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the more general result of Coleman et al.
It is clear that, unlike the axial case, the induced velocity at a particular point in
forward flight depends not merely on the local loading but on the way in which the
pressure gradient varies along the path of integration, which for the linearised analysis
is determined only by the rotor-disc incidence. Of course, in any given flight condition,
Pressure
gradient
x
–2 –1 0 1 2
Fig. 3.15 Pressure gradient along longitudinal axis of uniformly loaded rotor
90 Bramwell’s Helicopter Dynamics
the rotor thrust must be accountable in terms of the momentum changes in the air, but
care must be taken to ensure that all the forces acting on the air are properly taken
into account when applying the principle.
3.6 Induced power in forward flight
We saw in Chapter 2 that for a uniformly loaded rotor carrying a thrust T the
induced power is Tvi, where vi is the mean induced velocity. When the loading is
non-uniform, the induced power is always higher than this ‘ideal’ value. In particular,
for an axially symmetric loading the increase is about 13 per cent. A figure of
about 15 per cent greater than the ‘ideal’ value is generally accepted to account for
the non-uniformities of loading and induced velocity in both axial and forward
flight.
We now inquire as to whether this factor for forward flight is valid when we note
from Figs 3.6 and 3.7 the considerable distortion of the induced velocity field for
these cases.
As in the axial case, to find the induced power we calculate the energy being
supplied to the slipstream by the passage of a uniformly loaded rotor. In forward
flight the energy of the slipstream consists of two distinct contributions:
(i) the energy of the flow in the cylindrical wake (this is the only source of energy
in the axial case);
(ii) the energy imparted to the air outside the cylindrical wake due to the movement
of the wake through it. (We saw earlier in this chapter that the wake moves
normal to itself with velocity 2vi0 tan (χ/2).)
If E1 and E2 are the two contributions per unit length of wake, the power expended,
P, is V(E1 + E2), and this is the induced power. The mass flow through the wake at
the rotor is ρπR2vi cos χ, since πR2 cos χ is the cross-sectional area of the wake. The
absolute velocity of the air in the ultimate wake is 2vi sec2(χ/2), as discussed earlier,
so that
E 1 i R
= 2ρv2π 2 cos χ sec2(χ/2)
By considering the kinetic energy of the air outside the elliptic wake, it can be
shown2 that
E 2 i R
= 2ρv2π 2 tan2(χ/2)
for which we find that
P = Tvi
as in hovering and axial flight.
The extension of the above calculation to arbitrary loadings is much more difficult,
even when the relationship between the loading and the induced velocity is known
Rotor aerodynamics and dynamics in forward flight 91
completely, as in the work of Mangler and Squire referred to earlier. The case of the
rotor in high speed flight (χ = 90°) carrying Mangler’s loading can be calculated
quite easily2, however, and the result shows that the induced power is about 1.17
times greater than if the induced velocity were uniform.
One can also calculate the induced power by considering the in-plane
component of the blade thrust vector which is tilted backwards by the induced
velocity, as is considered in classical blade element theory, Chapter 2. If Δp is the
axisymmetrical radial pressure distribution, the thrust carried on an annulus of
width dr is
dT = 2πr Δp dr
and this thrust is shared by the b blades of the rotor. The elementary induced torque
is
dQi = r dTvi/Ωr = (vi/Ω) dT
where vi is the induced velocity at a given blade. The corresponding induced power
is
dPi = Ω dQi = vi dT
= 2πrΔpvi dr
From Mangler and Squire’s work, and also the work of Coleman et al., for
axisymmetrical pressure loadings the induced velocity can be expressed as
vi vi0
12
0 =1
= 4 [c – c cos n ]
n n Σ ∞ ψ
as in eqn 3.13 and where 2vi0c0 is the same as the induced velocity in axial flight for
the same loading. Because of the cosine terms, the mean value of the power with
respect to azimuth depends only on the first term of the series and we have
P pc r r
R
i i0
0
= 4π v ∫ Δ 0 d
= 4 i0 ( ) ( ) d
2
0
1
π vR ∫ Δpxc0xxx
Thus, when the rotor loading is axisymmetrical, the induced power can be calculated
from a knowledge only of the induced velocity distribution in axial flight fast enough
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