Читать книгу Wind Energy Handbook - Michael Barton Graham - Страница 62

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

(3.30)

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

(3.32)

It is similarly necessary to include the rotation induction factor to calculate the angle of slant φ_t of the vortices:

Thus Tan φ_t = (1 − a)/(1 + a′)λ