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

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 r is a distance measured from the wake edge. Distance d between the discs

should be that of the distance travelled by particle three between successive vortex

sheets. Glauert (1935) takes d as being the normal distance between successive

helicoidcal vortex sheets.

The helix angle of the vortex sheets is the ﬂow angle 

and so with N sheets

intertwining from N blades

d ¼

2R

sin 

2R

(1  a)

(3:75)

Prandtl’s model has no wake rotation but the discs may spin at the rotor speed

without affec ting the ﬂow at all, as it is inviscid, thus a9 is zero and W

is the

resultant velocity (not including the radial velocity) at the edge of a disc. Glauert

(1935a) argues that R

 r=W which is much more convenient to use.

W ¼

ﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ

(1  a)]

þ (r)



 r



R  r



ﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ

1 þ

(r)

(1  a)]

and

f () ¼



cos[e

((N=2)(1)=)

ﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ

1þ(º)

=(1a)

](3:76)

The Prandtl tip-loss factor for a three-blade rotor operating at a tip speed ratio of 6

is compared with the tip-loss factor of the helical vortex wake in the Figure 3.33.

1.5

0.5

0 0.2 0.4 0.6 0.8 1

r/R

Tip-loss factor

Prandtl

Vortex theor

Figure 3.33 Comparison of Prandtl Tip-loss Factor with that Predicted by a Vortex Theory

for a Three-blade Turbine Optimized for a Tip Speed Ratio of 6

THE EFFECTS OF A DISCRETE NUMBER OF BLADES 85

It should also be pointed out that the vortex theory of Figure 3.28 also predicts

that the tip-loss factor should be applied to the tangential ﬂow induction factor.

It is now useful to know what is the variation of circula tion along the blade. For

the previous analysis, which disregarded tip-losses, the blade circulation was

uniform (Equation (3.65) ). Following the same procedure from which Equation

(3.64) was developed

r(W 3 ˆ)sin  ¼ rˆU

(1  a

(r)) ¼ 4r



a(r)(1  a(r))

Recall that a

(r) is the ﬂow factor local to the blade at radius r, which is equal to a

and a(r ) is the average value of the ﬂow factor at radius r. Therefore

ˆ(r) ¼

4

1  a



af (r)(1  af (r))

(3:77)

ˆ(r) is the total circulation for all blades and is shown in Figure 3.34 and, as can be

seen, it is almost uniform except near to the tip. The dashed vertical line shows the

effective blade length (radius) R

¼ 0:975 if the circulation is assumed to be uniform

at the level that pertains at the inboard section of the blade.

The Prandtl tip-loss factor appears to offer an acceptable, simple solution to a

complex problem; not only does it account for the effects of discrete blades but it

also allows the induction factors to fall to zero at the edge of the rotor disc.

3.8.4 Blade root losses

At the root of a blade the circulation must fall to zero as it does at the blade tip and

so it can be presumed that a similar process occurs. The blade root will be at some

distance from the rotor axis and the airﬂow through the disc inside the blade root

0.6

0.4

0.2

0 0.2 0.4 0.6 0.8 1

Total blade circulation

r/R

Figure 3.34 Span-wise Variation of Blade Circulation for a Three-blade Turbine Optimized

for a Tip Speed Ratio of 6

AERODYNAMICS OF HORIZONTAL-AXIS WIND TURBINES

radius will be at the free-stream velocity. Actually, the vortex theory of Section 3.4

can be extended to show that the ﬂow through the root disc is som ewhat higher

than the free-stream velo city. It is usual, therefore, to apply the Prandtl tip-loss

function at the blade root as well as at the tip.

If 

is the normalized root radius then the root loss factor can be determined by

modifying the tip-loss factor of Equation (3.76).

() ¼



cos[e

N=2(

=)

ﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ

1þ(º)

=(1a)

](3:78)

If Equation (3.76) is now termed f

(r) the complete tip/root loss factor is

f () ¼ f

() f

()(3:79)

3.8.5 Effect of tip-loss on optimum blade design and power

With no tip-loss the optimum axial ﬂow induction factor is uniformly 1=3 over the

whole swept rotor. The presence of tip-loss changes the optimum value of the

average value of a which reduces to zero at the edge of the wake but, local to the

blade tends to increase in the tip region. If a(r) is taken as the azimuthal average at

radius r then locally, at the blade at that radius, the ﬂow factor will be a(r)= f (r). The

inﬂow angle  at the blade is then, from Equation (3.57),

tan  ¼

º

1 

1 þ

(3:80)

but Equation (3.56), which is the ratio of the non-dimensional rate of change of

1.5

0.5

0 0.2 0.4 0.6 0.8 1

r/R

Tip/root-loss factor

Figure 3.35 Span-wise Variation of Combined Tip/root-loss Factor for a Three-blade Tur-

bine Optimized for a Tip Speed Ratio of 6 and with a Blade Root at 20% Span

THE EFFECTS OF A DISCRETE NUMBER OF BLADES 87

angular momentum to the non-dimensional rate of change of axial momentum, is

not changed because it deals with the whole ﬂow through the disc and so uses

average values. If drag is ignored for the present, Equation (3.59) becomes

tan  ¼

ºa9(1  a)

a(1  a) þ (a9º)

