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P = 4 R2 /5
iT
π ρv3 (2.42)
The thrust from momentum considerations is
Τπ =ρ 4 2 d
0
1
iT
R∫ v2x3x
= 2
iT
ρπR v2 (2.43)
If the induced velocity vi is constant, we have, for the corresponding thrust T0,
T0 = 2ρπR2
vi
2 (2.44)
Comparing eqns 2.43 and 2.44 we see that for the thrusts to be the same we must
have
v v iT
2
i
= 2 2
Then
P = 8√2ρπR2
vi
3 /5
and, if P0 is the induced power when the induced velocity is constant,
P0 = 2ρπR2
vi
3
Hence
P/P0 = 4√2/5 = 1.131
that is, when the induced velocity is linear, the induced power is about 13 per cent
higher than if the induced velocity were constant; the latter condition corresponding
to the least induced power for a given thrust. For the linear induced velocity, the
torque coefficient would be
qc = δ/8 + 1.13√(s/2) tc
3/2 (2.45)
A typical value assumed for δ is 0.012. With typical values of 0.05 and 0.08 for
the solidity and thrust coefficient respectively, the two terms of qc are 0.0015 and
0.00403, showing that the induced power is more than two and a half times the
profile drag power.
52 Bramwell’s Helicopter Dynamics
Tests on aerofoils with rotor blade type of construction show that δ depends
considerably on incidence and can be represented in the form
δ = δ0 + δ1α + δ2α2 (2.46)
Bailey11 has suggested the values
δ = 0.0087 – 0.0216α + 0.4α2 (α in radians)
and has used them in the calculation of thrust, H-force, and torque coefficients in
hovering and vertical flight. The expressions which had to be calculated were very
lengthy and the results were given in tabular form. They are to be found in the book
by Gessow and Myers.12 Since, however, Bailey used constant induced velocity in
his calculations, it is rather doubtful whether the results he obtained would have been
much better than if δ had been assumed constant because, in forward flight especially,
the induced velocity differs considerably from the constant mean value, with
correspondingly large variations in local blade incidence.
Another parameter of great importance is Mach number, especially for current
helicopters which operate at higher tip speeds than formerly. With Mach number and
induced velocity properly taken into account, the calculations of thrust and torque
become more complicated; consideration of Mach number effects is provided in
Chapter 6. However, equations 2.28 and 2.45 give acceptable accuracy for many
performance problems.
2.6 Calculation of the inflow angle
When making rotor calculations, it is often useful to know the inflow angle when the
rotor geometry and operating conditions are given. We saw in the last section that the
elementary thrust dT on an annulus of the rotor disc, when there are b blades, is
d = – )d 12
T ρabcΩ2r2(θ φ r (2.47)
where it has been supposed that the local lift coefficient is given by CL = aα.
Now
φ = (Vc + vi)/Ωr
so that eqn 2.47 can be written
d = – ( + )/ ]d 122 2
T ρabcΩr [θ Vc viΩr r (2.48)
Momentum theory applied to the annulus gives
dT = 4πρ(Vc + vi)vir dr
and on eliminating dT from eqn 2.48 we have
v v i
2
c i
+ V + – – c = 0
abc abc r V
r
Ω Ω
8 8 Ω
2
π π θ 




(2.49)
Rotor aerodynamics in axial flight 53
Writing λ
i = vi/ΩR and λc = Vc/ΩR, as before, and putting σ = bc/πr, where σ is
the solidity based on the local radius, eqn 2.49 becomes
λ λ σ λ σ θ λ
i
2
c i
+ + – – c = 0
a x a x
8 8 x


 
 
(2.50)
In eqn 2.50 σ and θ are variables, so that variable twist and taper can be taken into
account. In hovering flight λc = 0 and eqn 2.50 reduces to
λ i
2 + (aσx/8)λ
i – (aσx2/8)θ = 0 (2.51)
Since φ = vi/Ωr = λ
i/x, eqn 2.51 can be written as
φ2 + (aσ/8)φ – (aσ/8)θ = 0
or
φ2 = (aσ /8)(θ – φ) (2.52)
Hence, given the local blade pitch angle and solidity, the local value of φ can be
　
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