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where vi0 is the induced velocity at the rotor centre (which is also the value along the
lateral axis) and χ is the angle between the wake and the rotor plane. On the lateral
axis, wP = wQ = w, say, and, by calculating the flow about the ellipse2, we find that
on the lateral axis outside the rotor
w = vi0 [1 – x(x2 – sin2 χ)–1/2] (3.16)
where x = r/R. This variation is shown in Fig. 3.14.
On the lateral axis, therefore, there is a strong upwash outside the rotor disc as
opposed to the downwash on the inside. The upwash also exists forward of the lateral
axis and for some distance behind it, but a downwash appears further to the rear at
a point depending on the incidence of the rotor.
v ′
°Q
u′
°P
Fig. 3.13 Symmetry relations for induced velocity in forward flight
P• •Q Q′• •P′ P+Q′• •P′+Q
88 Bramwell’s Helicopter Dynamics
3.5 General remarks on the forward flight case
We have seen that the mean induced velocity in forward flight can be given by
Glauert’s formula
vi = T/2ρAV′ (3.1)
and that it can be expressed in a form which resembles the ‘momentum’ formula of
climbing and hovering flight. Now, we assumed in Chapter 2 that this relationship
could be expressed in the differential form
dT = 4πrρ (Vc + vi)vi dr
and that from this expression the induced velocity for non-uniform loadings could be
derived. However, the axial flight case, which includes the conventional propeller, is
a special one for two reasons, namely,
(i) the induced velocity is in the same direction as the general flow, and
(ii) because of the symmetry of this, the flow is confined to well defined concentric
shells for which the mass flow can easily be calculated.
Under these circumstances, momentum principles can be confidently applied and
have been used already to obtain a number of results. But in forward flight neither of
these two conditions applies. We have already seen that, in the linearised problem of
the lifting rotor, the pressure field satisfies Laplace’s equation. In particular, it can be
shown2 that for the uniformly loaded rotor the pressure at any point is proportional
to the solid angle which the rotor disc subtends at that point. Further, the acceleration
and pressure gradient fields, for the linearised case, both have precisely the same
form as the velocity field of a vortex ring. Suppose, for simplicity, that the rotor is
moving in its own plane and that we wish to find the component of induced velocity
w normal to the rotor disc. As we have already seen, w can be calculated by the
integral in eqn 3.11. The interpretation of this integral is that we are integrating the
1
2
1 0 1
2
χ = 30°
60°
75°90°
r/R
Fig. 3.14 Induced velocity near uniformly loaded rotor on the lateral axis
w
vi0
Rotor aerodynamics and dynamics in forward flight 89
component of pressure gradient normal to the rotor along a path in the plane of the
disc. The shape of this pressure gradient component along the longitudinal axis of the
rotor is shown in Fig. 3.15.
Starting from a point to the left of Fig. 3.15, and a great distance from the leading
edge, we see that the pressure gradient is upward so that, if w is zero a long way in
front of the rotor, integration results in a gradual increase of upwash as the rotor is
approached. Just behind the leading edge the pressure gradient reverses and the
upwash diminishes until a downwash develops at a point between the leading edge
and the rotor centre, gradually increasing as the trailing edge is approached. Proceeding
further to the right, the pressure gradient becomes positive again as the trailing edge
is passed, gradually reducing the downwash until it becomes exactly twice that at the
rotor centre.
What has just been described is the qualitative evaluation of the downwash
corresponding to the case χ = 90° calculated by Coleman et al. and presented in Fig.
3.4. In fact, the integration for this case can be performed analytically and gives
simply
w
p
V
= – xK x
4
[2 4 ( )] Δ
πρ π m
1 > > 0
– 1 < < 0
x
x
w
p
V
= – K x
4
[2 (1/ )] Δ
πρ π m
1 < <
– 1 > > –
x
x
∞
∞
where Δp is the pressure step across the disc, x is normalised with respect to the rotor
radius, and K(x) is the complete elliptical integral of the first kind. This agrees with
　
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