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Wind Resource and Site Assessment 85

has a starting diameter equal to the rotor diameter and is linearly expanding as a

function of the downwind distance. The Ainslie model is based on a numerical

solution of the Navier-Stokes equations with an eddy viscosity closure in cylindri-

cal coordinates. The eddy viscosity is described by the turbulent mixing due to the

induced turbulence, generated within the shear layer of the wake, and the ambient

turbulence [ 33 ].

Other sources of losses could be related to availability, electrical losses, high

wind hysteresis, environmental conditions like icing and blade degradation and

curtailments, of which wind sector management is the most common one. As

explained earlier turbine loading is inﬂ uenced by the wake effects from nearby

machines. For some wind farms with particularly close spacing, it may be neces-

sary to shut down certain wind turbines for certain wind directions. Typically this

is done in a wind climate with a very dominant and narrow main wind direction.

7.3 Uncertainty

Uncertainty analysis is an important part of any assessment of the long-term energy

production of a wind farm. The most important contributors to the uncertainty are:

on-site wind speed measurements •

long-term correction •

ﬂ ow modelling •

performance of the wind turbine •

park losses •

Behind each point a vast number of aspects have to be considered and quantiﬁ ed

including anemometer calibration, mounting effects, inter-annual variability, qual-

ity of the long-term data, vertical and horizontal extrapolation, quality of the topo-

graphic input, power curve uncertainty, wake model, etc. Some inspiration can be

found in [ 34 ].

R eferences

[1] European Wind Energy Association. Wind Energy – The Facts. ISBN-978-1-

84407-710-6, 2009.

[2] IEC 61400-12-1. Wind Turbines Part 12-1: Power performance measure-

ments of electricity producing wind turbines , 2005.

[3] IEA. Recommended practices for wind turbine testing, Part 11: Wind speed

measurement and use of cup anemometry , 1999.

[4] Lindelöw, P., Courtney, M., Mortensen, N.G. & Wagner, R., Are Lidars good

enough? Accuracy of AEP predictions in ﬂ at terrain generated from measure-

ments by conically scanning wind sensing Lidars. Proc. of European Wind

Energy Conf. , Marseille, France, 2009.

[5] Brock, F.V. & Richardson, S.J., Meteorological Measurement Systems,

Oxford University Press, 2001.

86 Wind Power Generation and Wind Turbine Design

[6] Monna, W.A.A., Comparative investigation of dynamic properties of some

propeller vanes. WR 78-11, KNMI, The Netherlands, 1978.

[7] Bradley, S., Antoniou, I., von Hünerbein, S., Kindler, D., de Noord, M. &

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Emeis, S. & Warmbier, G., On the theory of Sodar measurement techniques.

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Kindler, D., de Noord, M. & Warmbier, G., The proﬁ ler inter-comparison

experiment (PIE). Proc. of European Wind Energy Conf. , London, UK, 2004.

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wind energy. A comparative guide . Proc. of European Wind Energy Conf. ,

Brussels, Belgium, 2008.

[11] Antoniou, I., Jørgensen, H.E., Mikkelsen, T., Pedersen, T.F., Warmbier, G. &

Smith, D., Comparison of wind speed and power curve measurements using

cup anemometer, a Lidar and a Sodar. Proc. of European Wind Energy Conf. ,

London, UK, 2004.

[12] Foussekis, D. & Georgakopoulos, T., Investigating wind ﬂ ow properties in

complex terrain using 3 Lidars and a meteorological mast. Proc. of European

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[13] Bingöl, F., Mann, J. & Foussekis, D., Lidar performance in complex terrain

modelled by WAsP Engineering . European Wind Energy Conf. , Marseille,

France, 2009.

[14] Dupont, E., Lefranc, Y. & Sécolier, C., A Sodar campaign in complex terrain

for data quality evaluation and methodological investigations. Proc. of Euro-

pean Wind Energy Conf. , Marseille, France, 2009.

[15] Petersen, E.L., Troen, I., Frandsen, S. & Hedegaard, K., Wind atlas for

Denmark – A rational method for wind energy siting. Risø-R-428, Risø

National Laboratory, Roskilde, Denmark, 1981.

