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leave the ends of this element, Fig. 6.2. If the rate of change of circulation is ∂Γ/∂t,
the circulation at an instant dt before would have been Γ – (∂Γ/∂t)dt, and the change
will have resulted in a spanwise vortex element or filament of strength (∂Γ/∂t)dt
being shed behind the element, as shown in Fig. 6.2. Since the strength of the bound
circulation at the earlier instant was Γ – (∂Γ/∂t)dt, the arrangement of vortex filaments
can be represented in the form shown in Fig. 6.3, i.e. the vortex wake can be regarded
as an assembly of infinitesimal vortex rings. The distribution of vortex rings is
equivalent to a layer of doublets whose axes are perpendicular to the vortex sheet,
and Baskin1 has used this to develop an induced velocity theory.
Fig. 6.1 Shed and trailing vortices from lifting blade
dΓ
Γ
Γ + dΓ
Γ
Γ
Γ + dΓ
Γ+ dΓ
Fig. 6.2 Spanwise change of bound circulation
Γ
Γ
Γ
Γ
d
d
d Γt
t
Γ Γ – d ∂
∂ t
t
Γ Γ – d ∂
∂ t
t
Γ
Γ
Fig. 6.3 Modelling of vortex wake
198 Bramwell’s Helicopter Dynamics
The total induced velocity at a point P on a given blade of the rotor will be the
combined effect of the velocities induced by the trailing and shed vortices of all the
blades and of the bound vortices of all the blades except the one in question. If Γ is
the strength of a vortex element whose length is ds, and l is the displacement vector
between the element and the point P, Fig. 6.4, the Biot–Savart law gives for the
corresponding increment of induced velocity
d = d
q Γ l 3s
4π ⋅ ×
l
(6.1)
Using this formula it would appear straightforward, in principle, to integrate the
contributions mentioned above. Unfortunately, as we also saw in the hovering case,
the vortices interact with one another and the wake pattern becomes very complicated.
This means that we are unsure of the positions of the vortex elements, particularly in
the more distant parts of the wake, or, in other words, l in eqn 6.1 is not known for
certain. Due to the forward flight case not having the axial symmetry of vertical
flight, the pattern of vortex interaction is different because, except at low speeds, the
flow through the rotor is determined largely by the component of forward flight
speed which is, of course, constant over the disc. Let us suppose that the induced
velocity is small compared with the forward speed (corresponding to a lightly loaded
rotor at high forward speed) and ignore the distortions due to vortex interaction. To
include partially the effect of the induced velocity we can calculate the ‘momentum’
value and add it to the component of the forward speed perpendicular to the rotor
disc. The trailing vortex lines then lie on the surfaces of skewed cylinders. This
represents, in effect, a rigid wake model by analogy with that for hovering flight. The
plan and side views of a trailing vortex line are shown in Figs 6.5 and 6.6.
If the x axis lies along the direction of motion and in the plane of the rotor, and the
z axis points upwards perpendicular to the rotor plane, as shown in Figs 6.5 and 6.6,
the co-ordinates of a trailing vortex element which was shed when the blade azimuth
angle was φ are
x = – r1 cos φ – Vt cos αD, y = x1R sin φ, z = Vt sin αD – vit
where t is the time taken for the blade to rotate from φ to ψ. Since ψ – φ = Ωt, we have
x = – r1 cos ψ – (V cos αD/Ω)(ψ – φ) = – r1 cos ψ – μDR(ψ – φ)
y = x1R sin φ
z = (V sin αD – vi)(ψ – φ)/Ω = λDR(ψ – φ)
l Γ
P
ds
Fig. 6.4 Induced velocity at P
Rotor aerodynamics in forward flight 199
where μD = V cos αD/ΩR, λD = (V sin αD – vi)/ΩR, x1 = r1/R
It can be seen that the plan view of a trailing vortex filament, under the given
assumptions, is a cycloid, and the side view shows the constant downward displacement
of the cycloid.
Returning to eqn 6.1, let
l = l1i + l2 j + l3 k
and ds = ds1i + ds2 j + ds3k
where i, j, k are unit vectors along the x, y, z, axes. Then, if wt denotes the downward
component of induced velocity due to a trailing vortex of strength Γ
d =
d – d
　
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