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trailing vortices shed by the individual rotor blades cut by the plane of the smoke
filaments.
A comprehensive and more easily controlled series of tests was conducted in a
wind tunnel by Heyson and Katzoff6. The method used was to place a grid of wool
tufts in a number of positions in the vicinity of the rotor and to record the tuft
deflections. An analysis of the deflections gave the induced-velocity field and it was
found that the theoretical distribution of Mangler and Squire was largely confirmed.
Heyson and Katzoff also made numerical calculations of the induced velocity field
by using the same basic uniform loading model as Coleman et al. but superimposing
them linearly to obtain symmetrical loadings having arbitrary radial distributions.
Examples of the induced velocity distributions along the longitudinal axis found by
Rotor aerodynamics and dynamics in forward flight 85
20°
15°
10°
5°
0
Non-dimensional longitudinal position x
Angle between local flow
direction and tip-path plane
Forward Aft
–0.8 –0.6 –0.4 –0.2 0 0.2 0.4 0.6 0.8 1.0
Note: Points were measured at lateral distances
between 0.25R and 0.4R on advancing
side of rotor
20°
15°
10°
5°
0
–1.0 –0.8 –0.6 –0.4 –0.2 0 0.2 0.4 0.6 0.8 1.0
Forward Aft
15°
10°
5°
0
–1.0 –0.8 –0.6 –0.4 –0.2 0 0.2 0.4 0.6 0.8 1.0
Forward Aft
x
x
μ = 0.188
μ = 0.167
μ = 0.138
Fig. 3.9 Longitudinal induced velocity distribution obtained from smoke photographs
Heyson and Katzoff and compared with that of Mangler and Squire are shown in
Fig. 3.10.
Heyson and Katzoff also investigated the velocity field behind the rotor and
found it to be remarkably similar to that of a circular wing, showing two distinct
trailing vortices. An example of the tuft pattern at a distance of about one diameter
behind the rotor, as seen by an observer looking forward through the grid, is shown
in Fig. 3.11.
Katzoff has obtained some useful symmetry relationships between the induced
velocity components for a uniform load. By superimposing a skew-symmetric wake
on the original one, Fig. 3.12, a two-dimensional elliptic wake is created by means of
which the following relationships can be deduced.
(i) If P and Q are two points within the rotor disc and symmetrically located on
either side of the lateral axis, the sum of the induced velocity components wP and
wQ, normal to the rotor plane, is equal to the normal component of the induced
velocity within the wake. Since this is constant, it follows that wP + wQ is
86 Bramwell’s Helicopter Dynamics
–1 –0.5 0 0.5 1 1.5 2 2.5 3
3
2
1
χ = 0.75°
μ = 0.095
–1
–1 –0.5 0 1 1.5 2 2.5 3
3
2
1
–1
χ = 82.3°
μ = 0.14
χ = 83.9°
μ = 0.232
–1 –0.5 0 1 1.5 2 2.5 3
3
2
1
–1
vi /vi0
Triangular loading
Measured
Fig. 3.10 Longitudinal induced velocity distribution obtained from wind tunnel tests for different values of μ
Uniform loading
Mangler loading
Fig. 3.11 Vortex pattern behind rotor
0.3
0.2
0.1
0
–0.1
–0.2
–0.3
–0.4
–0.5
–0.6
–0.7
–0.8
–0.9
1.4 1.2 1.0 0.8 0.6 0.4 0.2 0 –0.2 –0.4 –0.6 –0.8 –1.0 –1.2 –1.4
Advancing side Retreating side
Non-dimensional lateral distance, y
Distance behind hub, x = 1.07
Vertical distance below hub, z
x
Rotor aerodynamics and dynamics in forward flight 87
Fig. 3.12 Symmetry relations for induced velocity in forward flight
+ =
constant everywhere over the disc and, therefore, that the induced velocity
distribution at the rotor is skew-symmetric with respect to the lateral axis.
(ii) If P and Q are symmetrically located about the lateral axis but lie outside the
disc, we have
wP + wQ = v′ sin χ
where v′ is the longitudinal component of velocity in the ellipse plane at the
point corresponding to either P or Q, Fig. 3.13.
The components of velocity u′ and v′ about the elliptic wake arise from the selfinduced
motion of the wake itself through the surrounding air. If the velocity of the
wake normal to its axis is U, we have seen that Coleman et al. have found U to be
given by
U = 2vi0 tan (χ/2)
　
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