[16] Petersen, E.L., Mortensen, N.G., Landberg, L., Højstrup, J. & Frank, H.P.,

Wind power meteorology. Risø-R-1206, Risø National Laboratory, Roskilde,

Denmark, 1997.

[17] Troen, I. & Petersen, E.L., European wind atlas, ISBN 87-550-1482-8, Risø

National Laboratory, Roskilde, Denmark, 1989.

[18] Lange, B., Modelling the marine boundary layer for offshore wind power

utilization, Ph.D. thesis, University Oldenburg, Germany, 2002.

[19] Dellwik, E., Jensen, N.O. & Landberg, L., Wind and forests – general rec-

ommendations for using WasP, 2004, http://www.bwea.com/pdf/trees/Risoe.

pdf

[20] Mortensen, N.G., Bowen, A.J. & Antoniou, I., Improving WAsP predictions

in (too) complex terrain. Proc. of European Wind Energy Conf. , Athens,

Greece, 2006.

[21] Wood, N., The onset of ﬂ ow separation in neutral, turbulent ﬂ ow over hills.

Boundary Layer Meteorology , 76 , pp. 137–164, 1995.

Wind Resource and Site Assessment 87

[22] Pedersen, H.S. & Langreder, W., Forest-added turbulence: a parametric study

on turbulence. Journal of Physics , Conference Series 75 – 012062, 2007.

[23] Barthelmie, R., Hansen, O.F., Enevoldsen, K., Motta, M., Pryor, S., Højstrup,

J., Frandsen, S., Larsen, S. & Sanderhoff, P., Ten years of measurements of

offshore wind farms – what have we learnt and where are uncertainties? The

Science of Making Torque from Wind, Delft, 2004.

[24] Frandsen, S. & Thøgersen, M., Integrated fatigue loading for wind turbines

in wind farms by combining ambient turbulence and wakes. Wind Engineering ,

23( 6 ) , pp. 327–339, 1999.

[25] IEC 61400-1. Wind Turbines Part 1: Design requirements , 2005.

[26] Gumbel, E.J., Statistics of Extremes , Columbia University Press, 1958.

[27] Palutikof, J.P., Brabson, B.B., Lister, D.H. & Adcock, S.T., A review of

methods to calculate extreme wind speeds. Meteorological Applications , 6 ,

pp. 119–132, 1999.

[28] Langreder, W., Højstrup, J. & Svenningsen, L., Extreme wind estimates with

modest uncertainty – a contradiction? Proc. of Windpower Conf. , Chicago,

2009.

[29] Dekker, J.W.M. & Pierik, J.T.G., European Wind Turbine Standard II . ECN-

C-99-073, The Netherlands, 1999.

[30] Langreder, W., Kaiser, K., Hohlen, H. & Højstrup, J., Turbulence correc-

tions for power curves . Proc. of European Wind Energy Conf. , London, UK,

2004.

[31] Antoniou, I. & Pedersen, S.M., Inﬂ uence of turbulence, wind shear and low

level jets on the power curve and the AEP of a wind turbine. Proc. of Euro-

pean Wind Energy Conf. , Marseille, France, 2009.

[32] Ainslie, J.F., Calculating the ﬂ owﬁ eld in the wake of wind turbines. Journal

of Wind Engineering and Industrial Aerodynamics , 27 , pp. 213–224, 1988.

[33] Zigras, D. & Mönnich, K., Farm efﬁ ciencies in large wind farms. Proc. of

Windpower Conf. , Los Angeles, USA, 2007.

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CHAPTER 3

Aerodynamics and aeroelastics of wind turbines

Alois P. Schaffarczyk

University of Applied Sciences Kiel, Kiel, Germany .

Aerodynamics and aeroelastics of wind turbines are presented. First, the basic

results of analytical, numerical and experimental work are reviewed, then the

impact on commercial systems is discussed. A short section on non-standard wind

turbines is ﬁ nally included.

