Читать книгу Wind Energy Handbook - Michael Barton Graham - Страница 63

Just as a vortex is shed from each blade tip, a vortex is also shed from each blade root. If it is assumed that the blades extend to the axis of rotation, obviously not a practical option, then the root vortices will each be a line vortex running axially downstream from the centre of the disc. The direction of rotation of all of the root vortices will be the same, forming a core, or root, vortex of total strength Γ. The root vortex is primarily responsible for inducing the tangential velocity in the wake flow and in particular the tangential velocity on the rotor disc.

On the rotor disc surface the tangential velocity induced by the root vortex, given by the Biot–Savart law, is

so

(3.33)

This relationship can also be derived from the momentum theory – the rate of change of angular momentum of the air that passes through an annulus of the disc of radius r and radial width δr is equal to the torque increment imposed upon the annulus:

(3.34)

The torque per unit span acting on all the blades is given by the Kutta–Joukowski theorem. The lift per unit radial width L is

where (W × Γ) is a vector product, and W is the relative velocity of the air flow past the blade:

(3.35)

Equating the two expressions for δQ gives

If a^′ in Eq. (3.32) is now treated as being negligible with respect to 1 (which it is in normal circumstances) then:

At the outer edge of the disc the tangential induced velocity is

(3.36)

Equation (3.36) is exactly the same as Eq. (3.23) of Section 3.3.3.

If a′ is retained in Eq. (3.32), there is a small inconsistency here between vortex theory and the one‐dimensional actuator disc theory, which ignores rotation effects.