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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 Скачать книгу