where V is the tangential velocity component at the disc. If V = rΩ(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′rΩ 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.
3.4.9 Radial flow and the general flow field
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 U1 in the simple uniform actuator disc flow:
where r and x here are radial and streamwise coordinates non‐dimensionalised by the tip radius, a1 is the wake induction factor, and J0 and J1 are 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).
The limiting condition of the cylindrical wake model of the flow through an actuator disc occurs as the loading on the actuator disc is increased so that the wake induction factor a approaches a value of 0.5. At this value the streamwise velocity in the wake Uw(= (1 − 2a)U∞) falls to zero, and the wake is therefore predicted to expand indefinitely to an infinite cross‐section. Beyond this value the wake flow is predicted to be negative, and the theory must break down. The wakes of rotors and also of porous discs normal to a flow that similarly correspond to actuator discs all reach a state when the pressure in the wake region immediately downstream of the body has fallen sufficiently that steady streamline flow can no longer continue stably in the near wake region. Castro (1971) has studied the wake of a porous disc in detail, showing how a reverse flow bubble forms downstream in the wake and moves upstream towards the actuator disc as the loading increases further. This regime is known as the turbulent wake state for a turbine rotor and will be discussed further in the following Section 3.5 on BEM theory.
3.4.10 Further development of the actuator model
The one‐dimensional actuator disc model and associated vortex cylinder representation of the flow field is the simplest model of a HAWT that can provide useful results. This model may be developed in several ways to be more representative of the details of the flow.
Radial variation across the actuator disc may be considered as in BEM theory, Section 3.5 below.
Also, recognition may be given to the fact that the turbine has a finite number of blades, usually a small number such as two or three, each of which may be treated individually as a momentum sink actuator. In the simplest version taking average values, the forces on each blade are assumed to be radially constant. The lift and drag forces calculated from the flow angles at the blades with the relevant aerofoil section characteristics (as in Section 3.5.2) are converted into rotating axial and tangential momentum sinks projected onto a larger field grid computation. This is the basis of the simplest actuator line model (see Section 3.6).
With the development of large wind farms, particularly offshore, it has become important to simulate the flow through the whole