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was indeed fortunate that a reasonable loading shape could be obtained by using only
the first two pressure functions as, even with these, the calculations of the induced
velocity were very lengthy. Examples of the results of the calculations are shown in
Figs 3.6 and 3.7. It can be seen that the longitudinal variation of induced velocity is
far from being linear and that there is actually an upwash over a small portion of the
rear half of the rotor as well as at the leading edge. It can also be seen that the
downwash is very strong near the outer edges of the rear half of the rotor, reaching
values three or four times greater than the mean. The downwash distribution is
symmetric about the longitudinal axis.
Mangler and Squire also succeeded in expressing their results in the form of a
Fourier series, i.e. as
vi vi0
12
= 4 [c0 – =1 c( , D) cos n ]
n n Σ ∞ η α ψ (3.13)
where
c0 = 15η(1 – η2)/8 (3.14)
and for n ≥ 2, and even,
r/R
Blade
loading
0 1
Fig. 3.5 Rotor loading assumed by Mangler
Rotor aerodynamics and dynamics in forward flight 83
1.2
1.6
2.0 2.4 2.8
Relative
flow velocity
–1.44 0.4 0 –0.4 2.24
vi
/vi0
3.2
Fig. 3.6 Induced velocity distribution according to Mangler’s theory, αD = 15°
c
n
n
n
n n n
= (– 1) n 15
8
+
– 1
9 + – 6
– 9
+
3
– 9
( – 2)/2
2
2 2
2 2
η ⋅η η
 

 
× 




1 –
1 +
1 – sin
1 + sin
/2
D
D
η /2
η
α
α
n n
(3.15)
and c1
2 2 1/2 D
D
1/2
= – 15
256
(5 – 9 )(1 – )
1 – sin
1 + sin
π η η α
α


c3
2 3/2 D
D
3/2
= 45
256
(1 – )
1 – sin
1 + sin
π η α
α


where η2 = 1 – x2.
For odd values of n ≥ 5, cn = 0. The use of the mean induced velocity in eqn 3.13 (as
given by eqn 3.3) should not be taken to imply that eqn 3.13 may be used at the hover
or for slow speeds, since the Mangler and Squire theory assumed that vi << V.
Downwash
4
3
2
1
0
–1
–2
αD = 90° 30°
45°
15°
0°
90°
30°
45°
15°
0°
Trailing
edge
Leading
edge
Upwash
vi / vi0
Fig. 3.7 Longitudinal induced velocity distribution according to Mangler’s theory
–0.5 1
0 0.8 0.8
–1 x
84 Bramwell’s Helicopter Dynamics
3.4 Flight and wind tunnel tests
A series of pioneering attempts to measure the induced velocity in forward flight was
made in 1948 by the UK Royal Aircraft Establishment5 (now part of DERA, the
Defence Evaluation and Research Agency). Smoke generators were suspended from
a slow flying fixed-wing aircraft in front of the helicopter, a Sikorsky Hoverfly, and
Fig. 3.8 Flow pattern as revealed by smoke filaments in forward flight. Speed 53 mph. Plane of smoke
filaments 0.4R to starboard
were arranged to emit a band of smoke filaments which lay in a vertical plane. Figure
3.8 shows a typical example photograph taken by a third aircraft flying in formation
with the other two. Tests were conducted at a number of different speeds, and with
the plane of the filaments intersecting the rotor at various radii between 0.25R and
0.4R on the advancing side.
The smoke filaments gave a broad picture of the flow through the rotor but were
not really distinct enough to show the velocity distribution in detail. Figure 3.9
shows the induced velocity distribution along the longitudinal axis as deduced from
the deflection of the smoke filaments. The results would appear to indicate that the
distribution is roughly linear, but it is also clear from the photographs that the
filaments become too diffused to show the details predicted by Mangler and Squire’s
theory. The photograph clearly shows, however, an interesting phenomenon:
the series of ‘whorls’ appearing behind the tailrotor are the cross-sections of the
　
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