Burton T. (et. al.) Wind energy Handbook

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velocities and pressures above and below the aerofoil at the trailing edge must be

the same, the particles which meet there are not the same ones that parted company

at the leading edge; the particle whic h travelled above the aerofoil reaches the

trailing edge ﬁrst because it is speeded up by the circulatory ﬂow.

The pressure variation (minus the ambient static pressure of the undisturbed

ﬂow) around an aerofoil is shown in Figure A3.15. The upper surface is subject to

suction (with the ambient pressure subtracted) and is responsible for most of the lift

force. The pressure distribution is calculated without the presence of the boundary

layer.

Figure A3.16 shows the same distribution with the pressure coefﬁcient (C

p  p

=[1=2]rU

) plotted against the chord-wise co-ordinate of the aerofoil proﬁle:

the full line shows the pressure distribution if the effects of the boundary layer are

ignored, and the dashed line shows the actual distribution.

Figure A3.14 Flow Past an Aerofoil at a Small Angle of Attack

Figure A3.15 The Pressure Distribution Around the NACA0012 Aerofoil at Æ ¼ 58

DEFINITION OF LIFT AND ITS RELATIONSHIP TO CIRCULATION 165

The effect of the boundary layer is to modify the pressure distribution at the rear

of the aerofoil such that lower pressure occu rs there than if the bound ary is ignored.

The modiﬁed pressure distribution gives rise to pre ssure drag which is added to

the skin friction drag, also caused by the boundary layer.

A3.7 The Stalled Aerofoil

If the angle of attack exceeds a certain critical value (108 to 168, depending on the

Re), separation of the boundary layer on the upper surface takes place. This causes

a wake to form from above the aerofoil, reduces the circulation, reduces the lift and

increases the drag. The ﬂow past the aerofoil has then stalled (Figure A3.17). A ﬂat

plate will also develop circulation and lift but will stall at a very low angle of attack

because of the sharp leading edge. Arching the plate will improve the stalling

α (degrees)

1.5

0.5

0 5 10 15 20 25

NACA0012 Re=1 000 000

Stalled region

Attached flow region

Figure A3.16 The Pressure Distribution Around the NACA0012 Aerofoil at Æ ¼ 58

Figure A3.17 Stalled Flow Around an Aerofoil

166

AERODYNAMICS OF HORIZONTAL-AXIS WIND TURBINES

behaviour but a much greater improvement can be obtained by giving thickness to

the aerofoil together with a well-rounded leading edge.

A3.8 The Lift Coefﬁcient

The lift coefﬁcient is deﬁned as

Lift

(A3:8)

U is the ﬂow speed and A is the plan area of the body. For a long body, such as an

aircraft wing or a wind turbine blade, the lift per un it span is used in the deﬁnition

and the plan area is replaced by the chord length.

Lift=unit span

r(ˆ 3 U)

rUc sin ÆU

¼ 2 sin Æ (A3:9)

In practice,

¼ a

sin Æ (A3:10)

Without boundary layer

With boundary layer

0 0.2 0.4 0.6 0.8 1

2.2

1.8

1.4

0.6

0.2

-0.2

-0.6

-1

-C

x/c

Figure A3.18 C

 Æ Curves for a Symmetrical Aerofoil

THE LIFT COEFFICIENT 167

where a

, called the lift-curve slope dC

=dÆ, is about 5.73 (0.1/degree), rather than

2. Note that a

should not be confused with the ﬂow induction factor.

Lift, therefore, depe nds on two parameters, the angle of attack Æ and the ﬂow

speed U. The same lift force can be generated by different combinations of Æ and U.

The variation of C

with the angle of attack Æ is shown in Figure A3.15 for a typical

symmetrical aerofoil (NACA0012). Notice that the simple relationship of Equation

(A3.6) is only valid for the pre-stall region, where the ﬂow is attached, and because

the angle of attack is small (, 16 8) the equation is often written as

¼ a

Æ (A3:10a)

A3.9 Aerofoil Drag Characteristics

The deﬁnition of the drag coefﬁcient for an aircraft wing or a wind-turbine blade is

based not on the frontal area but on the plan area, for reasons that will become cl ear

later. The ﬂow past a body which has a large span normal to the ﬂow directio n is

basically two-dimensional and in such cases the drag coefﬁcient can be based upon

the drag force per unit span using the stream-wise chord length for the deﬁnition:

Drag=unit span

(3:11)

For a wing of large span the value of C

is roughly 0.01, at moderate Reynolds

numbers.

