ,3 D

= C

L ,2 D

+ f (2 p α – C

L ,2 D

) is often used.

100 Wind Power Generation and Wind Turbine Design

2.2 Optimum blade shape

Neglecting drag all relevant forces can be derived from lift via (compare to

Fig. 9 )

Bc C cos

( 27)

and torque

··.

Bc C sin r

( 28 )

Adding to eqn (10) the momentum balance for thrust, equation (17), and torque

equation (19) solution for a and a ′ is possible. The following ﬁ nal equations are

obtained:

(1 ) 4 ( )

Ccos

asin

−

( 29 )

acos

′

( 30 )

With s =BC/2pr solution for c · C

which is proportional to the circulation:

4(2

Bc C

sin cos )

=C=

vcos

−

(31 )

can be performed. Division by r/R = l

/ l

results in the desired ratio c / r against

r / R ( Fig. 13 ). Together with Glauert’s theory the more extended approach

of Wilson [ 18 ] and De Vries [ 17 ] , which includes also the lift to drag ratio and

tip losses:

−

(1 )

(1 ) 4 ( )

Ccos

aF aF

asin

( 32 )

′

acos

( 33 )

with F from eqn 12 is obtained. It can be seen that for C

/ C

= 100 and B = 3,

there are only small differences from the Glauert theory. Recently [ 40 ] it was

found that the widely used optimization approach by Betz may be overcome

by the older one of Joukowsky [ 36 ] thereby stating that a constantly loaded

rotor with a ﬁ nite number of blades may be superior to that with a Betz-type

load distribution.

Aerodynamics and Aeroelastics of Wind Turbines 101

3 Numerical CFD methods applied to wind turbine ﬂ ow

As a complement to analytical theories and experiments, CFD provides a third

approach in developing applied methods. In its purest form, only the differential

equations of Navier and Stokes (NS):

·0,v∇=

(34 )

fp v

rm=−∇+Δ=

( 35 )

together with suitable boundary conditions and a description of the blade geom-

etry is used. Unfortunately this ambitious goal cannot be reached at the pres-

ent time. The main obstacle is the emergence of turbulence at higher Reynolds

number (RN):

( 36 )

calculated from kinematic viscosity (v = 1.5 – 10

–5

–1

for air) and a typical

length L and a typical velocity v . Present wind turbines have a RN of several mil-

lion based on blade chord. Only Direct Numerical Simulation (DNS) solves the

(NS) without any further modeling. At the present time (2009), only airfoil calcu-

lations up to RN of a few thousands have been carried out at the price of months

0.05

0.1

0.15

0.2

0 0.2 0.4 0.6 0.8 1

c/R

r/R

Optimum Blade Shape C-L = 1 B = 3

Glauert

De Vries

Figure 13 : Optimum shape of a blade.

102 Wind Power Generation and Wind Turbine Design

of CPU time and terrabytes of data. After Large Eddy Simulation (LES), the next

stage of simpliﬁ cation is Reynolds averaged Navier Stokes equations (RANS). An

ensemble average, which is assumed to be a time average by an ergodic hypothesis

is carried out for all ﬂ ow quantities.

Turbulence then emerges as a never ending hierarchy of higher correlations

which has to be truncated by the so-called closure assumption. In wind-turbine

applications at the present time the k – w shear stress transport (SST) extension of

Menter isfrequently used. k is the turbulent kinetic energy (per unit mass) and w is

a local frequency scale. Unfortunately all these empirical turbulence models

describe only fully developed turbulent ﬂ ow and are not able to resolve the transi-

tional region from laminar to turbulent ﬂ ow. There are good reasons to believe that

parts of the blade must be laminar because otherwise the large L2D ratio cannot be

achieved for c

of the order of 0.5.

At UAS Kiel from 2000–2003 a program [ 65 ] comparing 2D-CFD simulation

with measurements was carried out. A 30% thick airfoil from Delft University was

chosen: DU-W-300-mod. For including transitional ﬂ ow properties to e

- method

of Stock was included in DLR’s structured CFD-Code FLOWer. The main ﬁ ndings

of this project were as follows:

Mesh generation with mostly orthogonal grids is very important. Therefore a •

hyperbolic type of generating equation, namely

0,xx yy

xh xh

( 37)

.xy yy A

xh xh

−=Δ

( 38 )

was chosen. For details see [ 62 ].

The • e

method has to be parameterized with an N (usually between 6 and 9)

which is related to a surrounding turbulence intensity by Mack’s correlation.

Comparison with wind tunnel measurements was difﬁ cult because the turbu-

lence intensity was not known exactly.

Prediction of •

max

, meaning computation of ﬂ ow separation (stall) was also

difﬁ cult, because the ﬂ ow started to become unsteady.

Transition points were predicted correctly as long as only laminar separation or •

Tollmien Schlichting (TS) instabilities triggered transition.

