Aerodynamics and aeroelastics of wind turbines - Wind Power Generation and Wind Turbine Design - page 93

of particular interest. Firstly, all turbines have a maximum c P and two kinds of
turbine can be distinguished, those with a maximum c P 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 P = 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

w

( 7 )

a ′ = .

2 Ω

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

3
4(1)

dM = r v − a a ′ dr

( 8 )

and the total power becomes

3

r r

⎛⎞ ⎛⎞

1

2

∫

c = 8 l a ′ (1 − a ) d . ( 9)

⎜⎟ ⎜⎟

P

⎝⎠ ⎝⎠

R R

0

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

orthogonality condition of Glauert [12] is the condition

′

2

aax ′ (1 + ′ ) = aa (1 − ) (10 )

with x : = wr / v 1 being the local TSR. Figure 4 gives a sketch of the arrangement of
the velocity vectors. Optimization results in the condition

13
14

− a

( 11 )

a ′=

+ a

Figure 4 : Velocity and force triangles.