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flight state, any disturbed blade motion will give rise to periodic lift forces and the
generation of a shed vortex wake in addition to the trailing vortices. The work of
Loewy and Jones was based on two-dimensional analysis which one would expect to
overestimate the effects somewhat, since the shed vortex lines in practice extend only
the length of the span. The mathematical model used is as indicated in Fig. 6.17. The
shed wake pattern is imagined to consist of the semi-infinite wake attached to the
blade and a vertical array of doubly infinite wakes representing those of other blades
and of the previous passages of the reference blade. The vertical spacing h between
each sheet is determined from the mean flow through the rotor. If the effect of the
lower sheets is averaged over the blade chord (lifting-line assumption), the lift on the
blade is then
L Vb Vb
n h
L zb
n
=
d
( – 1)
+
d
+
+ +
1
2 =1 –
2 2 2 0
ρ γ ξ 2
ξ
ρ γ ξ ξ
ξ
ρπ
∞ ∞
∞
+ ∝ ∫√ ∫
Σ ˙˙
which is the same as eqn 6.24 except for the addition of the second term which
denotes the contribution of those sheets below the blade. For oscillations which are
integer frequencies of the rotor speed, Miller found that
L
L
C k m,h
J Y
J Y Y J
F iF hk m
q
1 1
1 1 0 0
= ( , ) = 2 i R I
– i
– i + +i +[2i/e e –1)]
+ π ≡ (6.30)
where J0, Y0, J1, Y1 are Bessel functions of the first and second kinds, of orders zero
and unity and argument k; m = ω /Ω and k = ω b/ΩR. Figure 6.18 shows the comparison
of eqn 6.30 with Loewy’s exact lifting-surface results. The agreement is extremely
good for FR but there is poor agreement for the phase shift represented by FI. However
as pointed out by Miller, typical values of h are greater than unity and the error is
fairly small.
h
h
Fig. 6.17 Vortex sheets shed by oscillating aerofoil in hover
Rotor aerodynamics in forward flight 213
–FI
0.5
0
The case of an infinite number of blades leads to a particularly simple result. The
shed vorticity is then uniformly distributed vertically, and Miller found that, if the
frequency is nΩ, the induced velocity at the aerofoil becomes
w
nbc
R z
n z
n t nc R
=
i e
8
e
+
d d
i
2
0 –
i /2
2 2
Γ Ω
π λ ξ
ξ
∞ ξ
∞
+ ∝ ∫ ∫
where λ is the mean inflow ratio and Γn is the amplitude of the bound circulation.
From the blade element theory, if θ = θn einΩt is the blade pitch variation we have
Γ = πΩRc(θ – w/ΩR)
since the quasi-static circulation is Γq = πΩRcθn.
Elimination of w and writing s = bc/πR leads finally to
C = 1/(1 + sπ /4λ)
Thus, in this approximation, C is independent of the frequency. The analysis is
unable to predict the phase shift. Comparison of this method with Loewy’s results
shows very good agreement for values of h below 5, i.e. for typical values of h
occurring in practice.
Miller extended the approximate analysis just discussed to the three-dimensional
vertical flight case. It was assumed that the circulation was constant along the blade,
so that the trailing vortex wake consisted only of a tip vortex and a vortex along the
rotor axis. Applying the above approximations to the integrals 6.4 and 6.6 leads to the
result
w = (b/4πλR)(Γns sin nψ + Γnc cos nψ)
when the bound circulation is given by
Loewy
Eqn 6.30
h
k = 0.1
1.0
0.5
0 h
k = 0.5
1.0
0.5
0 5 10
0.5
0
k = 0.1
5 10
5 h 10 5 h 10
k = 0.5
Fig. 6.18 Comparison between exact and approximate oscillatory parameters
FR FR
–FI
214 Bramwell’s Helicopter Dynamics
Γn = Γns sin nψ + Γnc cos nψ
The lift deficiency function C turns out to be exactly that of the previous analysis,
namely,
C = 1/(1 + sπ/4λ)
It is interesting to note that, although the contribution of the shed wake in this case
is smaller on account of the finite length of the vortex lines, this is exactly compensated
for by the effect of the trailing vortex lines which, in the two-dimensional case, were
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Bramwell’s Helicopter Dynamics(108)