1 Introduction

Aerodynamics is a necessary tool for modeling the loads and power output of a

wind turbine. Unlike other related applications such as ship propellers [ 5 ] and

helicopters [ 6 ], there is no comprehensive and up-to-date presentation of this

important subject. The reader is given a short introduction to current knowledge.

A readable review of, especially, the German efforts during the 1950s and 1960s

was given by Hütter [ 33 ]. Hansen and Butterﬁ eld [ 26 ] and Hansen et al. [ 25 ] present

more up-to-date reviews.

It is assumed that the inﬂ ow velocity is more or less stationary, thereby omitting

turbulence as rapid variations above 1 Hz and also neglecting diurnal variation.

These distinctions are meant to be in the spirit of standard regulations, as for exam-

ple given by the IEC (International Electrotechnical Commission) or Germanischer

Lloyd (GL). Therefore no presentation of wake aerodynamics is found in this

chapter. The interested reader may ﬁ nd a review of these items in [ 64 ]. This rest of

this chapter is divided into seven sections.

Section 2 gives an account of analytical theories developed largely before the

emergence of digital computers, beginning with the global momentum theories of

Rankine [ 43 ] and Froude [ 23 ]. Several developments and extensions of these have

emerged only recently. Section 3 introduces the most important development of

the late 20th century: computational ﬂ uid dynamics (CFD). Therefore the reader

should be familiar with the basics of ﬂ uid dynamics [ 2 , 3 ] and viscous ﬂ uid ﬂ ow

[ 4 ], together with some of the basics of CFD. Section 4 is devoted to experimental

90 Wind Power Generation and Wind Turbine Design

work on wind turbines. Two distinct branches can be identiﬁ ed: so-called free-ﬁ eld

experiments, carried out in the open air and those performed under controlled

inﬂ ow conditions within wind tunnels. Considerations are restricted to NASA-

AMES blind comparison experiment and the European MEXICO (Measurements

and Experiments in controlled conditions) project, performed in Europe’s largest

wind tunnel in the Netherlands. After describing aeroelastics in Sections 5 and part 6

in general, the impact of this elaborate scientiﬁ c work on commercial wind tur-

bines is presented. This is a somewhat difﬁ cult and delicate task, as most of this

work and even the results are conﬁ dential. In practice, this means that only public-

domain work and non-standard turbines will be considered. Section 8 concludes

this discussion with a summary and an outlook for future developments after the

aerodynamics of some unconventional turbines are presented in Section 7.

2 Analytical theories

The ﬁ rst work which provided a simple complete model of the global ﬂ ow around

a wind turbine is the so-called actuator disk (AD) theory. It was ﬁ rst developed by

Rankine and Froude to describe the ﬂ ow around ship propellers. Figure 1 shows an

overview of all possible ﬂ ow states which can occur. Recently [ 58 ] it was possible to

reproduce all ﬂ ow states observed by Glauert from a numerical full-ﬁ eld AD model.

The main idea is the introduction of a slipstream ( Fig. 2 ) behind the rotor. Energy

is extracted by decelerating the inﬂ ow v

to v

at the rotor and v

far downstream.

Figure 1 : Flow states of propellers and wind turbines [ 12 ].

Aerodynamics and Aeroelastics of Wind Turbines 91

Applying the equations of conservation of mass, energy (Bernoulli’s equation) and

momentum:

213

(),

vvv=+

( 1)

(Froude’s law) and

4(1 ),Caa=−

(2 )

are obtained. C

is the non-dimensional power

(/2)

( 3 )

( 4 )

A turbine of diameter D has a swept area of A = (p / 4)D

. The most important param-

eter in eqn 2 is the so-called axial interference factor [ 12 ] often also called velocity

induction a = v

/ v

. Differentiating eqn 2 with respect to a one can easily show that

maximum energy is extracted when a = 1/3 and

etz

16 27 0.59.CC== =

This

law was found independently by Lanchester in 1915 and Betz in 1925. Using the

same arguments the main force on the turbine, the thrust in the wind direction:

(/2)

( 5 )

Figure 2: 1D momentum theory of Froude, Lanchester and Betz.

