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
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As in Figure 3.27, close to a blade tip a single concentrated tip vortex would on its own cause very high values of the flow factor a with an infinite value at the tip such that, locally, the net flow past the blade is in the upstream direction. This effect is similar to what occurs for the simple ‘horseshoe vortex’ model for a fixed wing aircraft showing that this model is not applicable at a blade or wing tip where a more detailed induced flow analysis is required. The azimuthal average of the axial induction a is uniform radially. Higher values of a tend to be induced close to the blades towards root and tip, becoming higher the closer to the tips. Therefore, low values relative to the average must occur in the regions between the blades. The azimuthal variation of a for a number of radial positions is shown in Figure 3.28 for a three blade rotor operating at a tip speed ratio of 6. The calculation for Figure 3.28 assumes a discrete vortex for each blade with a constant pitch and constant radius helix and is calculated from the effect of the shed wake vortices only.
At a particular radial position the ratio of the azimuthal average of a (which from here on will be written as
Figure 3.28 Azimuthal variation of a for various radial positions for a three blade rotor with uniform blade circulation operating at a tip speed ratio of 6. The blades are at 120°, 240°, and 360°.
Figure 3.29 Spanwise variation of the tip‐loss factor for a blade with uniform circulation.
From Eq. (3.20) and in the absence of tip‐loss and drag the contribution of each blade element to the overall power coefficient is
(3.77)
Substituting for a′ from Eq. (3.25) gives
From the Kutta–Joukowski theorem, the circulation Γ on the blade, which is uniform, provides a torque per unit span of
where the angle ϕr is determined by the flow velocity local to the blade.
If the strength of the total circulation for all three blades is still given by Eq. (3.69), in the presence of tip‐loss, the increment of power coefficient from a blade element is
in agreement with Eq. (3.78), except that the factor a(1 − a), which relates Γ to the angular momentum loss in the wake, must be expressed as
The high value of the axial flow induction factor ab at the tip, due to the proximity of the tip vortex, acts to reduce the angle of attack in the tip region and hence the circulation so that the circulation strength Γ(r) cannot be constant right out to the tip but must fall smoothly through the tip region to zero at the tip. Thus, the loading falls smoothly to zero at the tip, as it must for the same reason as on a fixed wing, and this is a manifestation of the effect of tip‐loss on loading. The result of the continuous fall‐off of circulation towards the tip means that the vortex shedding from the tip region that is equal to the radial gradient of the bound circulation is not shed as a single concentrated helical line vortex but as a distributed ribbon of vorticity that then follows a helical path. The effect of the distributed vortex shedding from the tip region is to remove the infinite induction velocity at the tip, and, through the closed loop between shed vorticity, induction velocity and circulation, converge to a finite induction velocity together with a smooth reduction in loading to zero at the tip. The effect on the loading is incorporated into the BEM method, which treats all sections as independent ‘2‐D’ flows, by multiplying a suitably calculated tip‐loss factor f(r) by the axial and rotational induction СКАЧАТЬ