Page 138 - Wind Energy Handbook

112 AERODYNAMICS OF HORIZONTAL-AXIS WIND TURBINES

difference in angle of attack can be attributed to flow expansion as depicted in
Figure 3.61.
The variation of the angle of attack makes the flow about a blade aerofoil
unsteady and so the lift will have a response of the kind discussed in Section 3.1.
The blade circulation will therefore vary during the course of a revolution, which
means that the vortex model is incomplete because it is derived from the assump-
tion that the circulation is constant.
There is clearly additional vorticity in the wake being shed from the blades’
trailing edges influencing the induced velocity, which is not accounted for in the
theory. The additional induced velocity would be cyclic so would probably not
affect the average induced velocity normal to the rotor disc but would affect the
amplitude and phasing of the angle of attack.
Further numerical analysis of the Coleman vortex theory reveals that at skew
angles greater than 458 higher harmonics than just the one per revolution term in
Equation (3.108) become significant in the flow expansion induced velocities. Only
odd harmonics are present, reflecting the anti-symmetry about the yaw axis.

3.10.5 Related theories

A number of refinements to the Glauert and Coleman theories have been proposed
by other researchers, mostly addressing helicopter aerodynamics but some have
been directed specifically at wind turbines. In particular, Øye (1992) undertook the
same analysis as Coleman and proposed a simple curve-fit to Equation (3.116)

1 3 5
F ö ( ì) ¼ ( ì þ 0:4ì þ 0:4ì ) (3:123)
2
Øye has clearly avoided the very large values that Equation (3.116) produces
close to the outer edge of the disc and Equation (3.122) is in general accordance with
the flow expansion functions shown in Figure 3.59.

γ
High angle of attack
χ

Low angle of attack

Figure 3.61 Flow Expansion Causes a Differential Angle of Attack