Wind Energy Handbook. Michael Barton Graham
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Название: Wind Energy Handbook
Автор: Michael Barton Graham
Издательство: John Wiley & Sons Limited
Жанр: Физика
isbn: 9781119451167
isbn:
Figure 3.7 Simplified helical vortex wake ignoring wake expansion.
3.4.2 Vortex cylinder theory
In the limit of an infinite number of blades and ignoring expansion the tip vortices form a cylinder with surface vorticity that follows a helical path with a helix angle ϕt, which is the same as the flow angle at the outer edge of the disc. The strength of the vorticity is
(3.27)
In the far wake the axial induced velocity uw is also uniform within the cylindrical wake and is
(3.28)
The ratio of the two induced velocities corresponds to that of the simple momentum theory and justifies the assumption of a cylindrical vortex sheet.
3.4.3 Relationship between bound circulation and the induced velocity
The total circulation on all of the multiplicity of blades is Γ, which is shed at a uniform rate into the wake in one revolution. So, from Figure 3.8 in which the cylinder has been slit longitudinally and opened out flat, we must have for the strength of the axial vorticity that
(3.29)
since irrespective of the vortex convection velocities the whole circulation Γ is distributed over the peripheral length 2πR.
Figure 3.8 The geometry of the vorticity in the cylinder surface.
To 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
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