Wind Energy Handbook. Michael Barton Graham

Wind Energy Handbook - Michael Barton Graham


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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:

upper U Subscript infinity Baseline left-parenthesis italic 1 minus italic 2 a right-parenthesis left-parenthesis inside right-parenthesis and upper U Subscript infinity Baseline left-parenthesis outside right-parenthesis

      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 = 2aΩR and just outside = 0. Therefore, the helical wake vortices (or vortex tube in the limit) rotate at aΩR. The result is that the pitch of the helical vortex wake (see Figure 3.8) is

      Using this value we obtain

      (3.31)g Subscript theta Baseline equals lamda upper Gamma left-parenthesis italic 1 plus a Superscript prime Baseline right-parenthesis slash italic 2 pi upper R left-parenthesis italic 1 minus a right-parenthesis

      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

StartFraction normal upper Gamma Over 4 pi r EndFraction equals a prime normal upper Omega r

      so

      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

upper L equals rho left-parenthesis upper W times normal upper Gamma right-parenthesis

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

      (3.35)delta upper Q equals rho upper W times normal upper Gamma r sine phi Subscript t Baseline delta r equals rho normal upper Gamma italic r upper U Subscript infinity Baseline left-parenthesis 1 minus a right-parenthesis delta r

      Equating the two expressions for δQ gives

a prime equals StartFraction normal upper Gamma Over 4 pi r squared normal upper Omega EndFraction a prime equals StartFraction upper U Subscript infinity Baseline Superscript 2 Baseline a left-parenthesis 1 minus a right-parenthesis Over left-parenthesis upper Omega r right-parenthesis squared EndFraction equals StartFraction a left-parenthesis 1 minus a right-parenthesis Over lamda Subscript r Baseline Superscript 2 Baseline EndFraction

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

      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)StartFraction italic d upper Q Over italic d r EndFraction delta r equals rho upper W normal upper Gamma r sine phi Subscript t Baseline delta r equals StartFraction rho Baseline 4 pi italic r upper U Subscript infinity <hr><noindex><a href=Скачать книгу