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the contraction is usually complete within a rotor diameter, and the curved part of the
contracting wake lies close enough to the blades to have a considerable influence on
the incidence near the tips, Fig. 2.29.
Unfortunately, although the amount of contraction, as we have just seen, can be
obtained quite easily, the calculation of the rate of contraction in the neighbourhood
of the rotor is very complicated and certainly cannot be found in closed form as with
Theodorsen’s other results. To obtain reliable information we need to make use of
experimental data, to be combined, if possible, with theory. These methods are described
in a later section.
2.10.4 The prescribed wake
The Goldstein–Lock theory assumes that the wake is a uniform or rigid helix. In
hovering and vertical flight of low axial velocity, the flow through the rotor is
dominated by the induced velocity which, as we have seen, is not generally uniform
over the rotor and in the wake. The vortex elements springing from the blade are
R
R0
tan–1Λ0
tan–1 Λ
R
R0
Fig. 2.28 Wake parameters in Theodorsen’s
calculations (propeller theory)
Fig. 2.29 Wake contraction for a hovering rotor
4
6
70 Bramwell’s Helicopter Dynamics
therefore transported downwards at different rates, rather than at a constant rate, as
assumed by the rigid wake model.
The reduction in bound circulation about the blade towards the tip as shown in
Fig. 2.23, which is quite rapid for the lower inflow angles, implies a concentration of
trailed vorticity here, that quickly rolls up to form a concentrated vortex emanating
from near the tip. Typically, the bound circulation also reduces inboard, but at a lesser
rate20 (in radial terms) and this leads to a trailing vortex sheet of opposite sign to that
of the tip vortex. The wake model shown in Fig. 2.30 consisting of an assembly of
discrete vortex elements can be considered to form a suitable representation, which
may be used to evaluate the induced velocity.
An initial calculation of the vertical induced velocity is made using momentum
theory. The vertical displacements of the vortex elements are then calculated, consistent
with the assumed induced velocity, and the pattern of the vortex wake is therefore
defined. Then, by applying the Biot–Savart law to the individual elements, the induced
velocity at the rotor disc can be calculated by summing the contributions of all the
elements.
Langrebe19 has made such calculations and compared them with those obtained
from the simple momentum theory, Fig. 2.31. It can be seen that the difference
Blade
Discrete
tip-vortex
filament
Discrete
inboard-vortex
filament
Fig. 2.30 Vortex wake representation (after Landgrebe)
0
10
20
30
Momentum (B = 0.97)
Prescribed wake
0 20 40 60 80 100
Axial induced velocity
m/s
Blade radial co-ordinate, r/R, per cent
Fig. 2.31 Comparison of induced velocity by momentum and prescribed wake methods
Rotor aerodynamics in axial flight 71
between the methods is quite small, in spite of the fact that the uniform helical wake
assumes quite a different loading distribution from those occurring in practice. The
indications are that Lock’s assumption, which is that Goldstein’s results for a uniform
helix can still be applied when the blade loading is not ideal, gives quite accurate
results.
The wake geometries of the theoretical methods described so far fail to take into
account completely the contraction of the wake and other distortions of the wake
when the blade loading does not conform to the ideal distribution. In the mid-1950s,
Gray21,22, from the results of smoke studies, concluded that the wake from a blade
consisted of strong tip vortex and an inner vortex sheet of opposite sense. This
arrangement is shown diagrammatically in Fig. 2.32.
It was observed that the outer part of the sheet moves faster than the inner part,
with the result that the sheet becomes more and more inclined to the rotor plane, and
that the outer part of the sheet moves faster than the tip vortex.
Landgrebe’s19 later series of smoke tests confirmed Gray’s results and also confirmed
that cross-sections of the tip vortices do not necessarily occur at the ends of the
corresponding sheet. It was also observed that the tip vortex from a blade moves
downwards relatively slowly until it passes beneath the following blade, from which
point it moves down more rapidly.
These results were confirmed by the experiments of Tangler et al.23, who investigated
the wake pattern by methods of flow visualisation and hot-wire anemometry. These
　
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