Page 78 - Wind Energy Handbook

52 AERODYNAMICS OF HORIZONTAL-AXIS WIND TURBINES

by a multiplicity of aerofoil blades each with radially uniform bound circulation
˜ˆ. From the tip of each blade a helical vortex of strength ˜ˆ convects downstream
with the local flow velocity (Figure 3.6). If the number of blades is assumed to be
very large but the solidity of the total is finite and small then the accumulation of
helical tip vortices will form the surface of a tube. As the number of blades
approaches infinity the tube surface will become a continuous tubular vortex sheet.
From the root of each blade, assuming it reaches to the axis of rotation, a line
vortex of strength ˜ˆ will extend downstream along the axis of rotation contribut-
ing to the total root vortex of strength ˆ. The vortex tube will expand in radius as
the flow of the wake inside the tube slows down. Vorticity is confined to the surface
of the tube, the root vortex and to the bound vortex sheet swept by the multiplicity
of blades to form the rotor disc; elsewhere in the wake and everywhere else in the
entire flow field the flow is irrotational.
The nature of the tube’s expansion cannot be determined by means of the
momentum theory and so, as an approximation, the tube is allowed to remain
cylindrical Figure 3.7. The Biot–Savart law is used to determine the induced
velocity at any point in the vicinity of the actuator disc. The cylindrical vortex
model allows the whole flow field to be determined and is accurate within the
limitations of the non-expanding cylindrical wake.

3.4.2 Vortex cylinder theory

The vortex cylinder has surface vorticity which follows a helical path with a helix
angle ö or, as it has been termed previously, the flow angle at the blade tip. The
strength of the vorticity is g ¼ dˆ=dn, where n is a direction in the tube surface
normal to the direction of ˜ˆ, and has a component g Ł ¼ g cos ö t parallel to the
rotor disc. Due to g Ł the axial (parallel to the axis of rotor rotation) induced velocity
at the rotor plane is uniform over the rotor disc and can be determined by means of
the Biot–Savart law as

z
∆Γ ∆Γ
∆Γ

∆Γ
Γ x
y Ω

U ∞ ∆Γ
∆Γ

Figure 3.6 Helical Vortex Wake Shed by Rotor with Three Blades Each with Uniform
Circulation ˜ˆ