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omitted.
The values of the lift deficiency function indicate that the effective lift slope of the
blade is much reduced and, under some conditions, could approach zero. This implies
correspondingly reduced damping of the blade flapping motion.
Because of the good agreement between Miller’s approximate two-dimensional
analysis and the exact analysis of Loewy, Miller argued that for calculations of the
forward flight case it is reasonable to consider both the shed and trailing wakes in
two parts: a ‘near’ shed wake which corresponds to the first quadrant of the wake
and a ‘far’ wake consisting of the remainder. Furthermore, the ‘near’ wake could
be regarded as straight and extended back to infinity, as was assumed by Willmer.
It was supposed that the classical two-dimensional unsteady theory of Theodorsen
could be applied to account for the shed vorticity in this section of the wake.
For the treatment of the shed vorticity in the far wake, it was assumed that the
chordwise variation of velocity at the blade could be neglected and that only a mean
velocity need be considered, as in Miller’s two-dimensional analysis. Calculations
showed that, above μ = 0.2, the far wake has little effect and the values of FR
and FI approach those of the classical two-dimensional theory. The results
suggested that a good approximation could be obtained by simply ‘fairing’ the
results found for the hovering case to those of the two-dimensional aerofoil theory,
Fig. 6.19.
1.0
0.5
0
0.1 0.2 0.3 μ
Complete shed and trailing wake
Shed wake only
Near shed wake only
– FI
FR
Fig. 6.19 Variation of oscillatory parameters in forward flight
Third harmonic
Rotor aerodynamics in forward flight 215
Calculations made by assuming infinite straight vortex filaments for the spiral
trailing vortices in the integral 6.4 showed little loss of accuracy but a great reduction
of computer time.
Calculations also showed that good accuracy was retained even when the trailing
wake was assumed to consist of only two discrete vortices, one springing from the tip
and another a little inboard of the mid-span. Such a vortex system has indeed been
found by Tararine10 from smoke tests, Fig. 6.20.
Comparison of Miller’s calculations with experimental air loads is shown in Figs
6.21 and 6.22.
The peak loading at an azimuth angle of about 90° is due to the tip vortex of the
preceding blade. This peak travels down the blade as it advances from 90° to 180°,
and can be detected in Figs 6.21 and 6.22. It might be thought that the peak could be
reduced by increasing the number of blades, but, although the vortex strength is
reduced, so are the blade spacings, with the result that the vortices pass closer to the
blade in question. According to Miller, the peak loading is influenced very little by
the number of blades.
A further conclusion of Miller’s work is that almost all the harmonic content of the
air loads is contained in harmonics above the second, and they are derived from the
effects of the far wake. These harmonics are practically independent of blade flapping
motion.
As was stated earlier, Miller’s work was based on the assumption of a rigid wake.
Fig. 6.20 Representation of shed and trailing vortices (after Tararine)
5
4
3
2
1
0 90° 180° 270° 360°
Theory
Experiment
x = 0.95
μ = 0.2
ψ
Fig. 6.21 Blade loading variation with azimuth angle
kN/m
216 Bramwell’s Helicopter Dynamics
x = 0.85
μ = 0.2
Theory
Experiment
5
4
3
2
1
0 90° 180° 270° 360°
ψ
Fig. 6.22 Blade loading variation with azimuth angle
6.2.3 Free wake model development
In the 1960s, Cornell Aeronautical Laboratory carried out a programme of work
aimed at improving the understanding of, and the ability to model, the wake of a
helicopter in steady state flight. As part of this, Piziali11 proposed a wake model that
embodied the shed and trailing vorticity in the form of a mesh of straight line vortex
filaments, as shown in Fig. 6.23. This occupied the near wake adjacent to the blade,
whilst the far wake was modelled simply by concentrated root and tip vortices, the
former being deleted for some studies. The mesh part of the wake could be truncated
at any chosen position downstream. Initial studies in the forward flight case placed
the wake mesh points on an undistorted skewed helix (i.e. a rigid wake), but later
models allowed distortions from this to occur for a fixed length of time, the distorted
　
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