Wind Energy Handbook. Michael Barton Graham
evaluate the strength of the azimuthal vorticity, we require the axial spacing over which it is distributed, i.e. the axial spacing of any tip vortex between one vortex and the next. Vortices and sheets of vorticity must be convected at the velocity of the local flow field if they are to be force‐free. This velocity can be evaluated as the velocity of the whole flow field at the vortex or vorticity element location less its own local (singular) contribution. In the case of a continuous sheet, it is the average of the velocities on the two sides of the sheet. For axial convection in the ‘far’ wake the two axial velocities are:
so that the axial convection velocity is U∞(1 − a). However, the vortex wake also rotates relative to stationary axes at a rate similarly calculated as halfway between the rotation rate of the fluid just inside the downstream wake = 2a′ΩR and just outside = 0. Therefore, the helical wake vortices (or vortex tube in the limit) rotate at a′ΩR. The result is that the pitch of the helical vortex wake (see Figure 3.8) is
Using this value we obtain
(3.31)
where λ = ΩR/U∞ the tip speed ratio and the rotation period = 2π/Ω.
So, the total circulation is related to the induced velocity factors
It is similarly necessary to include the rotation induction factor to calculate the angle of slant φt of the vortices:
Thus Tan φt = (1 − a)/(1 + a′)λ
3.4.4 Root vortex
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
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
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.
3.4.5 Torque and power
The torque on an annulus of radius r and radial width δr (ignoring a′ as actuator disc theory ignores rotation) is
(3.37)