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simple criterion has been objected to on the grounds that T/P is not a dimensionless
quantity. The standard measure of efficiency adopted in helicopter work is the figure
of merit M defined by
M = Tvi/P
where vi is the mean momentum induced velocity in hover. Since Tvi is the ideal
induced power, the figure of merit is the ratio of the induced power to the total power.
Since P = Tvi + Pp, where Pp is the profile drag power, the figure of merit can also
be written as
M = Tvi /(Tvi + Pp)
and, in non-dimensional form, as
M = (s/2)tc /q
3/2
√ c
It could well be argued that the figure of merit so defined is even less satisfactory
than the ratio T/P, because for constant thrust a high value can be achieved by
increasing the induced velocity (by reducing the radius, say), thereby increasing the
56 Bramwell’s Helicopter Dynamics
total power, which is the opposite of the desired effect. The reason for this rather
unsatisfactory feature of the figure M is that the induced power is regarded rather
like the ‘useful work’ of standard airscrew theory and takes no account of the fact
that induced power is no more desirable than profile power.
The difficulty of defining an efficiency factor for the helicopter is that many
parameters are involved and some of them cannot be arbitrarily varied because of
structural and mechanical as well as aerodynamic limitations. Nevertheless, some
useful conclusions can be drawn from an examination of the thrust/power ratio. Let
us write this ratio as
T
P
T
T A sA R
=
3/2/√(2ρ) + δρ Ω3 3/8
(2.57)
The first term in the denominator of eqn 2.57 is the induced power, and the second
term is the profile drag power.
Suppose the rotor radius is kept constant and the thrust is kept constant in such a
way as to keep the incidence at a favourable value. This means that the mean lift
coefficient and, hence, tc is kept constant. But since T = tcρsAΩ2R2, we must have
sΩ2 constant, so the only variable term in T/P is the profile drag power, which must
therefore be proportional to Ω. Thus T/P can be increased by reducing Ω, which also
requires s to increase; or, in other words, we need a low rotor speed and high solidity
if the radius is to be kept constant.
Suppose now we fix the thrust, solidity, and tip speed ΩR and vary the rotor
radius. Differentiating eqn 2.57 with respect to A gives
∂
∂
√
A √
T
P
T
s R T A
s R T A
= T P
/8 – /2 2
/8 + / (2 )]
= 0, for maximum /
3 3/2
3 3 3/2 2
δρ ρ
δρ ρ
Ω
Ω
3
[
i.e. δρ ρ s A R T A /8 = / (2 ) 3 12
Ω 3 √
that is, the profile power is half the induced power for maximum T/P. The figure of
merit for this condition is 2
3 .
Finally, for a given tip speed, solidity, disc area, and drag coefficient, we can write
T
P R
t
s t
= 1
/8 + ( /2)
c
c Ω δ √ 3/2
To find the optimum thrust coefficient we have
∂
∂
√ √
t √
T
P R
s t t s
c s t
c
3
2 c
c
2
= 1
/8 + ( /2) – ( /2)
/8 + ( /2) ]
= 0 Ω
δ
δ
3 2 3 2
3 2
/ /
[ /
or 12√(s/2)tc3/2 = δ/8
and, as above, the profile power is half the induced power and the figure of merit is
again 2
3 . If we take as typical values s = 0.05 and δ = 0.012, we find the optimum
value of tc to be 0.072.
Rotor aerodynamics in axial flight 57
2.9 Ground effect on the lifting rotor
When a rotor hovers near the ground, the presence of the ground has a considerable
effect on the induced-velocity distribution over the rotor and, hence, on the thrust and
power. At the ground the vertical component of velocity must vanish, and we can
expect that over the rotor the induced velocity would be less than in free air. The
reduction of induced velocity results in a proportionate reduction of induced power
for a given thrust and since, as we saw earlier, the induced power may be at least two
thirds of the total power, the improvement in performance may be quite remarkable;
indeed, some of the earlier, underpowered, helicopters could hover only with the help
of the ground.
A theoretical treatment of ground effect has been made by Knight and Hefner13.
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Bramwell’s Helicopter Dynamics(34)
