The Unsteady Aerodynamics of Turbine Yaw and Over-Speed Protection - Small Wind Turbines: Analysis, Design, and Application - page 151

and have been used to model a wind turbine tail fin by Ackerman [ 11 ], and Davis
and Hansen [ 12 ]. For a delta wing tail fin, K is given by

K ¼ pb = c ð 8 : 6 Þ

from slender body theory, see Eq. 8.94 of Katz and Plotkin [ 13 ]. Note also that
K = K p from Eq. 8.2 for any planform whose lift and drag are describable by the
Polhamus equations.
A linear second order equation for yaw performance can also be derived by
applying ''unsteady slender body'' theory (USB), Ebert and Wood [ 3 ]; the basic
material is in Chap. 13 of Katz and Plotkin [ 13 ]. Though still only valid for small
angles with drag neglected, this is, in principle, a more comprehensive treatment,
as it includes the time-variation of the trailing vorticity and does not ignore the
dynamic issues of a non-inertial co-ordinate system. The equation for the moment
about the turbine's yaw axis is

Ih ¼ K 1 h u Þ K 2 h K 0

2 u þ K 3 h

ð ð 8 : 7 Þ

where h is now the angle between the fin and an ''earth-fixed'' inertial system
whose actual origin is not important as attention is focused on yaw rates. u is the
wind direction in the same co-ordinates. As before, I is the total yaw moment of
inertia of the turbine, and
K 1 ¼ 1

4 pqb 2 U 2 ð 2 = 3 c þ x Þ¼ 1

4 pqb 2 U 2 r ;

4 pqb 2 U ð c 2 4 þ cx = 3 Þ;

K 2 ¼ 1 K 2 ¼ 1

4 pqb 2 U ð c þ x Þ 2 ;

ð 8 : 8 Þ

and

4 pqb 2 c ð c 2 5 þ x 2 3 þ cx = 2 Þ;

K 3 ¼ 1

where the tail boom length x = r - 2c/3. The term involving K 1 is the steady lift
and is equivalent to ( 8.6 ) for the lift slope with the centre of pressure being 2c/3
from the apex and distance r from the yaw axis. K 2 and K 2

0

are due to the
instantaneous downwash—the trajectory of the flow caused by the lift-produced
vorticity—and represents the main change from the pseudo-static equation. A term
in the time rate change of wind speed should also appear in ( 8.7 ) but many
applications of the equation to actual data as described below, suggest that it is not
important.
Neither theory is entirely satisfactory: the quasi-static method ignores the
unsteadiness of the tail fin wake and the ''added mass''—the effective mass of air
that moves with the fin—although the analysis presented below shows this error to
be small. USB has an added mass term and the unfortunate failing of predicting no
steady lift for a rectangular wing, because USB forces are generated by the
chordwise changes to the wing shape. Readers may be interested to know that USB
has been applied to the aerodynamics of bird tails [ 14 - 16 ].
Figure 8.2 suggests that x is considerably greater than c for most turbines,
which, combined with Eq. 8.8 for K 1 implies that the chord of the tail fin only has