Michael Graham - Wind Energy Handbook

Just as a vortex trails from the tip of an aircraft wing so does a vortex trail from the tip of a wind turbine blade. Because the blade tip follows a circular path, it leaves a trailing vortex as a helical structure that convects downstream with the wake velocity. For example, on a two blade rotor, unlike an aircraft wing, the bound circulations on the two blades shown in Figure 3.27are opposite in sign and so combine in the idealised case of the blade root being at the rotational axis to shed a straight line vortex along the axis with strength equal to the blade circulation times the number of blades. If as is usual in practice the blade root is somewhat outboard of the axis, the two blade root vortices form independent helices similar to the blade tips but of small radius, close together, and the combined straight line axis vortex is not a bad approximation of their effect.

For a single vortex to be shed from the blade at its tip, only the circulation strength along the blade span must be uniform right out to the tip with an abrupt drop to zero at the tip. As has been shown, such a uniform circulation provides optimum power coefficient. However, the uniform circulation requirement assumes that the axial flow induction factor is uniform across the disc. With an infinite number of blades, the tip vortices form a continuous cylindrical sheet of vorticity directed at a constant angle around the surface. Such a sheet is consistent with a uniform value of the axial induction factor over the disc. But, as has been argued above, with a finite number of blades rather than a uniform disc, the flow factor is not uniform. Sustaining uniform circulation until very close to the two ends (tip and root) of a blade results in a very large gradient of the blade circulation at the tips, which in turn induces large radial variations in the induced velocity factors a and a ′in those regions, with both tending to infinity in the limit of constant circulation up to the tip and root.

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 3.28for a three blade rotor operating at a tip speed ratio of 6. The calculation for Figure 3.28assumes 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 ) to the value a b(r) at the blade quarter chord is shown in Figure 3.29, being unity for most of the blade span, and only near the tip does it begin to fall to zero. This ratio is called the tip‐loss factor .

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

(3.78)

From the Kutta–Joukowski theorem, the circulation Γ on the blade, which is uniform, provides a torque per unit span of

where the angle ϕ ris 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

(3.79)

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 in terms of the azimuthally averaged axial flow induction factor , which = 1/3 for optimum operation. However, the final induction term (1 − a b) relates to the flow angle at the blade and must therefore be in terms of a b, the axial induction factor at the blade, with a b= / f , and therefore a b≈ except near the tips. The notation defined as here will be used in this section where required to distinguish them.

The high value of the axial flow induction factor a bat 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 factors a band that have been calculated by the uncorrected BEM method. Because the blade circulation must similarly fall to zero at the root of the blade, a similar ‘tip‐loss’ factor is applied there in the same way.