The radial distribution of power is
(3.38)
and, therefore, the total power is
(3.39)
Power coefficient:
(3.40)
Again, a result that is identical to that predicted by the simple momentum theory.
What is particularly interesting is that the residual rotational flow in the wake makes no apparent reduction in the efficiency of the power extraction.
3.4.6 Axial flow field
The induced velocity in the windwise (axial) direction can be determined both upstream of the disc and downstream in the developing wake, as well as on the disc itself. This velocity is induced by the azimuthal component of vorticity in the cylindrical wake sheet at radius R (which generates an axisymmetric axial back‐flow within the wake) as shown for a radial section 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 in this model falls discontinuously across the wake boundary from the external value 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 around the disc immediately outside of the wake. The induced velocity at the wake cylinder surface itself and hence its convection velocity is −½ a at the disc and −a in the far wake.
3.4.7 Tangential flow field
The tangential induced velocity is induced by three contributions: that due to the root line vortex along the axis (which generates a rising swirl from zero upstream to a constant value in the far wake), that due to the axial component of vorticity g sin ϕt in the cylindrical sheet at radius R, and that due to the bound vorticity, everywhere in the radial direction on the disc. 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 streamwise variations of the two summing to give a uniform velocity in the streamwise sense. The vorticity located on the surface of the wake cylinder makes a small contribution.
Figure 3.9 The radial and axial variation of axial velocity in the vicinity of an actuator disc,
Note that the bound vorticity (being the circulation on the rotor blades in response to the incident and induced flow) induces zero rotation at the disc and decays axially up and downstream. The discontinuity in tangential velocity at the disc is because the idealised changes are assumed to take place through a disc of zero thickness. In reality the azimuthal velocity rises rapidly but continuously as the flow passes through the rotor blades, which sweep through a disc and influence region of finite thickness as shown in Figure 3.5.
At the disc itself, because the bound vorticity induces no rotation and the wake cylinder induces no rotation within the wake cylinder either, it is only the root vortex that does induce rotation, and that value is half the total induced generally in the wake. Hence the root vortex induced rotation that is only half the rotational velocity is used to determine the flow angle at the disc. At a radial distance equal to half the disc radius, as an example, the axial variation of the three contributions is shown in Figure 3.10.
The rotational flow is confined to the wake, that is, inside the cylinder, and tends asymptotically to 2a′Ω well downstream of the rotor. There is no rotational flow anywhere outside the wake, neither upstream of the disc nor at radial distances outside the wake cylinder. Because of this there is no first order transverse effect of the proximity of a ground plane on the downstream convection of the vortex wake of a wind turbine as there is on the trailing vortices of a fixed wing aircraft. The rotational flow within the wake cylinder decreases radially from the axis to the wake boundary but is not zero at the outer edge of the wake, therefore there is an abrupt fall of rotational velocity across this cylindrical wake surface vortex sheet.
And because of this profile of rotation the cylindrical vortex sheet itself, therefore, rotates with the mean of the inside and outside angular velocities,
Figure 3.10 The axial variation of tangential velocity in the vicinity of an actuator disc at 50% radius,
Figure 3.11 The axial variation of tangential velocity in the vicinity of an actuator disc at 101% radius,
The contributions of the three vorticity sources to the rotational flow at a radius of 101% 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.
3.4.8 Axial thrust
The axial thrust T on the disc can be determined using the Kutta–Joukowski theorem: