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It was assumed that the circulation along the blade was constant, so that a vortex
trailed from the blade tip having the same circulation as that round the chord. The
spiral vortices from each blade were assumed to form a uniform vortex cylinder
reaching the ground. Now, it is well known that the presence of a plane ground can
be represented by an appropriate image system such that the flow normal to the plane
vanishes, which is the required boundary condition. In this case the image system is
a cylinder of opposite vorticity, Fig. 2.13. It is clear that the effect of the image
system is to produce an upflow tending to reduce the induced velocity at the rotor.
The treatment is very similar to that for a fixed wing flying near the ground, which
may be represented by a simplified horse-shoe vortex and its image system.
The ratio of the induced velocity to that which would have occurred in free air is
shown in Fig. 2.14 as a function of the radial position and the ratio of rotor height,
h, to rotor radius. The ratio of the corresponding powers is given in Fig. 2.15 as a
function of thrust coefficient and rotor height. A typical value of the ratio of the rotor
height to rotor radius when the helicopter is on the point of taking off is about 0.3,
Ground
plane
Fig. 2.13 Reflection of tip vortex in ground plane
58 Bramwell’s Helicopter Dynamics
1.0
0.8
0.6
0.4
0.2
0 0.2 0.4 0.6 0.8 1.0
3
2
3/2
1
1/2
h/R = 1/4
r/R
vi/vi∞
Fig. 2.14 Ground effect on mean induced velocity
1.0
0.8
0.6
0.4
0.2
0
1 2 3
tc /s = 3.5
1.5
0.5
h/R
P/P∞
Fig. 2.15 Ground effect on induced power
and it can be seen from Fig. 2.15 that the induced power is about half that which
would have occurred in free air, representing a rduction of about a third of the total
power.
The same improvement in performance can be presented in another way. From a
number of tests on model rotors, Zbrozek14 has derived curves of the thrust that can
be produced for a given power and has expressed the results as the ratio of the thrust
in ground effect to the thrust in free air as a function of the rotor height and thrust
coefficient, Fig. 2.16.
Hovering in ground effect (IGE) confers considerable operational benefits at high
altitude when the power available may not be sufficient to hover out of ground effect
(OGE). A take-off at altitude, for example, may be initiated followed by transition to
forward flight IGE until a speed is reached such that the power required becomes less
than the power available (see Chapter 4) and a climb out may be performed.
Rotor aerodynamics in axial flight 59
2
1.5
1
0 1 2
0.2
tc = 0.05
T/T∞
h/R
0.1
Fig. 2.16 Ground effect on thrust
2.10 Rotor wake models
As stated at the beginning of this chapter, since the only flow through the rotor in
hovering flight is due to the velocity field created by the bound vortices and the
trailing vortex sheets, and since the distribution of the trailed vorticity is determined
by this velocity field, the problem of calculating the flow becomes extremely complex
and a purely analytical solution is not feasible.
A hierarchy of methods of analysis has developed of which the simplest is the
uniform or rigid wake model, mentioned at the start of this chapter. This consists of
a helical surface representing a vortex sheet trailed from each blade and moving
axially at constant velocity. From the point of view of an observer on the rotating
blade, the wake configuration remains fixed. A more refined model is that of the
prescribed wake which embodies improvements to the wake description and velocity
field, including those that have been observed from experiment. The most refined of
all models is that of the free wake, whose configuration interacts with and is consistent
with the velocity field. These are now considered in more detail.
2.10.1 The rigid wake and the methods of Goldstein, Lock, and Theodorsen
We referred earlier to the theorem of Betz which states that the induced power of an
airscrew or rotor is least when the vortex wake springing from the blades moves
axially as if it were a rigid helical surface. A proof of Betz’s theorem has been given
by Theodorsen5.
To appreciate the implications of this result we consider the analogous but much
simpler problem of a fixed wing and its vortex wake. It is well known that the
induced drag of a wing is least when the induced velocity of the wing is constant
along the span, in which case the spanwise loading is elliptical. Sufficiently far
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