曝光台 注意防骗
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control volume, and the continuity relation, eqn 2.5, has been used.
With p2 = p∞ and vi =
12
v2 , we see from eqn 2.11 that
P = T(Vc + vi) (2.14)
The first term on the right of eqn 2.14 is the useful work done in climbing at speed
Vc. The term Tvi is the induced power, i.e. the work done producing the (unwanted)
slipstream. In hovering flight,
P = Tvi = T3/2/√(2ρA)
If the thrust of the helicopter is 45 000 N, with the disc loading of 250 N/m2
referred to above, the induced power Pi is
Pi = 45 000 × 10.2 kW
1000
= 453 kW
and this would represent about 60 per cent of the total power in hovering flight, the
rest being used to overcome the blade drag and tailrotor and transmission losses. The
induced power calculated above is a rather optimistic value because it has been
assumed that the induced velocity is uniformly distributed over the disc and that this
can be shown to be the optimum distribution. As we shall see shortly, for the induced
velocity distributions likely to occur in practice, the induced power may be 10–15 per
cent larger than the ‘ideal’ value just calculated.
The contraction ratio is the ratio of the radius of the final wake to that of the disc.
The continuity equation gives at once
R2/R = √(vi/v2)
= 1/√2
when p2 = p∞.
Now let us assume that the pressure in the ultimate slipstream is given from eqn
2.3 as
p2 p
12
2
= ∞ + ρv2
where we have taken the wake velocity w to be the same as the local velocity
increment v2. Then from eqn 2.10 we get
v vv v v v 2
2
2
= i 2 + 2 ( + )/( + )
12
Vc i Vc 2 (2.15)
If we define ki = vi/v2 and V c = Vc/vi, eqn 2.15 can be written
Vc = (3 – 2/ki)/(1 – 2ki)
Rotor aerodynamics in axial flight 41
In hovering flight, Vc = 0, we find ki
2
3 = , i.e. the final slipstream velocity is only
3
2 times the induced velocity instead of twice the induced velocity when p2 = p∞. As
the axial velocity Vc increases, ki varies as shown in Fig. 2.5.
It can be seen that ki → 12
as the axial velocity increases indefinitely. In practice,
however, Vc is unlikely to exceed 2.
In general, the thrust is, from eqn 2.6,
T = A(Vc + i ) 2 + A(V + )/(V + )
12
ρ v v ρv c vi c v2 2
2
= ρA(Vc + v2)v2 (2.16)
from eqn 2.15.
In particular, in hovering flight we find from eqn 2.16 that
vi
2
3 = √(T/ρA)
or, at sea-level, vi = 0.604 √wD m/s when the disc loading wD is given in N/m2.
This result shows that the induced velocity is about 6 per cent lower, for a given
disc loading, than when p2 = p∞.
From eqn 2.13 the thrust power P is
P = A(Vc + i )(Vc + ) + A (V + )
12
2 2
12
ρ v v v ρv c vi 2
2
= ρA(Vc + vi )(Vc + v2)v2
= T(Vc + vi )
as in the previous case. However, as we have just seen, the induced velocity is less
than for the case p2 = p∞, so the induced power is correspondingly lower. In hovering
flight we see that the induced power is about 6 per cent lower.
It was stated earlier that, in practice, it appears that the wake ‘overpressure’ is
somewhat less than the ‘ideal’ value given by eqn 2.3; consequently the difference
between the induced velocities and the induced powers for the two cases considered
Fig. 2.5 Variation of final slipstream velocity factor with axial velocity factor
0.65
0.60
0.55
0.50
0 1 2 3 4 5 6
ki
Vc
42 Bramwell’s Helicopter Dynamics
would be expected to be less than the 6 per cent calculated above. In what follows,
it will be assumed that the wake pressure and the ambient pressure are equal, since
the considerable simplification it affords justifies the acceptance of the fairly small
inaccuracies just mentioned, particularly as it is not certain what the wake pressure
should be. This assumption conforms, of course, to the classical actuator disc theory.
One should be aware, therefore, that some of the quantities calculated by this theory
differ by a few per cent from those calculated with a wake ‘overpressure’. However,
more exact rotor analyses, which require a knowledge of the geometry of the vortex
wake and which will be discussed later in this chapter, will depend on the contraction
ratio and the ratio of the velocity in the final wake to that at the rotor disc. As we have
seen, the effect of the ‘overpressure’ on these quantities is considerable, although it
is usually taken into account only indirectly through wake visualisation methods.
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