Page 79 - Wind Energy Handbook

VORTEX CYLINDER MODEL OF THE ACTUATOR DISC 53

z

∆Γ

x
Γ
y

φ
U ∞
∆Γ

Figure 3.7 Simplified Helical Vortex Wake Ignoring Wake Expansion

g Ł
u d ¼ ¼ aU 1 (3:27)
2
In the far wake the axial induced velocity is also uniform within the cylindrical
wake and is

u w ¼ g Ł ¼ 2aU 1 (3:28)
The ratio of the two induced velocities corresponds to that of the simple momentum
theory and justifies the assumption of a cylindrical vortex sheet.

3.4.3 Relationship between bound circulation and the induced
velocity

The total circulation on all of the multiplicity of blades is ˆ which is shed at a
uniform rate into the wake in one revolution. So, from Figure 3.8 in which the
cylinder has been slit longitudinally and opened out flat,

δn

φ ∆Γ = dΓ δn
dn
2 R
δs

Figure 3.8 The Geometry of the Vorticity in the Cylinder Surface