Page 82 - Wind Energy Handbook

56 AERODYNAMICS OF HORIZONTAL-AXIS WIND TURBINES

Power coefficient
4a(1 a) 2 2
C P ¼ ¼ 4a9 t (1 a) (3:40)
(1 þ a9 t )

The reduced efficiency compared with the simple actuator disc result, C P ¼
2
4a(1 a) , is caused by the energy required to spin the wake, as a rigid body, with
an angular velocity 2a9 t Ù. It should be noted that any additional rotational energy is
accounted for by the loss of static pressure caused by the pressure gradient which
balances the centrifugal forces on the rotating fluid.
The general momentum theory of Glauert (1935a) includes the expansion of the
wake and shows that no contribution at all is actually required from the kinetic
energy of the free-stream flow to maintain wake rotation; all the kinetic energy of
rotation is derived from static pressure energy.

3.4.6 Axial flow field

By means of the Biot-Savart law the induced velocity in the wind-wise (axial)
direction at the actuator disc can be determined both upstream of the disc and
downstream in the developing wake, as well as on the disc itself. The flow field (net
velocity) is axisymmetric and a radial cross section is shown in Figure 3.9. Both
radial and axial distances are divided by the disc radius with the axial distance
being measured downstream from the disc and the radial distance being measured
from the rotational axis. The velocity is divided by the wind speed.
The axial velocity within the wake is sharply lower than without and is radially
uniform at the disc and in the far wake, just as the momentum theory predicts.
There is a small acceleration of the flow immediately outside of the wake. The
1
induced velocity on the wake cylinder itself is a at the disc and a in the far wake.
2

1
0.8
0.6
Axial flow
velocity 0.4
2
0.2 1 1.5
0.5 2
0
-2 1
-1
0 1
1 0.8 0
Axial distance 2 0.6
2 0.4 -1
1.5 0.2
1
0.5 0 0 -2
Radial distance
Figure 3.9 The Radial and Axial Variation of Axial Velocity in the Vicinity of an Actuator
Disc, a ¼ 1=3