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2.3 Vertical descent and the vortex ring state
The results obtained so far have been made possible only because it has been assumed
that there has been a definite flow through the rotor with a well-defined slipstream.
In vertical descent, however, it is clear that the relative upward flow will, if it
becomes large enough, prevent a slipstream from forming, and some of the air will
recirculate the rotor in what is known as the vortex ring state, Figs 2.6(a) and (b). The
vortex ring state occurs when the rate of descent is of the same order as the induced
velocity in hovering flight. It can also occur in forward flight and in either case leads
to very high descent rates together with uncommanded pitch and roll excursions.
Recovery is by reducing the collective pitch and attaining a forward flight velocity
component, thereby moving the rotor into unrecirculated air. Flight tests describing
the condition have been made by Brotherhood.7,8
At higher rates of descent the recirculation ceases and a well defined slipstream
develops again, but the wake widens after passing through the rotor and the vortex
wake develops on the upper side of the rotor, Fig. 2.7(a). In contrast to the verticalclimb
case, the air slows down on passing through the rotor and the condition is
(a) (b)
Fig. 2.6 Vortex ring flow in vertical descent: (a) slow rate of descent; (b) faster rate of descent
Rotor aerodynamics in axial flight 43
known as the windmill brake state. There is a transitional state between this and the
vortex ring state in which the rotor acts rather like a bluff body, producing a turbulent
wake downstream (i.e. on the upper side of the rotor). This is generally known as the
turbulent wake state and is shown in Fig. 2.7(b).
In the absence of a well-defined slipstream, the momentum theory can no longer
be readily applied, since the mass flow and velocity changes cannot be easily defined.
A simple theoretical relationship between the induced velocity and the axial velocity
of the rotor in such cases is no longer possible. We are then forced to obtain the
induced velocity experimentally by inferring it from the results of the blade element
theory, to be discussed in section 2.3, in connection with the measured rate of descent
and blade collective pitch angle.
To calculate the induced velocity in hovering and climbing flight, we make use of
eqn 2.11. Then, if v0 is the induced velocity in hovering flight, or ‘thrust velocity’,
we define
v i = vi/v0 and Vc = Vc/v0
so that eqn 2.11 can be written
v i (Vc + vi ) = 1 (2.17)
For vertical descent velocities which are large enough for a slipstream to be
developed again, i.e. the windmill-brake state, eqn 2.11 must be written as
2ρA|Vc + vi |vi = T
the modulus sign indicating that the mass flow, represented by the term Vc + vi, must
be positive (which it is certain to be in climbing flight). The correct result for descending
flight can be expressed as
(a) (b)
Fig. 2.7 (a) Windmill brake state; (b) Turbulent wake state
44 Bramwell’s Helicopter Dynamics
2ρA(Vc + vi)vi = –T
or, in non-dimensional form, as
v i (Vc + vi ) = –1 (2.18)
From eqns 2.17 and 2.18, and using values of the induced velocity obtained from
flight and wind tunnel tests for the vortex ring state, we can describe the complete
curve of vi as a function of Vc , Fig. 2.8.
The broken lines of Fig. 2.8 are the continuations of eqns 2.17 and 2.18 into
regions for which the vortex ring state renders them invalid. Of particular interest is
the state of ‘ideal autorotation’ in which there is zero mean flow through the rotor so
that Vc = – vi. This is given by the intersection of the curve of vi against Vc with the
line v i = – Vc , shown chain-dotted in Fig. 2.8. We find that this occurs when
Vc = – 1.8. The condition of ‘ideal autorotation’ is equivalent to the motion of a
circular plate broadside on to the stream which destroys the momentum of the air
approaching it. The thrust of the rotor in this condition can be equated to the drag of
such a disc, so that, if CD is the drag coefficient,
T = CD VcA = 2 A
2
0
2 12
ρ ρv
or
CD Vc
= 4/ 2
Substituting the value of Vc found above, we have
CD = 4/(1.8)2 = 1.23
which is close to the drag coefficient of a circular plate. Thus, in ‘ideal autorotation’
the rotor behaves rather like a parachute.
2.4 The swirl velocity
So far the rotational or ‘swirl’ velocity has been omitted from the calculations. Since
Fig. 2.8 Variation of induced velocity in vertical flight
Vc
Turbulent
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