(3:81)

Hence

(1  a)ºa9

a(1  a) þ (ºa9)

1 



º 1 þ



which becomes



( f  1)

 º



(1  a)a9 þ a(1  a)1



¼ 0(3:82)

A great simpliﬁcation can be made to Equation (3.82) by ignoring the ﬁrst term

because, clearly, it disappears for much of the blade, where f ¼ 1, and for the tip

region the val ue of a9

is very small. For tip speed ratios greater than 3 neglecting

the ﬁrst term makes negligible difference to the result.



a9 ¼ a 1 



(3:83)

As before, Equation (3.57) still applies

da9

a ¼

1  a

From Equation (3.83)

a9 ¼



1  2



Consequently

(1  a)1 2



¼ º



which combined with Equation (3.83) gives



( f þ 1)a þ

f ¼ 0

88 AERODYNAMICS OF HORIZONTAL-AXIS WIND TURBINES

a ¼

f 

ﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ

1  f þ f

(3:84)

The radial variation of the average value of a, as given by Equation (3.84), and the

value local to the blade a= f is shown in Figure 3.36. An exact solution would also

have the local induced velocity falling to zero at the blade tip.

Clearly, the required blade design for optimal operation would be a little differ-

ent to that which corresponds to the Prandtl tip-loss factor because a= f , the local

ﬂow factor, does not fall to zero at the blade tip. The use of the Prandtl tip-loss

factor leads to an approximation, but that was recognized from the outset.

The blade design, which gives optimum power output, can now be determined

by adapting Eq uations (3.66) and (3.67) accordingly



ºC

4º



ﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ

1 



þ º 1 þ



1  a

1 

Introducing Equation (3.83) gives



ºC

4a(1  a)

ﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ

1 



þ º 1 þ

a 1 





(3:85)

The blade geometry parameter given by Equation (3.85) is sho wn in Figure 3.37

compared with the design which excludes tip-loss. As can be seen, only in the tip

region is there any difference between the two designs.

0.6

0.4

0.2

0 0.2 0.4 0.6 0.8 1

Axial flow induction factor

r/R

Azimuthal average

Local to the blade

Figure 3.36 Axial Flow Factor Variation with Radius for a Three-blade Turbine Optimized

for a Tip Speed Ratio of 6

THE EFFECTS OF A DISCRETE NUMBER OF BLADES 89

Similarly, the inﬂow angle distribution, shown in Figure 3.38, can be determined

by suitably modifying Equation (3.66)

tan  ¼

1 

º 1 þ

a 1 





(3:86)

0.5

0 1 2 3 4 5 6

Blade geomerty parameter

With tip-loss

Without tip-loss

Local speed ratio

Figure 3.37 Variation of Blade Geometry Parameter with Local Speed Ratio, with and

without Tip-loss for a Three-blade Rotor with a Design Tip Speed Ratio of 6

0 1 2 3 4 5 6

Inflow angle

With tip-loss

Without tip-loss

Local speed ratio

Figure 3.38 Variation of Inﬂow Angle with Local-Speed Ratio, with and without Tip-loss

for a Three-blade Rotor with a Design Tip Speed Ratio of 6

AERODYNAMICS OF HORIZONTAL-AXIS WIND TURBINES

Again, the effects of tip-loss are conﬁned to the blade tip.

The power coefﬁcient for an optimized rotor, operating at the design tip speed

ratio, without drag and tip-losses is equal to the Betz limit 0.593 but with tip-loss

there is obviously a reduced optimum power coefﬁcient. Equation (3.20) determines

the power coefﬁcient distribution along the blade, see Figure 3.39.

The power coefﬁcient

R

¼ 8º

a9(1  a)

d (3:87)

for which a9 and a are obtained from Equations (3.83) and (3.84).

The maximum power coefﬁcient that can be achieved in the presence of both

drag and tip-loss is signiﬁcantly less than the Betz limit at all tip speed ratios. As is

shown in Figure 3.42 drag reduces the power coefﬁcient at high tip spee d ratios but

the effect of tip-loss is most signiﬁcant at low tip speed ratios bec ause the separation

of the helicoidal vortex sheets is large.

3.8.6 Incorporation of tip-loss for non-optimal operation

The blade element–momentum Equations (3.51), (3.51a) and (3.52) are used to

determine the ﬂow induction factors for non-optimal operation. With tip-loss

included the BEM equations have to be modiﬁed. The necessary modiﬁcation

0 0.2 0.4 0.6 0.8 1

With tip-loss uniform circulation

With tip-loss optimized

No tip-loss

r/R

0.5

R dCp/dr

Figure 3.39 Span-wise Variation of Power Extraction in the Presence of Tip-loss for Three

Blades with Uniform Circulation and of Optimized Design for a Tip Speed Ratio of 6

THE EFFECTS OF A DISCRETE NUMBER OF BLADES 91

depends upon whether the azimuthally averaged values of the ﬂow factor s are to

be determined or the maximum (local to a blade element) values. If the former

alternative is chosen then, in the momentum terms the ﬂow factors remain

unmodiﬁed but in the blade element terms the ﬂow factors must appear as the

average values divided by the tip-loss factor. Choosing to determine the maximum

values of the ﬂow factors means that they are not modiﬁed in the blade elemen t

terms but are multiplied by the tip-loss factor in the momentum terms. The latter

choice allows the simplest modiﬁcation of Equations (3.51) and (3.52).

