Page 151 - Wind Energy Handbook

THE METHOD OF ACCELERATION POTENTIAL 125

yaw angle. For 308 of yaw the magnitudes of the measured and calculated tilting
moments are comparable. The measured mean tilting moment is quite definitely
non-zero and positive but the calculated mean moment is much smaller although
still positive. A positive tilt rotation would displace the upper part of the rotor disc
in the downwind direction. In the theory the small mean tilting moment is caused
by the wake rotation velocities.
A theory based upon computational fluid mechanics should provide a much
more accurate prediction of the aerodynamics of a wind turbine in yaw. However,
the severe computational time limitations associated with CFD solutions precludes
their use in favour of the simple theory outlined in these pages.

3.11 The Method of Acceleration Potential

3.11.1 Introduction

An aerodynamic model that is applied to the flight performance of helicopter rotors,
and which can also be applied to wind turbine rotors that are lightly loaded, is that
based upon the idea of acceleration potential. The method allows distributions of
the pressure drop across an actuator disc that are more general than the, strictly,
uniform pressure distribution of the momentum theory. The model has been
expounded by Kinner (1937), inspired by Prandtl, who has developed expressions
for the pressure field in the vicinity of an actuator disc, treating it as a circular wing.
To regard a rotor as a circular wing requires an infinity of very slender blades so
that the solidity remains small.
Kinner’s theory, which is derived from the Euler equations, assumes that the
induced velocities are small compared with the general flow velocity. If u, v and w
are the velocities induced by the actuator disc in the x-, y- and z-directions,
respectively, and which are very much smaller than the free-stream velocity in the
x-direction U 1 , then the rate of change of momentum in the x-direction of a unit
volume of air will be in response to the pressure gradient in that direction

@(U 1 þ u) @(U 1 þ u) @(U 1 þ u) @ p
r (U 1 þ u) þ v þ w ¼ (3:144)
@x @ y @z @x

The free-stream velocity U 1 does not change with position therefore, for example,
@(U 1 )=@x ¼ 0. Also, U 1 (u, v, w) and so, for example, v(@u=@ y) can be ignored
in comparison with U 1 (@u=@x). The momentum equation in the x-direction then
simplifies to

@u @ p
rU 1 ¼ (3:145a)
@x @x
Similarly, in the y- and z-directions, the momentum equations are also simplified