Page 104 - Wind Energy Handbook

78 AERODYNAMICS OF HORIZONTAL-AXIS WIND TURBINES

3.8 The Effects of a Discrete Number of Blades

3.8.1 Introduction

The analysis described in the previous Sections of this chapter assumes that there is
a sufficient number of blades on the rotor for every fluid particle passing through
the rotor disc to interact with a blade, i.e., that all fluid particles undergo the same
loss of momentum. With a small number of blades some fluid particles will interact
with them but most will pass between the blades and, clearly, the loss of momen-
tum by a particle will depend on its proximity to a blade as the particle passes
through the rotor disc. The axial induced velocity will, therefore, at any instant,
vary around the disc, the average value determining the overall axial momentum of
the flow and the larger value local to a blade determining the forces on the blade.

3.8.2 Tip-losses

If the axial flow induction factor a is large at the blade position then, by Equation
(3.42), the inflow angle ö will be small and the lift force will be almost normal to the
rotor plane. The component of the lift force in the tangential direction will be small
and so will be its contribution to the torque. A reduced torque means reduced
power and this reduction is known as tip loss because the effect occurs only at the
outermost parts of the blades.
In order to account for tip losses, the manner in which the axial flow induction
factor varies azimuthally needs to be known but, unfortunately, this requirement is
beyond the abilities of the blade element–momentum theory.
Just as a vortex trails from the tip of an aircraft wing so does a vortex trail from
the tip of a wind turbine blade. Because the blade tip follows a circular path it
leaves a trailing vortex of a helical structure, as is shown in Figure 3.27, which
convects downstream with the wake velocity. For a two-blade rotor, unlike an
aircrafts wings, the bound circulations on the two blades are opposite in sign and
so combine to shed a straight line vortex along the rotational axis with strength
equal to the blade circulation times the number of blades.
For a single vortex to be shed from the blade tip the circulation strength along the
blade span must be uniform and, as has been shown, uniform circulation is a
requirement for optimized operation. However, the uniform circulation require-
ment assumes that the axial flow induction factor is uniform across the disc and, as
has been argued above, with discrete blades rather than a uniform disc the flow
factor is not uniform.
In the case of Figure 3.27, very close to the blade tips the tip vortex causes very
high values of the flow factor a such that, locally, the net flow past the blade is in
the upstream direction. The average value of a, azimuthally, is radially uniform
which means that if high values occur in the vicinity of the blades then low values
occur elsewhere. The azimuthal variation of a at various radial positions is shown
in Figure 3.28 for a three blade rotor operating at a top speed ratio of 6. The