Page 85 - Wind Energy Handbook

ROTOR BLADE THEORY 59

velocity is predicted at the disc edge for the flow to pass around and continue
radially inwards on the downstream side. In practice there would be insufficient
static pressure available to fuel an infinite, or even a very high, velocity and so some
discontinuity in the flow must occur. The presence of even the smallest amount of
viscosity would produce a thick boundary layer towards the disc edge because of
the high radial velocity. The viscosity in the boundary layer would absorb much of
the available energy and dissipate it as heat so that as the flow accelerated around
the disc edge the maximum velocity attainable would be limited by the static
pressure approaching zero. Instead of the flow moving around the edge it would
separate from the edge and continue downstream leaving a very low pressure
region behind the disc with very low velocity—a stagnant region.
In the case of the permeable rotor disc there would be some flow through the
disc, which would behave as described above, but the separation of the flow at the
disc edge would produce an additional low pressure in the wake.
The problem of the infinite radial velocity at the rotor disc edge arises because of
the assumption of an infinite number of rotor blades. If the theory is modified such
that there are only a few blades the infinite radial velocity disappears. However, if,
for a given rotor, the tip speed ratio is increased, with a consequent increase of the
axial flow induction factor, the radial velocity at the tip rises sharply and the
problem of edge separation returns, which is what actually occurs, see Section 3.6.

3.4.9 Conclusions

Despite the exclusion of wake expansion, the vortex theory produces results largely
in agreement with the momentum theory and enlightens understanding of the flow
through an energy extracting actuator disc.

3.5 Rotor Blade Theory

3.5.1 Introduction

The aerodynamic lift (and drag) forces on the span-wise elements of radius r and
length är of the several blades of a wind turbine rotor are responsible for the rate of
change of axial and angular momentum of all of the air which passes through the
annulus swept by the blade elements. In addition, the force on the blade elements
caused by the drop in pressure associated with the rotational velocity in the wake
must also be provided by the aerodynamic lift and drag. As there is no rotation of
the flow approaching the rotor the reduced pressure on the downwind side of the
rotor caused by wake rotation appears as a step pressure drop just like that which
causes the change in axial momentum. Because the wake is still rotating in the far
wake the pressure drop caused by the rotation is still present and so does not
contribute to the axial momentum change.