Page 172 - Wind Energy Handbook

146 AERODYNAMICS OF HORIZONTAL-AXIS WIND TURBINES

3.13.3 Unsteady yawing and tilting moments

For unsteady flow in yaw the normal unsteady acceleration distribution on the disc
is required to have the same form of linear variation as the velocity, given in
Equation (3.182). In terms of flow factors
@a ¼ @a 0 þ @a s ì sin ł þ @a c ì cos ł (3:202)
@ô @ô @ô @ô

The condition that causes a yawing moment arises from the anti-symmetric
pressure distribution of Section 3.11.4 and can be obtained from Equations (3.175).
For the whole field surrounding the rotor disc the pressure distribution is
!
3 1 p ffiffiffiffiffiffiffiffiffiffiffiffiffi p ffiffiffiffiffiffiffiffiffiffiffiffiffi 1 1 p ffiffiffiffiffiffiffiffiffiffiffiffiffi i
2
2
p(v, ç, ł) ¼ D v 1 v 2 3iç 1 þ ç tan 3i 1 þ ç þ p ffiffiffiffiffiffiffiffiffiffiffiffiffi sin ł
2
2 ç 1 þ ç 2
which, on the disc, produces the pressure shown in Figure 3.72. The coefficient D 1
2
is related to the yawing moment coefficient in Equation (3.179)
5
1
iD ¼ C mz (3:179)
2
4
Therefore
!
15 p ffiffiffiffiffiffiffiffiffiffiffiffiffi p ffiffiffiffiffiffiffiffiffiffiffiffiffi 1 1 p ffiffiffiffiffiffiffiffiffiffiffiffiffi 1
2
2
p(v, ç, ł) ¼ C mz v 1 v 2 3ç 1 þ ç tan 3 1 þ ç þ p ffiffiffiffiffiffiffiffiffiffiffiffiffi sin ł
8 ç 1 þ ç 2
(3:203)
As before, the pressure in Equation (3.203) is non-dimensionalized by the free-
2
stream dynamic pressure (1=2)rU .
1
Applying the differential operator given in Equation (3.196) to Equation (3.203),
from Equation (3.192) we get at the rotor disc, where ç ¼ 0,
45 U 2
@u s 1
¼ ð C mzD ì sin ł (3:204)
@t 32 R
In terms of non-dimensional time and velocity

@a s ¼ 45 ðC mzD ì sin ł (3:205)
@ô 32

Similarly, if there is a tilting moment then the corresponding acceleration is

@a c 45
¼ ðC myD ì cos ł (3:206)
@ô 32
The radial variation is linear and so no linearization adjustment is necessary. Again,