The drag coefﬁcient of an aerofoil also varies with angle of attack. Figure A3.16

shows that on the upper surface pressure is rising as the ﬂow moves towards the

trailing edge, this is called an adverse pressure gradient and seeks to slow the air

down. If the air is slowed to a standstill stall will occur and the pressure drag will rise

Attached flow region

Stalled region

NACA0012 Re=1 000 000

α (degrees)

0 5 10 15 20 1 25

0.5

0.4

0.3

0.2

0.1

Figure A3.19 Variation of C

with Æ for the NACA0012 Aerofoil

168

AERODYNAMICS OF HORIZONTAL-AXIS WIND TURBINES

sharply. The strength of the adverse pressure gradient increases with angle of attack

and so it can be expected that the drag will rise with angle of attack. Figure A3.19

shows the variation of C

with Æ also for the symmetrical NACA0012 aerofoil.

The lift/drag ratio (shown in Figure A3.20) has a signiﬁcant affect upon the

efﬁciency of a wind turbine and it is desirable that a turbine blade operates at the

maximum ratio .

A3.10 Variation of Aerofoil Characteristics with

Reynolds Number

The nature of the ﬂow pattern around an aerofoil is determined by the Reynolds

number and this signiﬁcantly affects the values of the lift and drag coefﬁcients. The

Attached

flow region

Stalled region

NACA0012 Re=1 000 000

α (degrees)

0 5 10 15 20 25

Figure A3.20 Lift/Drag Ratio Variation for the NACA0012 Aerofoil

Laminar boundary

layer

Decreasing Re

NACA0012

1 x 10

5 x 10

1 x 10

2 x 10

3 x 10

0 2 4 6 8

α (degrees)

0.015

0.02

0.015

0.01

0.005

Figure A3.21 Variation of the Drag Coefﬁcient with Reynolds Number at Low Angles of

Attack

VARIATION OF AEROFOIL CHARACTERISTICS WITH REYNOLDS NUMBER 169

general level of the drag coefﬁcient increases with decreasing Reynolds number

and below a critical Reynolds number of about 200 000 the boundary layer remains

laminar causing a sharp rise in the coefﬁcient. The affect on the lift coefﬁcient is

largely concerned with the angle of attack at which stall occurs. As the Reynolds

number rises so does the stall angle and, because the lift coefﬁcient increases

linearly with angle of attack below the stall, the max imum value of the lift coefﬁ-

cient also rises. Characteristics for the NACA0012 aerofoil are shown in Figures

A3.21 and A3.22.

A3.11 Cambered Aerofoils

Cambered aerofoils, such as the NACA4412, shown in Figure A3.23, have curved

chord lines and this allows them to produce lift at zero angle of attack. Generally,

cambered aerofoils have higher maxim um lift/drag ratios than symmetrical aero-

foils for positive angles of attack and this is the reason for their use.

The classiﬁcation of the NACA four-digit ran ge of aerofoils, which were com-

monly used on wind turbines, is very simple and is illustrated in Figure A3.24: from

left to right, the ﬁrst digit represents the amount of camber as a percentage of the

chord length, the second digit represents the perce ntage chord position, in units of

10 percent, at which the maximum camber occurs and the last two digits are the

Decreasing Re

NACA0012

x 10

0 5 10 15 20 25

α (degrees)

0.4

0.3

0.2

0.1

x 10

5 x 10

1 x 10

3 x 10

Increasing Re

NACA0012

x 10

0 5 10 15 20 25

α (degrees)

1.5

0.5

x 10

5 x 10

1 x 10

3 x 10

Figure A3.22 Variation of the Drag and Lift Coefﬁcients with Reynolds Number in the Stall

Region

Figure A3.23 The Proﬁle of the NACA4412 Aerofoil

170

AERODYNAMICS OF HORIZONTAL-AXIS WIND TURBINES

maximum thickness to chord ratio, as a percentage of the chord length, which, in

this family of aerofoils, is at the 30 percent chord position. The cambered chordline,

now called the camber line, comprises two parabolic arcs that join smoothly at the

point of maximum camber. For other ranges of aerofoils the reader should refer to

Abbott and von Doenhoff (1959).

