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2.5 Blade element theory in vertical flight
The relationship developed in the previous sections between the thrust and the induced
velocity requires that either the thrust or the induced velocity is known. We now
consider the lift characteristics of the blade regarded as an aerofoil to obtain a further
relationship between thrust and induced velocity, thereby enabling both to be evaluated.
The calculations follow closely the standard methods of aerofoil theory but the rotor
Rotor aerodynamics in axial flight 47
analysis is simplified considerably because the blade incidence and inflow angles are
usually so small that the familiar small angle approximations may be made.
Consider an element of blade of chord c with width dr at a radius r from the axis
of rotation. The geometric pitch angle of the blade element relative to the plane of
rotation is θ, the climbing speed is Vc, and the local induced velocity is vi. The
direction of the flow relative to the blade makes an angle φ (usually called the inflow
angle) with the plane of rotation, Fig. 2.11, and φ is given by
tan φ = (Vc + vi)/Ωr
or, for small φ,
φ = (Vc + vi)/Ωr
The lift on the blade elements is
d = d 12
L ρW2CLc r
≈ d 12
ρΩ2r2CLc r
since, for small φ, W2 ≈ Ω2r2.
Let us suppose that the lift slope a of the section is constant so that, if the section
incidence α is measured from the no-lift line, we can write
CL = aα = a(θ – φ)
Empirical data suggests a lift slope of about 5.7. The elementary lift is now
d = ( – ) d 12
L ρΩ2r2aθ φc r
Since φ is usually a small angle, we can write dL ≈ dT, where dT is the elementary
thrust, the force perpendicular to the plane of rotation. The total thrust is therefore
T ab c r r
R
= ( – ) d 12
2
0
ρ Ω ∫ θ φ 2 (2.25)
where b is the number of blades.
dT dL
dD
φ
W
Vc + v i
φ
θ
Ωr
Fig. 2.11 Force components on blade
48 Bramwell’s Helicopter Dynamics
Defining
λ c = Vc/ΩR, λi = vi/ΩR, x = r/R
eqn 2.25 can be written
T ab R c x x x = [ – ( + ) ] d 12
2 3
0
1
2
ρ Ω ∫ θ λ c λ i (2.26)
If the chord, induced velocity, and ‘collective’ pitch angle θ are constant along the
blade, eqn 2.26 can be integrated easily to give
T acb R = [ – ( + )] 12
2 3 1
3
12
ρ Ω θ0 λ c λ i (2.27)
where θ0 is the constant (collective) pitch angle.
Defining a thrust coefficient by
tc = T/ρsAΩ2R2
where s = bc/πR is the rotor solidity, eqn 2.27 gives
tc = (a/4)[2θ0/3 – (λc + λ
i )] (2.28)
In American work, the thrust coefficient is usually defined by
CT = T/ρAΩ2R2
so that the two thrust coefficients are related by tc = CT /s.
From the momentum theory, the induced velocity and the thrust are related by
T = 2ρA(Vc + vi)vi (2.11)
which can be written in non-dimensional form as
λ λλ i
2
c i
12
+ – stc= 0 (2.29)
the positive root being the correct one to take. With λc being given, eqns 2.28 and
2.29 can be solved for tc if θ0 is known, or the required pitch angle θ0 can be
calculated if tc is given. In hovering flight we have simply
tc = (a/4)(2θ0/3 – λ
i) (2.30)
and
stc i
= 2λ 2 (2.31)
Equations 2.28 and 2.29 have been obtained on the assumption that the blade pitch
and chord were constant along the blade and that the downwash velocity had the
constant ‘momentum’ value given by eqn 2.11. Modern helicopter blades usually
have constant chord and approximately linear twist, and, if we assume that a linear
variation of induced velocity is quite a good approximation to that obtaining in
practice, eqn 2.26 can again be integrated quite easily. Let us write the local blade
pitch as θ0 – θ1x and the local induced velocity as vi = viTx, where θ1 is the blade
Rotor aerodynamics in axial flight 49
‘washout’ angle and viT is the downwash velocity at the blade tip. Then eqn 2.26
integrates to give
t a
c = 0 1 c iT
4
2
3
– 3
4
– – 2
3
θ θ λ λ ( )
(2.32)
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