Michael Graham - Wind Energy Handbook

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 U w(= (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 3.5on BEM theory.

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 3.5below.

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 wind farm to calculate the effect of multiple wakes interacting with each other and with the incident atmospheric boundary layer (ABL) and impinging on downstream rotors. Wake interactions have a very significant effect on power generated by turbines downstream of others (see, e.g. Argyle et al. 2018) and on the buffeting of downstream rotors. The usual method of carrying out these computations is to embed actuator models of the turbines within much larger numerical grid representations of the flow through and surrounding the whole wind farm. This outer large‐scale flow is solved numerically on the grid by conventional, and now well‐established, CFD Reynolds averaged Navier–Stokes (RANS) or higher fidelity but much more computationally expensive large eddy simulation (LES) computer codes. The actuator model embedded in the grid to represent the action of each turbine may be at the simplest level of an actuator disc model, in which the thrust force on the rotor disc is inserted as a momentum sink, i.e. a step change in momentum in the streamwise direction across grid cells that are intersected by the rotor disc. However, it is usually found desirable to go to a higher level of representation including swirl and embed an actuator line model for each turbine blade in the grid. The rotating actuator lines are now the momentum sinks of both axial and azimuthal forces including the radial variations, which are projected at each timestep onto the adjacent grid points (see, for example, Soerensen and Shen 2002).

Despite the exclusion of wake expansion, the vortex theory produces results in agreement with the momentum theory and enlightens understanding of the flow through an energy extracting actuator disc. However, the infinite radial velocity predicted at the outer edge of the disc is further evidence that the actuator disc is physically unrealisable.

The aerodynamic lift (and drag) forces on the spanwise elements of radius r and length δr of the several blades of a wind turbine rotor are responsible for the rate of change of axial and angular momentum of all of the air that passes through the annulus swept by the blade elements. In addition, the force on the blade elements caused by the drop in pressure associated with the rotational velocity in the wake must also be provided by the aerodynamic lift and drag. As there is no rotation of the flow approaching the rotor, the reduced pressure on the downwind side of the rotor caused by wake rotation appears as a step pressure drop just as is that which causes the change in axial momentum. Because the wake is still rotating in the far wake, the pressure reduction associated with the rotation is still present and so does not contribute to the axial momentum change.

It is assumed that the forces on a blade element can be calculated by means of two‐dimensional (2‐D) aerofoil characteristics using an angle of attack determined from the incident resultant velocity in the cross‐sectional plane of the element. Applying the independence principle (see Appendix A3.1), the velocity component in the spanwise direction is ignored. Three‐dimensional (3‐D) effects are also ignored.

The velocity components at a radial position on the blade expressed in terms of the wind speed, the flow factors, and the rotational speed of the rotor together with the blade pitch angle will determine the angle of attack. Having information about how the aerofoil characteristic coefficients C land C dvary with the angle of attack, the forces on the blades for given values of a and a ′can be determined.

Consider a turbine with B blades of tip radius R each with chord c and set pitch angle β measured between the aerofoil chord‐line and the plane of the disc. (Note that in referencing the pitch to the blade chord line the zero incidence lift coefficient must be included). Both the chord length, section profile (thickness and camber), and the pitch angle may vary along the blade span. Let the blades be rotating at angular velocity Ω and let the wind speed be U ∞. The tangential velocity experienced by the blade element shown in Figure 3.13is (1 + a ′) r Ω. The actuator disc is infinitesimally thin; the change in tangential velocity is abrupt, but it is only the component induced by the root vortex that contributes. This varies smoothly across the region of the actuator disc ( Figure 3.10). The bound velocity induced by the vorticity on the disc does not contribute.

Figure 3.14shows all the velocities and forces relative to the blade chord line at radius r .

From Figure 3.14, the resultant relative velocity at the blade is

(3.43)

Figure 3.13 A blade element sweeps out an annular ring.

Figure 3.14 Blade element velocities and forces: (a) velocities, and (b) forces.