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The function for the tip‐loss factor f(r) is shown in Figure 3.29 for a blade with uniform circulation operating at a tip speed ratio of 6 and is not readily obtained by analytical means for any desired tip speed ratio. Sidney Goldstein (1929) did analyse the tip‐loss problem for application to propellers and achieved a solution in terms of Bessel functions, but neither that nor the vortex method with the Biot–Savart solution used above is suitable for inclusion in the BEM theory. Fortunately, in 1919, Ludwig Prandtl, reported by Betz (1919), had already developed an ingenious approximate solution that does yield a relatively simple analytical formula for the tip‐loss function.

Prandtl's approximation was inspired by considering that the vortex sheets could be replaced by material sheets, which, provided they move with the velocity dictated by the wake, would have no effect upon the wake flow. The theory applies only to the developed wake. To simplify his analysis Prandtl replaced the helicoidal sheets with a succession of discs, moving with the uniform, central wake velocity U_∞(1 − a) and separated by the same distance as the normal distance between the vortex sheets. Conceptually, the discs, travelling axially with velocity U_∞(1 − a), would encounter the unattenuated free‐stream velocity U_∞ at their outer edges. The fast flowing free‐stream air would tend to weave in and out between successive discs. The wider apart successive discs the deeper, radially, the free‐stream air would penetrate. Taking any line parallel to the rotor axis at a radius r, somewhat smaller than the wake radius R_w (∼ rotor radius R), the average axial velocity along that line would be greater than U_∞(1 − a) and less than U_∞. Let the average velocity be U_∞(1 − af(r)), where f(r) is the tip‐loss function, has a value less than unity and falls to zero at the wake boundary. At a distance from the wake edge the free stream fails to penetrate, and there is little or no difference between the wake‐induced velocity and the velocity of the discs, i.e. f(r) = 1.

A particle path, as shown in Figure 3.32, may be interpreted as an average particle passing through the rotor disc at a given radius in the actual situation: the azimuthal variations of particle axial velocities at various radii are shown in Figure 3.28, and a ‘Prandtl particle’ would have a velocity equal to the azimuthal average of each. Figure 3.32 depicts the wake model.

The mathematical detail of Prandtl's analysis is given in Glauert (1935a), and because it is based on a somewhat strangely simplified model of the wake will not be repeated here. It has, however, remained the most commonly used tip‐loss correction because it is reasonably accurate and, unlike Goldstein's theory, the result can be expressed in closed solution form. The Prandtl tip‐loss factor is given by

(3.80)

R_w − r is a distance measured from the wake edge. Distance d between the discs should be that of the distance travelled by the flow between successive vortex sheets. Glauert (1935a), takes d as being the normal distance between successive helicoidal vortex sheets.

The helix angle of the vortex sheets ϕ_s is the flow angle assumed to be the same as ϕ_t, the helix angle at the blade tip, and so with B sheets intertwining from B blades and assuming that the discs move with the mean axial velocity in the wake, U_∞(1 – ):

(3.81)

Figure 3.32 Prandtl's wake‐disc model to account for tip‐losses.

Prandtl's model has no wake rotation, but whether the discs are considered to spin is irrelevant to the flow field, as it is inviscid, thus a^′ is zero and W_s is the resultant velocity (not including the radial velocity) at the edge of a disc. Glauert (1935a) argues that , which is more convenient to use,

so

and

(3.82)

Although the physical basis of this model is not correct, it does quite effectively represent a convenient approximation to the attenuation towards the tips of the real velocities induced by the helicoidal vortex sheets.

The Prandtl tip‐loss factor for a three blade rotor operating at a tip speed ratio of 6 is compared with the tip‐loss factor of the helical vortex wake in Figure 3.33.

It should also be pointed out that the vortex theory of Figure 3.28 also predicts that the tip‐loss factor should be applied to the tangential flow induction factor.

It is now useful to know what the variation of circulation along the blade is. For the previous analysis, which disregarded tip‐losses, the blade circulation was uniform [Eq. (3.69))].

Following the same procedure from which Eq. (3.68) was developed:

Recall that a_b(r) is the flow factor local to the blade at radius r and (r) is the average value of the flow factor at radius r.

Figure 3.33 Comparison of Prandtl tip‐loss factor with that predicted by a vortex theory for a three blade turbine optimised for a tip speed ratio of 6.

Figure 3.34 Spanwise variation of blade circulation for a three blade turbine optimised for a tip speed ratio of 6.

Therefore,

(3.83)

Γ(r) is the total circulation for all blades and is shown in Figure 3.34, and, as can be seen, it is almost uniform except near to the tip. The dashed vertical line shows the effective blade length (radius) R_ef = 0.975 if the circulation is assumed to be uniform at the level that pertains at the blade sections away from the tip.

The Prandtl tip‐loss factor that is widely used in industry codes appears to offer an acceptable, simple solution to a complex problem; not only does it account for the effects of discrete blades, it also allows the induction factors to fall to zero at the edge of the rotor disc.

A more recently derived tip‐loss factor that has been calibrated against experimental data and appears to give improved performance was given by Shen et al. (2005). The spanwise distribution of axial and tangential forces is multiplied by the factor

where g₁ = 0.1 + exp.{−c₁(Bλ‐c₂)} with c₁ = 0.125 and c₂ = 21.0.

This formulation is similar to Glauert's (1935a) original simplification of the Prandtl tip‐loss correction but introduces a variable factor g₁ rather than unity. Wimshurst and Willden (2018) suggest that a better fit is given in the above by using different constants for the axial force correction (c₁ ∼ 0.122 and c₂ ∼ 21.5) and for the tangential force correction to be similarly defined but with (c₁ ∼ 0.1 and c₂ ∼ 13.0).