Page 108 - Wind Energy Handbook

82 AERODYNAMICS OF HORIZONTAL-AXIS WIND TURBINES

parallel to one another and will form a surface of discontinuity of velocity in a
radial sense within the wake; the axial velocity components will be equal. The
surface of discontinuity is called a vortex sheet. A similar phenomenon occurs with
aircraft wings and a textbook of aircraft aerodynamics will explain it in greater
detail.
A deeper understanding of the mechanism of tip-loss can be obtained by
following the path of air particles. An air particle approaches the spinning rotor,
‘senses’ high pressure ahead and slows down accordingly. The high pressure on
the upwind side of the rotor blades is effectively smeared around the whole disc.
Slowing down also causes the particle to move outwards to maintain the mass flow
rate. When the particle reaches the rotor plane it will either be close to a blade or
not and its axial velocity will be affected accordingly, as shown in Figure 3.28. If the
particle passes through the rotor plane close to blade then it will also be strongly
affected by the blade’s pressure field.
A particle which passes close to and in front of a blade will leave the trailing edge
having accelerated in the tangential direction; it will then pass downstream, on the
upwind side of the vortex sheet being shed from the trailing edge and so will also
be moving radially outwards. The particle, therefore, migrates outward to the edge
of the vortex sheet around which it is swept on to the downwind side and migrates
inward with a radial velocity which reduces to zero at a radial point on the sheet
where the shed vorticity is zero. The particle then continues downstream with the
velocity of the axial and tangential velocities of the vortex sheet.
A second particle which passes a blade close to the downwind, low pressure,
surface is accelerated tangentially in the opposite direction to the blade motion and
then slows down, leaving the trailing edge with the same axial and tangential
velocity components as the first particle but on the downwind side of the vortex
sheet so it will have, in addition, a radially inwards velocity. The second particle
will, depending on its radial position, migrate inwards until the radial velocity
becomes zero.
A third particle which passes between two blades will be moving axially at a
greater velocity than the first two particles, will not be strongly affected by the
pressure fields of the blades but, because of the solid blockage presented by the
blades (see Figure 3.5), will be directed into a helical path. Being faster, axially, than
the vortex sheet ahead the particle will begin to catch up, as it does so the influence
of the vortex sheet will move it outwards, around the edge of the sheet and then
inwards, just like the first particle. Unlike the first particle, however, the third
particle will still be moving faster than the vortex sheet and so will move axially
away from the sheet, approaching the next sheet downstream and repeating the
motion around the edge of that sheet. The particle will proceed downstream over-
taking and hopping around each vortex sheet in turn.
The third particle does not lose as much axial momentum as particles one and
two and is therefore affected by the so-called tip-loss. The affect is greater the closer
the third particle is to the edge of the rotor disc as it passes through the disc.
A fourth particle passes between the blades but at a radial position, closer to the
axis of rotation, where its axial velocity is equal to that of the vortex sheets. If the
particle passes midway, say, between two blades then it remains midway between
the two corresponding vortex sheets as it moves downstream and does not undergo