92 Wind Power Generation and Wind Turbine Design

is seen to be

() 4(1 )ca a a=−

at Betz’ value of

Betz T

1/ 3 ( 1/ 3) 8 / 9 ~ 0.9.aca===

. This shows that a wind turbine is heavily loaded at the optimum condition.

The following limitations apply to the theory:

it is implicitly assumed that there is • no slipstream as there are no radial components

calculation of the full details of the slipstream expansion cannot be performed •

as the theory does not consider the radial velocity component

the axisymmetric disk is assumed to be inﬁ nitesimally thin. •

As already discussed by Betz [ 9 ] and further by Loth McCoy [ 70 ] in the context

of a double AD for vertical axis wind turbines there is a possibility to beat Betz to

some extent (roughly to 0.64 for a double AD). A recent discussion for beating

Betz with general devices was given by Jamieson [ 34 ].

Wind turbines are rotating machines, and a very important dimensionless number

is the tip speed ratio (TSR) deﬁ ned as

wind

( 6 )

where Ω is the angular velocity of the turbine.

Figure 3 gives a graph of various c

against l . Apart from its own data, data

from the classical literature, for example [ 14 , 18 ], was also included. Two items are

Figure 3 : c

as function of TSR, various types of wind turbines.

Aerodynamics and Aeroelastics of Wind Turbines 93

of particular interest. Firstly, all turbines have a maximum c

and two kinds of

turbine can be distinguished, those with a maximum c

around 6–10 (so-called fast

running machines) and those with maximum below 4 (so-called slow running

machines). Secondly the curve must tend towards c

= 0 when l = 0. This was

discussed by Glauert [ 12 ] and stems from the fact that in addition to the axial

induction a , a second a′ has to be introduced, where a′ is deﬁ ned by

′

( 7 )

Here w is the local angular velocity of the ﬂ ow. It comes from Newton’s third law as

applied to angular momentum. If dr is an increment of radius, the torque is now

4(1) dM r v a a dr=−

′

( 8 )

and the total power becomes

8(1) .

caad

⎛⎞ ⎛⎞

=−

′

⎜⎟ ⎜⎟

⎝⎠ ⎝⎠

∫

( 9)

The two parameters a and a ′ have to be optimized. A third equation, the so-called

orthogonality condition of Glauert [ 12 ] is the condition

(1 ) (1 )aaxaa+=−

′′

(10 )

with x: = wr/v

being the local TSR. Figure 4 gives a sketch of the arrangement of

the velocity vectors. Optimization results in the condition

−

′

( 11 )

Figure 4 : Velocity and force triangles.

94 Wind Power Generation and Wind Turbine Design

which shows that 1/4 < a < 1/3 must hold. These losses as compared to Betz’ ideal

limit are called swirl losses. Figure 5 shows the quantitative dependence. Two

other mechanisms have to be introduced, to model all effects shown in Fig. 3 . They

are the so-called tip-losses and proﬁ le-drag losses. An AD was deﬁ ned as a com-

pact disk, formally having inﬁ nitely many blades. To estimate the effect of a ﬁ nite

number of blades, the two models are used. One is based on conformal mapping

of the ﬂ ow around a stack of plates to that of a rotor with a ﬁ nite number of blades

(given by Prandtl [ 42 ]) and the second is based on the theory of propeller ﬂ ow of

Goldstein [ 24 ]. A recent investigation has been made by Sørensen and Okulov

[ 39 , 40 ]. Usually a reduction factor F is introduced to account for the decreasing

forces on the blade towards the tip:

exp{ }F arccos f ,

=−

( 12 )

with

BR r

−

=+ ( 13 )

l as expressed by eqn 6.

Comparison with measurements by Shen et al. [ 55 ] resulted in a new empirical

tip-loss model for use in AD and CFD simulations. Recently Sharpe [ 53 ] has

revised the arguments of Glauert and extended them slightly. Mikkelsen et al. [ 38 ]

applied his numerical AD method to investigate this effect. His ﬁ ndings were that

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

0 2 4 6 8 10

27Cp/16

TSR

Glauert’s optimum rotor

Figure 5:

Betz

/cc as a function of TSR.