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are shown in Table 2.2.
For the small inflow angles, i.e. the small vortex sheet spacings, which normally
occur in practice, it might be expected that Prandtl’s much simpler analysis would be
adequate. The same calculations were made using Prandtl’s formula and the results
are compared with those of the Goldstein–Lock analysis, Table 2.2, from which it can
be seen that the differences are indeed very small.
The effect on the blade loading distribution x2CL is shown in Fig. 2.24, and the
loss of thrust, or ‘tip loss’, amounts to about 412
per cent of the total thrust. The loss
would be larger for two blades and less for four or more blades. The figure also
shows clearly that the difference between the Prandtl and the Goldstein–Lock analyses
is very small.
2.10.2 The tip loss factor
The above calculations show how we can estimate the loss of thrust near the blade
b = ∞
Goldstein–Lock
Prandtl
0.4
0.3
0.2
0.1
0
0.2 0.4 0.6 0.8 1.0
r/R
x2CL
Fig. 2.24 Calculations of tip effect
Table 2.2 Variation of factor k and inflow angle φ near blade tip
x 0.9 0.95 0.96 0.97
Goldstein k 0.97 0.85 0.79 0.68
φ 0.0536 0.0532 0.0540 0.0567
Prandtl k 0.97 0.85 0.81 0.73
φ 0.0536 0.0532 0.0537 0.0551
66 Bramwell’s Helicopter Dynamics
tips. It would clearly be desirable to have a means of calculating the tip loss which
is simpler than the strip analysis described above. For this purpose Prandtl devised a
tip loss factor which gave the ratio of the mean induced velocity over the rotor to an
effective velocity at the blades themselves.
If the number of blades were infinite, the vortex sheets would be indefinitely close
together and all the air between them would be carried down with the sheets. The air
outside the sheets would remain at rest. When the number of blades is finite, however,
the spaces between the sheets allows some of the air to escape upwards round the
edges so that, for a given velocity of the sheets, the average downwash velocity is
somewhat less than at the sheets themselves, and this latter velocity corresponds to
the induced velocity at the blades. Expressed in another way, if a finite-bladed rotor
carries a given thrust, the mean induced velocity at the blades is higher than the value
of the induced velocity calculated on the basis of infinite blades.
Prandtl regarded the defect of mean velocity as equivalent to a shortening of the
lengths of the sheets, i.e. of a reduction of the radius of the blades from R to an
effective value Reff. By finding the ratio between the mean velocity between the
sheets and the velocity of the sheets themselves, Prandtl showed17 that
R – Reff = (1.386/b)xR sin φ
or Reff/R ≈ 1 – (1.386/b) sin φ (2.68)
since, near the tip, x can be taken as unity.
For hovering flight it is usual to take
sin φ = λ
i = √(CT /2) (2.69)
so that to a good approximation eqn 2.68 can be written as
Reff /R = B = 1 – √CT /b (2.70)
This expression, and others similar to it, has often been used to determine the ‘tip
loss factor’ B. For the three-bladed rotor considered earlier, B is about 0.980.
Now the idea of Prandtl’s tip loss factor is to represent the increased induced
velocity at the blades by calculating the mean induced velocity from the simple
momentum theory but based on a reduced radius Reff = BR. The tip loss factor is
intended to apply only to the calculation of the induced velocity, but in most textbooks
and in many technical papers it has been interpreted as meaning that the outer portion
of the blade R – Reff is incapable of carrying lift. This means that the thrust integral
would be written
T T r r T x x
BR B
= (d /d ) d = (d /d ) d
0 0 ∫ ∫
Clearly this interpretation is quite different from the one intended by Prandtl. If the
upper limit of the integral of eqn 2.26 is B instead of unity, we have, for constant
induced velocity and hovering flight,
T b ac R B B = ( /3 – /2) 12
2 3 3
0
2
ρ Ω θ λ i
Rotor aerodynamics in axial flight 67
giving
tc = (aB2/4)(2Bθ0 /3 – λ
i) (2.71)
instead of
tc = (a/4) (2θ0 /3 – λ
i) (2.28)
Using the values for θ0 and λ
i of section 2.5, we find that eqn 2.71 leads to a
reduction of thrust of about 7.7 per cent compared with the thrust given by eqn 2.28.
Thus the use of B as an upper limit to the thrust integral considerably overestimates
the ‘tip loss’. In addition, the expressions for thrust and other rotor quantities become
　
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