Tip-Loss Factor

The ideas of a so-called tip-loss factor have been introduced in Chapter 2 and can be used to account for the effects on the rotor thrust and induced power because of the locally high induced velocities produced at the blade tips by the trailed tip vortices. The basic form of the BET permits a finite lift to be produced at the blade tip, which, of course, is physically unrealistic. The Prandtl tip-loss factor В is used to represent this loss of blade lift and can be considered as an “effective” blade radius, Re = BR. In practice, В is found to take a value between 0.95 and,0.98 for most helicopter rotors.

When the tip loss is included in the calculation of rotor thrust using the BET, one approach is to consider the outer portion of the blade, R — Re, to be incapable of carrying lift. This is the approach suggested by Gessow (1948) and Gessow & Myers (1952) and followed by Payne (1959), Johnson (1980), and others. In this case the result for the lift is given by integrating the segment lift over the effective blade span as

fB <7 9 aCi, „

CT = / – Ctr2 dr = / —- (Or" – Xr) dr. (3.33)

Jo 4 Jo 2

For untwisted blades (в — во) and uniform inflow assumptions (A. = ■s/Cj/2), this becomes

which can be compared to the result given previously in Eq. 3.22 with no tip losses (or Eq. 3.34 with В = 1). For a rotor with a twist distribution of the form в — (9tiP/r (as will be shown, this is known as “ideal” twist), then

oCia

4

In either of Eq. 3.34 or Eq. 3.35 because В is between 0.95 and 0.98, we find a 6-10% reduction in rotor thrust resulting from tip-loss effects for a given blade-pitch setting under the stated assumptions.

C*A – • __________ 1 _!_______ лЛ________ i.!„ 1___________________ J _ J________________________ J г______________ i. L _ n_______________ li.1 i-1____________________ J:_____________________ ^ J

dUlcuy spCctRing, liic np-ioss cquanun ueauccu ниш uic riauuu нісшу аь iusl uisuusscu in Chapter 2 should be applied to the calculation of an increased inflow (for the same total thrust). Therefore, to assume that the outboard part of the blade, R — Re, is ineffective in carrying lift is not the correct interpretation of Prantdl’s theory. This fact has also been pointed out by Bramwell (1976). The correct interpretation is to consider that for the same thrust the induced inflow will be increased to a value

that is, Vh (or A./,) is increased by a factor В 1 compared to the case with no assumed tip losses. For untwisted blades and uniform inflow with tip losses alone, the thrust becomes

which can be compared to Eq. 3.34 with the alternative interpretation of tip loss. Eor ideal twist and uniform inflow, the thrust now becomes

(3.38)

compared to the result in Eq. 3.35, which will overpredict the effects of tip loss.

Because of tip-loss effects, a real rotor will always have a higher overall average induced velocity compared to that given by momentum theory and so the induced power will also be increased relative to the simple. momentum result. Tip loss constitutes an additional source

of nonuniform inflow, and would normally be factored into the value of к. Using the BET, the induced power can be written as

‘=! гЛ і

kdCr = I – aXCy2 dr,
Jo 2

where A. must be calculated for each element. Using untwisted rectangular blades and uniform inflow assumptions, then with a tip loss the total power can be approximated by

where the induced power factor к includes the effects of both tip loss and nonuniform inflow over the remainder of the blade. As suggested in Chapter 2 by the simple momentum analysis, к approximately equals 1.25 for the rotor used in this case. In general, we can write that

1

к = — +kx + k0,

where Kx accounts for the effects of nonuniform inflow and k0 accounts for other miscel­laneous induced losses. Figure 3.3 shows the variation in power required versus collective pitch for four rotors with increasing solidity. Again, the data are taken from Knight and Hefner (1937). The calculations have used a tip-loss factor, В — 0.97, к = 1.25, and CdQ = 0.011. The result in Eq. 3.40, with the stated assumptions, correlates well with the measured data but now shows a slight underprediction at the higher collective pitch angles. For the most part, this is because of the higher drag coefficients associated with boundary layer thickening and the onset of blade stall, the effects of which tend to increase the profile power above that obtained by assuming a constant profile drag coefficient.