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tests indicated very clearly the movement of the tip vortices and the way in which
they eventually interact and diffuse after maximum wake contraction has occurred.
The ultimate wake appears to be unstable and moves downstream in a confused
manner. The instability of a helicopter wake was implied by the results of Levy and
Forsdyke24, who investigated the motion of a helical vortex. They showed that the
Ω
Γ
Γ
Γ
Tip vortex
Vortex sheet
Fig. 2.32 Landgrebe’s calculation of wake velocity in hovering flight
Blade
72 Bramwell’s Helicopter Dynamics
vortex would be stable only if the tangent of the helix pitch angle were greater than
0.3; in general this is not satisfied by a hovering helicopter rotor.
The axial and radial co-ordinates normalised on R of the tip vortex, derived from
Landgrebe’s results for a particular case, are shown in Fig. 2.33 and for two radial
positions of the inner sheet in Fig. 2.34. The change of downward velocity of the tip
velocity just referred to is seen in Fig. 2.33 as a sudden change of slope of the axial
displacement.
Landgrebe reduced these results to formulae giving the radial and axial co-ordinates
of a tip vortex in terms of the azimuth angle, and corresponding results for the inner
sheets. For example, the axial displacement of the tip vortex is given by
zΤ = k1ψw, for 0 ≤ ψw ≤ 2π /b
= ( ) + ( – 2 / ), for 2 / zΤ wψ = 2 /bπ k2ψw πb ψw > πb
where ψw is the wake azimuth angle relative to the blade,
0.2
0.4
0.6
0.8
1.0
0 90° 180° 270° 360° 450° 540° 630° 720°
Tip-vortex co-ordinates
Wake azimuth angle, ψw
Fig. 2.33 Tip vortex co-ordinates as a function of wake azimuth angle
0
0.2
0.4
0.6
0.8
0° 180° 360° 540° 720°
At r = 1.0
At r = 0
Inboard vortex
sheet co-ordinates
Wake azimuth co-ordinate, ψw
Fig. 2.34 Inboard vortex sheet co-ordinates (after Landgrebe)
r
–zT
Tip vortex co-ordinates and –rzT
Axial co-ordinate, –zT
Rotor aerodynamics in axial flight 73
k1 = – 0.25(tc + 0.001θ1),
and k2 = – (1 + 0.01θ1)√CT,
where θ1 is the blade twist in degrees, and CT = stc in which s is the solidity.
The formula for the radial co-ordinate of the tip vortex is
r = 0.78 + 0.22 e –Λψ w
where Λ = 0.145 + 27CT
The two formulae define the boundary of the wake, at least for the part near the
rotor which remains stable, and form the basis for an experimentally prescribed
wake. It is interesting to note that Landgrebe’s results indicate a final slipstream
contraction ratio of 0.78, which is closer to the value of 0.816 predicted by Theodorsen’s
ideal wake theory than to the 0.707 of the classical momentum theory.
These experiments show how rapidly the wake contracts under the rotor. Using the
results given in Fig. 2.33 we can draw the wake boundary for that case, as shown
below in Fig. 2.35. It can be seen that the contraction is practically complete within
only about half a rotor radius, most of it occurring within a distance of 20 per cent of
the radius.
The importance of these figures is that they show that the vortices at the boundary
of the slipstream are displaced well inboard of the blade tip while still close to the
rotor. This means that the inwardly displaced vortices induce an upwash in the tip
region instead of the strong downwash which would occur if the wake contraction
were small or occurred relatively slowly as with the conventional propeller.
Landgrebe calculated the blade incidence distribution using the wake geometry
deduced from the smoke tests and found that the upwash results in a sharp rise of
incidence just inboard of the tip, Fig. 2.36. Because of the high Mach numbers
occurring at the tip, even in hovering flight, locally high incidences are very undesirable,
since they may lead to shock stall with corresponding loss of lift and increase of drag.
Further, the locally high tip incidence increases the spanwise loading gradient and
intensifies the already strong tip vortex.
z/R
r/R
Fig. 2.35 Contraction of rotor wake using Fig. 2.33 results
74 Bramwell’s Helicopter Dynamics
2.10.5 Free wake analysis
Amongst the earliest work on free wakes in the hover is that of Clark and Leiper.25
An initial wake geometry was assumed which is based on the mean induced velocity
calculated from the simple momentum theory. The wake is broken into a number of
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