In addition, drag effects for example from Carborundun or Zig-Zag band were •

difﬁ cult to predict [ 28 ].

The situation becomes more complicated when whole blades are investigated.

Correct prediction of overall power curves is seen to be very difﬁ cult. The picture

is uneven [ 37 ] but parametric studies such as for wings including winglets [ 35 ]

give valuable insight into the ﬂ ow ﬁ eld and comparative changes. This is especially

important when discussing performance enhancements.

Much better results can be achieved when no turbulence has to be included in

the discussion. This is always the case, when basic studies have been performed

Aerodynamics and Aeroelastics of Wind Turbines 103

which cannot easily be checked by hand calculations. An example of such methods

are the modiﬁ cation of the AD and actuator line method initiated by J. Sørensen

and further developed by R. Mikkelsen. Another promising ﬁ eld of application is

site-assessment and wind-farm optimization. The second torque conference at

Copenhagen in 2007 gives an impressive report of the progress which has been

made in this ﬁ eld. One has to bear in mind that the usual problems with turbulence

modeling are the same as in other ﬁ elds mentioned above.

4 Experiments

4.1 Field rotor aerodynamics

From 1992 to 1997, the International Energy Agency (IEA) conducted large-scale

comparisons of measurements on research-type wind turbines. The project was

termed Annex 14 [ 52 ] and was designed to

validate and develop design codes, •

investigate design principles for stall controlled turbines. •

Five turbines from ECN in the Netherlands, IC/RAC in the UK, NREL in the

USA, RISO in Denmark and DUT in the Netherlands were considered. The diam-

eter ranged from 10 to 27 m. The proﬁ les used were quiet diverse, ranging from the

NACA 44xx and NACA 632xx to the NLF 0416 and the S809. One objective was

to investigate the correlation of measured performance and 2D-proﬁ le data from

wind tunnel measurements.

-0.5 0 0.5 1 1.5

-0.8

-0.6

-0.4

-0.2

0.2

0.4

0.6

0.8

Figure 14 : 2D mesh generated by a hyperbolic generator.

104 Wind Power Generation and Wind Turbine Design

As a result many problems with consistent interpretation of all the data were

identiﬁ ed. Especially the deﬁ nition of a local angle of attack was very difﬁ cult.

This of course is also the case when 3D-CFD investigations are discussed. Several

approaches have been tried, one being a correlation between the location of the

stagnation pressure line and AOA. A second is the so-called inversed BEM, where

and c

are correlated to the local ﬂ ow angle. One may object that both methods

rely on 2D assumptions for connecting forces and velocities so they may fail when

3D effects are strong. A somewhat detailed review is given by van Rooij [ 45 ] and

Shen et al. [ 54 ].

It may be useful to note that a new program subsequent to the above-mentioned

experiments will be started in early 2009 which will be undertaken on the research

wind turbine E30 at the University of Applied Sciences Flensburg. During this

experiment, using an aerodynamic glove, shear stresses which indicate the state

of the boundary layer (laminar, transitional or fully developed turbulent) will be

measured for the ﬁ rst time [ 51 ].

4.2 Chinese-Swedish wind tunnel investigations

In a joint effort of Chinese and Swedish aerodynamicists, a 4.75-m diameter wind

turbine was investigated in the late 1980s [ 21 , 44 ].

4.3 NREL unsteady aerodynamic experiments in the NASA AMES-wind tunnel

As a result of the rather disappointing ﬁ ndings from open air experiments a big

effort was mounted by NREL, the US National Renewable Energy Laboratory.

A 10-m diameter two-bladed rotor was put into the world’s largest wind-tunnel,

the NASA-AMES wind tunnellocated in California. The main advantage was a

complete control over all inﬂ ow conditions. Most important was the so-called

blind comparison in which a variety of design and analysis tools were used to

predict the power and forces based on 2D data. Figure 15 gives an overview of

all simulation for the shaft torque, which is proportional to the power, see [ 56 ] for

broad discussion. Several important conclusions can be drawn from the results of

this large-scale experiment:

10 tested aeroelastic codes (see Section 5) showed extremely large disagree-•

ment even in the attached ﬂ ow regime. The measured low-speed shaft torque

being 800 Nm , is scattered in the predictions from 200 up to 1400 Nm (factor

of 7). This situation is only a little bit better at 10 m/s (30%) but became worse

at the same level as before in the deep-stalled case.

So-called performance codes (only three, being descended from the famous PROP •

Code) showed strong variations in the deep-stalled condition with one remarkable

exception which gives almost the same result as one CFD investigation.

Two wake codes give also non-uniform results. •

Most impressive were the CFD results of Sørensen [ 59 ]. There, without any •

2D-proﬁ le data, results with the same degree of accuracy of the best BEM