Читать книгу Wind Energy Handbook - Michael Barton Graham - Страница 66

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_t 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, , and so the rotation of the flow relative to the disc is (1 + a^′)Ω. The helix angle ϕ_t takes 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.