Page 143 - Wind Energy Handbook

THE AERODYNAMICS OF A WIND TURBINE IN STEADY YAW 117

The rate of change of momentum will use either Equation (3.105), Glauert’s
theory, or Equation (3.111), the vortex cylinder theory; in both equations the flow
induction factor a should be replaced by af to account for Prandtl tip loss.
For Glauert’s theory

1 2 p ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
äF m ¼ rU 4af 1 af(2 cos ª af)räłär (3:134a)
1
2
Or, for the vortex theory,

1 2 ÷ 2 ÷
äF m ¼ rU 4af cos ª þ tan sin ª af sec räłär (3:134b)
1
2 2 2
The Equations (3.134) do not include the drop in pressure caused by wake rotation
but to do this would require greater detail about wake rotation velocities from the
vortex theory. The algebraic complexity of estimating the wake rotation velocities is
great and even then fluctuation of bound circulation is ignored. The drop in
pressure caused by wake rotation, however, is shown to be small in the non-yawed
case and so it is assumed that it can safely be ignored in the yawed case.
The moment of the blade element force about the wake axis is

1 2 äł
äM b ¼ rW Nc(cos ł sin ÷C x þ cos ÷C y rär
2 2ð

where

C y ¼ C l sin ö C d cos ö

therefore

1 2 2
äM b ¼ rW ó r (cos ł sin ÷C x þ cos ÷C y )r äräł (3:135)
2

The rate of change of angular momentum is the mass flow rate through an
elemental area of the disc times the tangential velocity times radius.

2
äM m ¼ rU 1 (cos ª af)räłär2a9 fÙr-

where
2
2
2
2
2
r- ¼ r (cos ł þ cos ÷ sin ł)
therefore
1 2 2 2 2 2
äM m ¼ rU ºì4a9f(cos ª af)(cos ł þ cos ÷ sin ł)r äräł (3:136)
1
2