1  a



4 sin







4 sin





1  a

1  af

(3:51b)

a9 f

1 þ a9



4 sin  cos 

1  a

1  af

(3:52a)

There remains the problem of the breakdown of the momentum theory when wake

mixing occurs. The helicoidal vortex sheets may not exist and so Prandtl’s approx-

imation is not physically appr opriate. Nevertheless particles which pass between

blades will no doubt still lose less momentum than those which interact with a

blade and so the application of a tip-loss fact or is necessary. Prandtl’s approxima-

tion is the only practical method available and so is co mmonly used . In view of the

manner in which the experimental results of Figure 3.16 were gathered it is the

0.6

0.5

0.4

0.3

0.2

0 2 4 6 8 10

Design tip speed ratio

Maximum power coefficient

Zero drag

L/D = 120

L/D = 80

L/D = 40

Figure 3.40 The Variation of Maximum C

with Design º for Various Lift/Drag Ratios and

Including Tip-losses for a Three-bladed Rotor

AERODYNAMICS OF HORIZONTAL-AXIS WIND TURBINES

average value of a which shou ld determine at which stage the momentum theory

breaks down.

3.9 Calculated Results for an Actual Turbine

The blade design of a turbine operating at constant uniform rotational speed and

ﬁxed pitch is given in Table 3.1 and the aerofoil characteristics are shown in Figure

3.41.

The complete C

 º curve for the design is given in Figure 3.15.

Using the abov e data the following results were obtained.

The blade is designed for optimum performance at a tip speed ratio of about 6

and, ideally, the angle of attacks uniform along the span at the level for which the

lift/drag ratio is a maxim um, about 78 for the aerofoil concerned. At the lowest tip

speed ratio shown in Figure 3.42 the entire blade is stalled and for a rotational

speed of 60 r.p.m. the corresponding wind speed will be 26 m/s which is about the

cut-out speed. For the highest tip speed ratio shown the corresponding wind speed

will be 4.5 m/s, the cut-in speed. Maximum power is developed at a tip speed ratio

of 4 at a wind speed of 13 m/s and, clearly, much of the blade is stalled.

The axial ﬂow induction factor is not uniform along the span at any tip speed

ratio, indicating that the blade design is an engineering comp romise, but at the tip

speed ratio of 6 there is a value a little higher than 1=3. The ﬂow factors shown in

Figure 3.43 are those local to the blade and so the average value of axial ﬂow factor

will be close to 1=3 at a tip speed ratio of 6.

Generally, the axial ﬂow factor increases with tip speed ratio while the tangen tial

Table 3.1 Blade Design of a 17m-diameter Rotor

Radius r

(mm)

 ¼ r=R Chord c

(mm)

Pitch 

(degree)

Thickness/chord

ratio of blade (%)

1700 0.20 1085 15.0 24.6

2125 0.25 1045 12.1 22.5

2150 0.30 1005 9.5 20.7

2975 0.35 965 7.6 19.5

3400 0.40 925 6.1 18.7

3825 0.45 885 4.9 18.1

4250 0.50 845 3.9 17.6

4675 0.55 805 3.1 17.1

5100 0.60 765 2.4 16.6

5525 0.65 725 1.9 16.1

5950 0.70 685 1.5 15.6

6375 0.75 645 1.2 15.1

6800 0.80 605 0.9 14.6

6375 0.85 565 0.6 14.1

7225 0.90 525 0.4 13.6

8075 0.95 485 0.2 13.1

8500 1.00 445 0.0 12.6

CALCULATED RESULTS FOR AN ACTUAL TURBINE 93

ﬂow factor decreases with tip speed ratio. The angular velocity of the wake in-

creases sharply with decreasing radius because it is determined by the root vortex.

The importance of the outboard section of the blade is clearly demonstrated in

Figure 3.44. The dramatic effect of stall is shown in the difference in torque

distribution between the tip speed ratio of 4 and the tip speed ratio of 2. Note, also,

-10 0 10 20 30

Lift coefficient

-10 0 10 20 30

1.5

0.5

-0.5

0.4

0.3

0.2

0.1

12%

15%

18%

21%

25%

Drag coefficient

le of attack

Angle of attack

Figure 3.41 The Aerodynamic Characteristics of the NACA632XX Aerofoil Series

0.2 0.4 0.6 0.8 1

r/R

Angle of attack in degrees

Stall angle

Lamda = 2

Lamda = 4

Lamda = 6

Lamda = 8

Lamda = 10

Lamda = 12

Figure 3.42 Angle of Attack Distribution for a Range of Tip Speed Ratios

AERODYNAMICS OF HORIZONTAL-AXIS WIND TURBINES