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where λiT = viT/ΩR.
Now it is generally accepted that eqn 2.11 can be expressed in differential form as
dT = 4πrρvi(Vc + vi) dr (2.33)
where 2πr dr is the area of the annulus of width dr over which the thrust dT is
distributed. It can be shown9 that eqn 2.33 is not strictly valid but it has given
successful results in airscrew work and may be regarded as sufficiently accurate for
most purposes. It appears to be true10 for the linearised problem in which we take
Vc + vi ≈ Vc and dT = 4πrρviVc dr. Then putting v1 = viTx in eqn 2.33 and integrating
gives
T = ρπR2(viT
2 + 4viTVc/3)
or, in coefficient form,
λ iT
2 + 4λiTλc/3 – stc = 0 (2.34)
Numerical solutions of eqn 2.34 show that the values of λiT are very nearly equal
to √2λi (λ
i being the constant momentum value of eqn 2.29) for a wide range of λc
and is exactly equal to √2λi for the hovering condition (λc = 0). Thus, when we
assume the induced velocity is linear, which, as we have said, is good approximation
to real conditions, viT can be replaced with good accuracy by √2λi . Substituting for
λiT in eqn 2.32 gives
t a
c = 0 1 c i
4
2
3
– 3
4
– – 2 2
3
(θ θ) λ √ λ
(2.35)
But θ0 – 3
4θ1 is the blade pitch angle at 3
4R and 2√2/3 = 0.943; hence, if we take
θ0 as the value of θ at the 3
4 radial position and approximate 2√2/3 by unity, we can
use the simple equations 2.28 and 2.29 or 2.30 and 2.31 for all cases. These
approximations mean that the thrust will be underestimated by about 2 or 3 per cent
relative to eqns 2.32 and 2.34, but, since the blade lift slope and the actual induced
velocity will not be known precisely, further refinement is hardly justified.
It can easily be verified that if the blade planform also has linear taper, eqn 2.28
still holds, with the exception of some very small terms, if the chord is taken as that
at 3
4 R as well as the blade pitch angle.
A useful relationship between the thrust coefficient and the blade lift coefficient
can be obtained since, for constant blade chord,
T bc R xC x L = d 12
2 3
0
1
ρ Ω ∫ 2
50 Bramwell’s Helicopter Dynamics
or tc xCL x
12
0
1
= ∫ 2 d
= CL/6 (2.36)
where CL= 3 xCL dx
0
1
2 ∫
If the lift coefficient is constant along the blade, then
tc = CT /s = CL/6
Usually the rotor operates at a mean CL of between 0.35 and 0.6, giving typical
values of tc within the range of 0.06 to 0.1.
The rotor torque can be calculated in a similar way to the rotor thrust. From
Fig. 2.11, the torque dQ of a blade element about the axis of rotation is
dQ = r(dD + φ dL)
= ( + )d 12
ρΩ2r3cδ φCL r (2.37)
where δ is the local blade section drag coefficient. If δ is assumed to be constant, eqn
2.37 can be integrated to give
Q bc R bc R x C x L = /8 + d 2 4 12
2 4
0
1
δρ Ω ρ Ω ∫ 3φ (2.38)
Defining a torque coefficient qc by
qc = Q/ρsAΩ2R3
eqn 2.37 can then be written in coefficient form as
qc x CL x
12
0
1
= δ/8 + ∫ 3φ d (2.39)
Assuming constant induced velocity, φ = (λc + λ
i)/x, so that eqn 2.39 becomes, on
using eqn 2.36,
qc = δ/8 + (λc + λ
i)tc (2.40)
For the special case of hovering flight, λc = 0,
qc = δ/8 + λ
itc
= /8 + ( /2) c
δ √s t3/2 (2.41)
The first term of eqn 2.41 represents the torque required to overcome the profile
drag; the second represents the torque to overcome the induced drag of the blades. It
can be seen that the second term is the non-dimensional form of the hovering power
calculated in section 2.1 from energy and momentum considerations.
Using momentum principles we can find the effect of a non-uniform induced
Rotor aerodynamics in axial flight 51
velocity distribution on the induced power. Let us assume that eqn 2.14 holds in
differential form; then in hovering flight we can write
dP = dTvi = 4 πrρ i3dr v
where vi is the local induced velocity. If we take the linear induced velocity distribution
vi = viTx, we have
d = 4 2 d
iT
P πRρv3x4 x
so that
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Bramwell’s Helicopter Dynamics(31)
