Page 80 - Wind Energy Handbook

54 AERODYNAMICS OF HORIZONTAL-AXIS WIND TURBINES

ˆ
g ¼ (3:29)
2ðR sin(ö t )

hence

ˆ cos ö t ˆ ÙR(1 þ a9 t )
g Ł ¼ ¼ (3:30)
2ðR sin ö t 2ð U 1 (1 a)
therefore

ˆ ÙR(1 þ a9 t )
2aU 1 ¼ (3:31)
2ðR U 1 (1 a)

So, the total circulation is related to the induced velocity
2
4ðU a(1 a)
ˆ ¼ 1 (3:32)
Ù(1 þ a9 t )

3.4.4 Root vortex

Just as a vortex is shed from each blade tip a vortex is also shed from each blade
root. If it is assumed that the blades extend to the axis of rotation, obviously not a
practical option, then the root vortices will each be a line vortex running axially
downstream from the centre of the disc. The direction of rotation of the all the root
vortices will be the same forming a core, or root, vortex, of total strength ˆ. The root
vortex is primarily responsible for inducing the tangential velocity in the wake flow
and in particular the tangential velocity on the rotor disc.
On the rotor disc surface the tangential velocity induced by the root vortex, given
by the Biot–Savart law, is

ˆ
Ùra9 ¼
4ðr
ˆ
a9 ¼ (3:33)
4ðr Ù
2
This relationship can also be derived from the momentum theory: the rate of change
of angular momentum of the air which passes through an annulus of the disc of
radius r and radial width dr is equal to the torque increment imposed upon the
annulus

dQ ¼ rU 1 (1 a)2ðr dr2a9Ùr 2 (3:34)

By the Kutta–Joukowski theorem the lift per unit radial width is

L ¼ r(W 3 ˆ)