Page 127 - Wind Energy Handbook

THE AERODYNAMICS OF A WIND TURBINE IN STEADY YAW 101

undergoes a rate of change of momentum in the forward direction as well, thus
balancing the drag. The drag is termed induced drag as it comes about by the
backward tilting of the lift force caused by the induced velocity and has nothing to
do with viscosity, it is entirely a pressure drag. Equation (3.92) should also be
modified to replace V by W, the resultant velocity at the disc, and the area S will be
in a plane normal to W. Also W has a direction which lies close to the plane of the
rotor and so the lift force L will be almost the same as the thrust force F, which is
normal to the plane of the rotor, and by the same argument the induced velocity is
almost normal to the plane of the rotor
2F
u ¼ (3:100)
2
ð(2R) rW
It can be assumed that a wind turbine rotor at high angles of yaw behaves just like
the autogyro rotor.
At zero yaw the thrust force on the wind turbine rotor disc, given by the
momentum theory, is

1
F ¼ ðR 2 r4u(U 1 u) (3:101)
2
where U 1 now replaces V, so the induced velocity is

2F
u ¼ (3:102)
2
ð(2R) r(U 1 u)
The area S now coincides in position with the rotor disc.
Putting W ¼ U 1 u to represent the resultant velocity of the flow at the disc in
Equation (3.102) then gives exactly the same Equation as (3.100) which is for a large
angle of yaw. On the basis of this argument Glauert assumed that Equation (3.100),
which is the simple momentum theory, could be applied at all angles of yaw, area
2
S ¼ ðR , through which the mass flow rate is determined, always lying in a plane
normal to the resultant velocity. The rotation of the area S is a crucially different
assumption to that of the theory of Section 3.10.1 (which will now be referred as the
axial momentum theory) and allows for part of the thrust force to be attributable to
an overall lift on the rotor disc.
Thus
2
F ¼ rðR W2u (3:103)

where
q ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
2
W ¼ U sin ª þ (U 1 cos ª u) 2 (3:104)
2
1
Thrust is equal to the mass flow rate times the change in velocity in the direction of
the thrust. Both F and u are assumed to be normal to the plane of the disc. The
thrust coefficient is then