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the point P and at rest relative to the undisturbed air. As the vortex sheets leave the
blade and move downwards, the observer would be aware of a periodic flow. Now
Bernoulli’s equation for unsteady flow is
p q t + + / = constant 12
ρ 2 ρ∂φ∂
where q is the local fluid velocity and φ is the velocity potential of the flow. At a great
distance from the wake both q and φ tend to zero and p approaches the ambient value
p∞, therefore
p q t p + + / = 12
ρ 2 ρ∂φ∂ ∞ (2.1)
This equation holds throughout the flow field, including points between the sheets,
since only the sheets themselves represent regions for which the flow is not irrotational.
If z denotes the distance along the rotor axis, r the radial distance, and ψ the angular
co-ordinate of the point P, the periodicity of the flow enables us to write
φ = f (z – wt, r, ψ)
°P
Fig. 2.2 Vortex sheet leaving blade
36 Bramwell’s Helicopter Dynamics
because φ is constant for a point z0, r, ψ which moves with the wake, and z0 = z – wt.
Hence
∂φ/∂t = –w∂φ /∂z
But
∂φ /∂ z = qz
where qz is the fluid velocity component in the axial direction, therefore
p q wq p z + – = 12
ρ2 ρ ∞ (2.2)
It will be shown shortly that the flow component in the axial direction is small
compared with the rotor tip velocity, and this means that the vortex sheets are nearly
parallel to the rotor plane and are also fairly close together. The flow about these
sheets, as seen by the stationary observer, is indicated in Fig. 2.3.
Except near the edges of the sheets, where there may be a considerable radial flow,
the velocity between the sheets is very nearly equal to the velocity of the sheets
themselves, i.e. qz and q are both approximately equal to w. Hence, for the rotor case,
eqn 2.2 becomes
p p w = + 12
2
∞ ρ (2.3)
showing that the pressure in the wake is generally higher than the ambient value p∞.
The above result can be obtained in another way. If the observer is moving with
the sheets, the total head pressure at a point a great distance from the sheets is clearly
p∞ + 12
ρw2 . Within the sheets, which are assumed to be almost parallel and close
together, the flow is relatively at rest, but since the total head is constant, whether
within the sheets or without, we have
p p w = + 12
2
∞ ρ
as before.
The fact that the vortex sheet theory gives rise to an ‘overpressure’ in the wake has
been remarked upon by Theordorsen,5 who derived eqn 2.2, but since his work was
concerned with propellers in their normal operating state, for which the ‘overpressure’
is extremely small, its significance in relation to the helicopter rotor seems to have
been overlooked.
Blade
w
Fig. 2.3 Flow about adjacent vortex sheets
Rotor aerodynamics in axial flight 37
In practice, the helicopter blade is usually designed to give a favourable loading
in forward flight and, as a result, the ‘ideal’ loading and helical wake is not achieved
in hovering and axial flight. It appears6 that the wake pressure is somewhat overestimated
by eqn 2.3, and one should expect a value which is somewhere between that given by
eqn 2.3 and the ambient value p∞.
In the application of the classical actuator disc theory, we shall take an arbitrary
value of the pressure in the final wake at first and then investigate the special case (i)
where the wake pressure is equal to the ambient pressure p∞ and (ii) the value given
by eqn 2.3 above.
Let us take a cylindrical control surface surrounding a control volume whose
radius is R1, which encloses the rotor, radius R, and its slipstream, Fig. 2.4. Although
the slipstream does not extend upstream of the rotor, it is convenient to imagine that
it does so for the purpose of applying momentum principles. Far upstream of the
Fig. 2.4 Control volume for rotor in axial flight
Control Volume
R2
p∞
p2 Vc + v2
Vc
p + Δp Vc + v
i
p
R
R1
p∞ Vc
38 Bramwell’s Helicopter Dynamics
rotor, the air velocity relative to the rotor is the rate of climb Vc and the pressure is
p∞. As the air approaches the rotor, the airspeed increases to Vc + vi at the rotor itself.
Because the airflow is continuous there is no sudden change of velocity at the rotor,
but there is a jump of pressure Δp which accounts for the rotor thrust T = ΔpA, A
being the rotor disc area πR2. The slipstream velocity continues to increase downstream
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