The angle of attack Æ is measured form the chord line which is now deﬁned as

the straight line joining the ends of the camber line. Note that the lift at zero angle

of attack is no longer zero; zero lift occurs at a small negative angle of attack. With

most cambered aerofoils the zero lift line is approximately at A8 where A is the

percentage camber. The behaviour of the NACA4412 aerofoil is shown in Figure

A3.25 for angles of attack below and just above the stall. Note that the lift at zero

A% chor

Chord line

10 x B% chord

30% chord

Zero lift line

Camber line

NACA AB12 Aerofoil

12% thickness/chord ratio

Figure A3.24 Classiﬁcation of the NACAXXXX Aerofoil Range

-10 0 10 20 30

α (degrees)

-1

-10 0 10 20 30

α (degrees)

0.6

0.4

0.2

-0.5 0 0.5 1 1.5

0.02

0.015

0.01

0.005

-10 -5 0 5 10

α (degrees)

150

100

-50

Figure A3.25 The Characteristics of the NACA4412 Aerofoil for Re ¼ 1:5  10

CAMBERED AEROFOILS 171

angle of attack is no longer zero; zero lift occurs at a small negative angle of attack

of approximately 48.

The centre of pres sure, which is at the

chord position on symmetrical aerofoils

lies aft of the

chord position on cambered aerofoils and moves towards the trailing

edge with increasing angle of attack. However, if a ﬁxed chordwise position is

chosen then the resultant force through that point is accompanied by a pitching

moment (nose-up positive, by convention). If a pitching moment coefﬁcient is

deﬁned as

Pitching=unit span

(3:12)

then there will be a position, called the aerodynamic centre, for which dC

=dC

¼ 0.

Theoretically, the aerodynamic centre lies at the

chord position and is close to this

point for most practical aerofoils. The value of C

depends upon the degree of

camber but for the NACA4412 the value is 0:1, note that pitching moments are

always negative in practice (nose down) despite the sign convention. Above the

stall there is no aerodynamic centre, as deﬁned, and so the pre-stall position

continues to be used to determine the pitching moment coefﬁcient, which then

becomes dependent upon Æ.

172 AERODYNAMICS OF HORIZONTAL-AXIS WIND TURBINES

Wind-turbine Performance

4.1 The Performance Curves

The performance of a wind turbine can be characterized by the manner in which

the three main indicators—power, torque and thrust—vary with wind speed. The

power determines the amount of energy captured by the rotor, the torque

developed determines the size of the gear box and must be matched by whatever

generator is being driven by the rotor. The rotor thrust has great inﬂuence on the

structural design of the tower. It is usually convenient to express the perform-

ance by means of non-dimensional, characteristic performance curves from which

the actual performance can be determined regardless of how the turbine is

operated, e.g., at constant rotational speed or some regime of variable rotor

speed. Assuming that the aerodynamic performance of the rotor blades does not

deteriorate the non-dimensional aerodynamic performance of the rotor will

depend upon the tip speed ratio and, if appropriate, the pitch setting of the

blades. It is usual, therefore, to display the power, torque and thrust coefﬁcients

as functions of tip speed ratio.

4.1.1 The C

– º performance curve

The theory described in Chapter 3 gives the wind turbine designer a means of

examining how the power developed by a turbine is governed by the various

design parameters. The usual method of presenting power performance is the non-

dimensional C

– º curve and the curve for a typical, modern, three-blade turbine

is shown in Figure 4.1.

The ﬁrst point to notice is that the maximum value of C

is only 0.47, achieved at

a tip speed ratio of 7, which is much less than the Betz limit. The discrepancy is

caused, in this case, by drag and tip losses but the stall also reduces the C

at low

values of the tip speed ratio (Figure 4.2).

Even with no losses included in the analysis the Betz limit is not reached because

the blade design is not perfect.

4.1.2 The effect of solidity on performance

At this stage, the other principal parameter to consider is the solidity, deﬁned as

total blade area divided by the swept area. For the three-blade machine above the

solidity is 0.0345 but this can be altered readily by changing its number of blades.

The solidity could also have been changed by changing the blade chord.

The main effects to observe of changing solidity are as follows, see Figure 4.3.

0.5

0.4

0.3

0.2

0.1

0 5 10 15

Figure 4.1 C

– º Performance Curve for a Modern Three-blade Turbine

All losses included

No drag or stall losses

No losses at all

0 2 4 6 8 10 12 14 16

0.6

0.4

0.2

Drag losses

Tip losses

Stall

losses

Figure 4.2 C

– º Performance Curve for a Modern Three-blade Turbine Showing Losses

174

WIND-TURBINE PERFORMANCE