Michael Graham - Wind Energy Handbook

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, , and so the rotation of the flow relative to the disc is (1 + a ′)Ω. The helix angle ϕ ttakes this additional rotation into account, as determined from Eq. (3.30).

Figure 3.10 The axial variation of tangential velocity in the vicinity of an actuator disc at 50% radius, , λ = 6.

Figure 3.11 The axial variation of tangential velocity in the vicinity of an actuator disc at 101% radius, , λ = 6.

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.

The axial thrust T on the disc can be determined using the Kutta–Joukowski theorem:

where V is the tangential velocity component at the disc. If V = r Ω (1 + a′) , then, using Eq. (3.32)ignoring any additional inflow at the disc caused by the centrifugal pressure reduction due to wake swirl discussed at the end of Section 3.3.2:

(3.41)

Integration of Eq. (3.41)over the entire disc gives the thrust coefficient as

(3.42)

That is, the same as for the simple momentum theory and so in balance with the rate of change of axial momentum. Note that if the induced tangential velocity a ′ r Ω is included in V as it is in blade‐element/momentum (BEM) theory and the blade circulation is constant from the axis to the tip, there is a singularity in the axial force on the blade section at the axis as there is also at the outer tip. This points to the failure of a simple constant strength bound vortex model at the blade ends as discussed in the section on tip‐loss corrections.

Although the vortex cylinder model has been simplified by not allowing the cylinder to expand, the vortex theory nevertheless predicts flow expansion. A radial velocity is predicted by this theory as in Figure 3.12, which shows a longitudinal section of the flow field through the rotor disc. The theory is in fact a ‘small disturbance theory’ in which the singularities in the flow field (the vortex sheets in the present case) are placed on the surfaces they would lie on in the limit of vanishingly small disturbance by the rotor.

The radial velocity field that is predicted is largest on any given streamline at the actuator disc rising from zero at the axis to a weak logarithmic infinite value at the edge of the disc, which is the path of the blade tips. The infinite radial velocity at the edge is associated with non‐zero disc loading right up to the edge. This is not realistic, being a consequence of assuming the rotor to consist of an infinite number of blades whose effect is ‘smeared’ uniformly over the disc, but being a weak singularity does not significantly affect the rest of the flow field. In applying the more detailed BEM theory the tip region is corrected by a tip correction factor to recognise that in reality the blade loading must fall to zero at the blade tips.

An alternative method of deriving the velocity field of the actuator disc has been given more recently by Conway (1998). This method takes the approach of building up the flow field from a sum of Bessel functions that are fundamental solutions of a cylindrical potential flow. The method has advantages if it is required to calculate the velocity at general points throughout the flow field both within the bounding streamtube that forms the boundary of the wake and outside it. For the streamwise velocity U 1in the simple uniform actuator disc flow:

where r and x here are radial and streamwise coordinates non‐dimensionalised by the tip radius, a 1is the wake induction factor, and J 0and J 1are Bessel functions of the first kind.

Figure 3.12 Flow field through an actuator disc for a = 1/3 .

This flow field may also be computed by solving the axisymmetric flow equations numerically either as inviscid Euler equations or as the full Navier–Stokes equations to compute the effects of viscous (or turbulent) mixing in the wake of the rotor (see section on computational fluid dynamics [CFD] in Chapter 4). Both stream function – vorticity and primitive variable (velocity – pressure) formulations have been used to do this; see, e.g. Mikkelsen (2003), Soerensen et al. (1998), Madsen et al. (2010).