Michael Graham - Wind Energy Handbook

It is similarly necessary to include the rotation induction factor to calculate the angle of slant φ tof the vortices:

Thus Tan φ t= (1 − a )/(1 + a′ ) λ

Just as a vortex is shed from each blade tip, a vortex is also shed from each blade root. If it is assumed that the blades extend to the axis of rotation, obviously not a practical option, then the root vortices will each be a line vortex running axially downstream from the centre of the disc. The direction of rotation of all of the root vortices will be the same, forming a core, or root, vortex of total strength Γ. The root vortex is primarily responsible for inducing the tangential velocity in the wake flow and in particular the tangential velocity on the rotor disc.

On the rotor disc surface the tangential velocity induced by the root vortex, given by the Biot–Savart law, is

so

(3.33)

This relationship can also be derived from the momentum theory – the rate of change of angular momentum of the air that passes through an annulus of the disc of radius r and radial width δr is equal to the torque increment imposed upon the annulus:

(3.34)

The torque per unit span acting on all the blades is given by the Kutta–Joukowski theorem. The lift per unit radial width L is

where ( W × Γ) is a vector product, and W is the relative velocity of the air flow past the blade:

(3.35)

Equating the two expressions for δQ gives

If a ′in Eq. (3.32)is now treated as being negligible with respect to 1 (which it is in normal circumstances) then:

At the outer edge of the disc the tangential induced velocity is

(3.36)

Equation (3.36)is exactly the same as Eq. (3.23)of Section 3.3.3.

If a′ is retained in Eq. (3.32), there is a small inconsistency here between vortex theory and the one‐dimensional actuator disc theory, which ignores rotation effects.

The torque on an annulus of radius r and radial width δr (ignoring a′ as actuator disc theory ignores rotation) is

(3.37)

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.

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.

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 ϕ tin 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.