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the leading edge of the rotor, although giving some dependence on the wake angle.
Another result obtained by Coleman et al. is that, since the vortex rings composing
the wake lie at an angle to the wake axis, there will be a component of induced
velocity causing the wake to move downwards normal to its axis and that, in the
ultimate wake, this component of wake velocity is 2vi0 tan (χ/2). Also, if the disc
incidence is αD, the above result leads to the following relationship between the
mean induced velocity and the ultimate wake angle
vi0 /V = cos (χ + αD)/2 tan (χ/2) (3.6)
Rotor aerodynamics and dynamics in forward flight 81
3.3 The method of Mangler and Squire
One of the more complete induced velocity calculations in which the rotor is treated
as a lifting surface with a pressure jump is that of Mangler and Squire.4 They considered
a rotor loading distribution which closely resembled that of a typical rotor and succeeded
in obtaining an exact solution for the induced velocity for any point on the rotor.
Their method was quite different from Coleman’s and it will be instructive to discuss
it in some detail.
If the induced velocity field is regarded as a small perturbation superimposed
upon an otherwise uniform velocity field, and if the x axis is taken along the direction
of the uniform velocity V, Euler’s equations of motion for an elemental volume of
fluid (e.g. Ref. 1, Chapter 2, p. 150) can be linearised to read
V∂u/∂x = – (1/ρ) ∂p/∂x (3.7)
V∂v/∂x = – (1/ρ) ∂p/∂y (3.8)
V∂w/∂x = – (1/ρ) ∂p/∂z (3.9)
Differentiating eqns 3.7 to 3.9 with respect to x, y, z respectively and then adding
gives
V
x
u
x y
w
z
p
x
p
y
p
z
∂
∂
∂
∂
∂
∂
∂
∂
∂
∂
∂
∂
∂
∂
+ + = – 1 + +
2 v
ρ 2
2
2
2
2
But, by continuity,
∂u/∂x + ∂v /∂y + ∂w/∂z = 0
so that
∂ 2p/∂x 2 + ∂ 2p/∂ y2 + ∂ 2p/∂z2 = 0 (3.10)
Hence, we have the rather interesting result that for small disturbances the pressure
field satisfies Laplace’s equation. The first part of the problem, then, is to find a
solution of Laplace’s equation which also satisfies the given pressure discontinuity
across the rotor disc. Secondly, having obtained such a solution for p, any one of the
induced velocity components can be calculated by integrating the appropriate one of
eqns 3.7 to 3.9. Thus, if the disturbance far in front of the rotor is assumed to be zero,
the velocity component w normal to the flight direction is given by
w V p/ z x
x
= – (1/ρ ) ( ) d
∞ ∫ ∂ ∂ (3.11)
where x corresponds to the chosen point P in the field. The integration, in this case,
is performed with the values of y and z appropriate to the path of integration; e.g. if
the path of integration were in the plane of the rotor, z would be zero and y would be
constant and have the value corresponding to the field point P.
The other two velocity components, u and v, would be obtained by similar
integrations.
82 Bramwell’s Helicopter Dynamics
Although the method has been illustrated here by the use of Cartesian co-ordinates
for the sake of simplicity, Mangler and Squire used elliptical co-ordinates since they
allowed the boundary conditions to be satisfied more easily and because these functions
give the necessary discontinuity across the disc. In boundary problems of this kind,
the solution appears as a multiply infinite series; in this case the solution for the
pressure at the rotor disc appears as an infinite series of pressure mode ‘shapes’
which have to be chosen to match some prescribed pressure distribution corresponding
to the rotor loading. It was found that the first two pressure shapes at the disc plane
were given by
( – u = (1 – )
3
4
pl p)1 ρV2CT √ x2
( – u = – (1 – 5 /2) (1 – )
12
pl p)2 ρV2CT x2 √ x2
and a particular linear combination gives
(pl – pu)/ρV2CT = 15x2√(1 – x2)/8 (3.12)
This distribution is shown in Fig. 3.5 and is such that the load vanishes at the edge
and the middle of the rotor, thereby representing a typical rotor loading very well. It
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