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very complicated when the upper limit B is applied to the forward flight cases. A
much simpler method of accounting for the tip loss is to apply the reduced radius to
the calculation of the induced velocity in the manner conceived by Prandtl. Thus the
induced velocity is simply increased by 1/B2 and the expression for the thrust coefficient
becomes
tc = (a/4)(2θ0/3 – λ
i /B2) (2.72)
This expression gives a thrust reduction of 4 per cent, which is quite close to the
value of 412
per cent from the strip theory.
It is suggested that eqn 2.72 should be used as the simplest and most accurate
method of allowing for tip loss. In forward flight the value of B given by eqn 2.70 no
longer applies, as the vortex wake is skewed relative to the rotor disc which makes
it necessary to adopt a different analysis.
2.10.3 Theodorsen’s theory
The analyses of Prandtl and Goldstein–Lock described above can be justly criticised
in that the wake contraction has not been taken into account. It was assumed that the
induced velocity in the wake was twice that at the rotor disc but that the wake was
a uniform helix having the same diameter as the rotor. Because of this restriction, the
analysis was assumed to apply to ‘light loadings’. Theodorsen5 was the first to make
an attempt to take the wake contraction into account. On the assumption that the ideal
helical wake was being created, Theodorsen used Goldstein’s circulation results to
establish relationships between the power (or thrust) of the propeller and the far wake
parameters. By considering the efficiency of a blade element, the ratio of the induced
velocity in the wake to that at the propeller could be found and the slipstream
contraction could also be calculated. Thus, with the helix angle and diameter of the
final wake known, the corresponding values of the induced velocity and helix angle
at the blade could be found. From Goldstein’s results the ideal circulation and, hence,
the required blade geometry could then be calculated. Actually, Theodorsen’s method
works in the reverse sense to that of Goldstein–Lock, as he begins with the final wake
and calculates the (ideal) propeller which generates it.
Later, in 1969, Theodorsen18 extended his method to the analysis of static propellers
and hovering rotors. The full theory is given in his paper and the book previously
referred to, but the results for hovering flight are interesting and are given in the
68 Bramwell’s Helicopter Dynamics
figures below. In these figures the common parameter is Λ0, the tangent of the wake
helix angle, defined by
Λ0 = w/ΩR0
where w is the velocity of the far wake and R0 is its radius. With CT (= stc) assumed
known for the rotor, Λ0 can be read off from Fig 2.25. Then, from Figs 2.26 and 2.27
we can find the contraction ratio η and ν, the ratio of the final wake velocity to that
at the rotor. Since for most helicopters CT is not likely to exceed 0.01, it can be seen
that η will vary only between about 0.816 and 0.835 and ν from about 1.5 to 1.65.
The theory depends on the assumption that the ideal wake exists; as we noted in
section 2.1, however, these values may be different for the non-uniform wakes likely
to occur in practice.
In his 1969 paper, Theodorsen does not explain how his results for hovering flight
were to be used, but it is reasonable to assume that they would be applied in the same
way as for the conventional propeller described in his book. Theodorsen’s consideration
of the wake contraction is not complete, however. The implication of his method is
that, although the local wake helix angle and induced velocity are correctly estimated,
the wake at the propeller is still assumed to be cylindrical since the theory makes use
0.020
0.015
0.010
0.005
0 0.1 0.2 0.3
6
4 b = 2
CT
Λ0
Fig. 2.25 Thrust coefficient as a function of wake helix angle (i.e. velocity)
0.9
0.85
0.80
0 0.1 0.2 0.3
Λ0
b = 2
6 ∞
Fig. 2.26 Contraction ratio as a function of wake velocity
η
4
Rotor aerodynamics in axial flight 69
2.0
1.8
1.6
1.4
ν
0 0.1 0.2 0.3
Λ0
b = 2
10
Fig. 2.27 Induced velocity ratio as a function of wake velocity
of Goldstein’s results, Fig. 2.28. This is probably quite justified for the conventional
propeller, since not only does the contraction amount to only a few per cent at most
but also the high axial velocity means that the contraction is complete only at a
considerable distance from the disc. For the hovering rotor, however, it is known that
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