Page 83 - Wind Energy Handbook

VORTEX CYLINDER MODEL OF THE ACTUATOR DISC 57

3.4.7 Tangential flow field

The tangential induced velocity is determined not only by the root vortex but also
by the component of vorticity g sin ö on the wake cylinder and the bound vorticity
on the rotor disc. At a radial distance equal to half the disc radius, as an example,
the axial variation of the three contributions are shown in Figure 3.10.
The bound vorticity causes rotation in opposite senses upstream and downstream
of the disc with a step change across the disc. The upstream rotation, which is in the
same sense as the rotor rotation, is nullified by the root vortex, which induces
rotation in the opposite sense to that of the rotor. The downstream rotation is in the
same sense for both the root vortex and the bound vorticity the stream-wise
variations of the two summing to give a uniform velocity in the stream-wise sense.
The vorticity located on the surface of the wake cylinder makes a small contri-
bution.
At the disc itself the bound vorticity induces no rotation, see Figure 3.10, the
wake cylinder induces no rotation either and so it is only the root vortex which does
induce rotation and that value is half the total induced generally in the wake. It is
now clear why only half the rotational velocity is used to determine the flow angle
at the disc.
The rotational flow is confined to the wake, that is, inside the cylinder. There is
no rotational flow anywhere outside of the wake, neither upstream of the disc or
outside the cylinder. The rotational velocity within the cylinder falls with increasing
radius but is not zero at the outer edge of the wake, therefore there is an abrupt fall
of rotational velocity across the cylindrical surface. The contributions of the three
vorticity sources to the rotational flow at a radius of 101 percent of the disc radius
are shown in Figure 3.11; the total rotational flow is zero at all axial positions but
the individual components are not zero.

0.06
Total
0.04
Tangential flow induction factor 0.02 Total Vortex cylinder
Root vortex
Bound vortex

0

-0.02

-0.04
-1 -0.5 0 0.5 1
ξ
Axial distance
Figure 3.10 The Axial Variation of Tangential Velocity in the Vicinity of an Actuator Disc at
50% Radius, a ¼ 1=3, º ¼ 6