Page 161 - Wind Energy Handbook

THE METHOD OF ACCELERATION POTENTIAL 135

ð ð ð ð
1 2ð 1 1 1 1 p ffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2ð
2
C mz ¼ ì sin łp( ì, ł)ì dì dł ¼ 6iD 2 ì 3 1 ì dì sin ł dł (3:178)
2
ð 0 0 ð 0 0
which gives
5
1
iD ¼ ðC mz (3:179)
2
4
To establish a relationship between the yawing moment coefficient and the axial
velocity induced by the pressure distribution of Equation (3.176) the velocity distri-
bution has to be obtained by integrating Equations (3.145). Unfortunately, no
analytical solution has been determined for the anti-symmetric case, as Mangler
and Squire have done for the symmetric case. Numerical values of induced velo-
cities need to be calculated from Equations (3.145) using the pressure distribution
defined by Equations (3.174) and (3.175).
Pitt and Peters (1981) have determined the axial velocity distribution for values of
the yaw angle from 08 to 908: the yaw angle fixes the far upstream conditions where
the integration commences. The velocity distribution found corresponds to that of
Equation (3.167) for the axi-symmetric case. Pitt and Peters again impose the form
of Equation (3.170) and determine the average value of the axial induced velocity a 0
and the value of a s , using the same method of Equation (3.171); in both cases, of
course, numerical integration is necessary.
The values of a 0 , are not zero, as might have been expected from the anti-
symmetric pressure distribution, but are equal and opposite to the values of a s
found for the corrected axi-symmetric pressure distribution of Equation (3.166). The
variation of the two coefficients a 0 and a s with yaw angle ª is determined
numerically but, using the Mangler and Squire analytical forms for guidance,
analytical variations can be inferred. Pitt and Peters found that the linearized axial
induced velocity distribution is

15 ª
a 0 ¼ ð tan C mz (3:180)
128 2
and
ª
a S ¼ 1 tan 2 C mz (3:181)
2

Pitt and Peters also include a cosine term in the linearized axial induced flow factor
1
representation of Equation (3.170) which will only arise if C 6¼ 0
2
a ¼ a 0 þ a S ì sin ł þ a c ì cos ł (3:182)

In which case there will be an additional pressure distribution given by
p ffiffiffiffiffiffiffiffiffiffiffiffiffiffi
1
1
1
1
p( ì, ł) ¼ P ( ì)Q (i0)C cos ł ¼ 6iC ì 1 ì cos ł (3:183)
2
2
2
2
2