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chordwise velocity variations and apply the idea of lifting line theory to the high
aspect ratio blade. The shed wake integral, on the other hand, really demands lifting
surface techniques since the chordwise vorticity distribution behind the aerofoil induces
a considerable chordwise velocity distribution as well as difficulties of singularities
in the region of the trailing edge. This matter will be dealt with in more detail in
section 6.2.2.
The induced velocity contributions given by eqns 6.4, 6.5, and 6.6 must now be
related to the circulation which gave rise to them. If W is the resultant chordwise
velocity at the blade, the local loading is
ψ φ
Fig. 6.8 Shed vortex element
∂
∂
∂
∂φ
Γ Γ φ
t
dt = d
202 Bramwell’s Helicopter Dynamics
d /d = 12
L r ρW2CLc (6.7)
and the Kutta–Zhukowsky relation is
dL/dr = ρWΓ
Equating eqns 6.7 and 6.8 gives
Γ = 12
WCLc
Let us suppose that the lift slope is constant. Then
CL = aα
and writing
α = α0 + α
i
where α0 is the incidence in the absence of the induced velocity and α
i is the downwash
angle relative to the blade, we have
Γ = ( + ) 12
aα0 αiWc
But
αi
≈ w/W
therefore
Γ = + ( + + ) 12
0
12
aWα c ac wt wb ws (6.9)
where wt, wb, ws are given by eqns 6.4, 6.5, and 6.6.
Equation 6.9 is the integral equation for the circulation of the reference blade.
Although the equation has been simplified by the assumption of a rigid wake, no
standard method exists for solving it. One possible way which suggests itself is to
assume a simple induced velocity distribution, such as a uniform or Glauert type, and
make a first approximation to the circulation. From this approximation the integrals
eqns 6.4 to 6.6 are calculated, and new values of Γ are obtained from eqn 6.9. The
process is repeated until satisfactory convergence has been obtained. It cannot be
certain that convergence will always occur.
It is not necessary, of course, to develop an integral equation such as eqn 6.9.
Some methods begin by assuming a prescribed wake model based upon experimental
data or reasonable physical arguments and performing piecewise calculations of their
contributions to the induced velocity, as discussed in the following section.
6.2.1 Prescribed wake model development
One of the first attempts to depart from the idea of the rotor as a lifting surface, and
to use reasonable physical arguments to model the vortex wakes from individual
blades in such a way as to make the problem tractable, was that of Willmer3. The
main feature that allowed effective computation, particularly in view of the speed of
Rotor aerodynamics in forward flight 203
computers at that time, was his principle of ‘rectangularisation’. Consider the trailing
wake from two successive blades, Fig. 6.9. Willmer argued that the radius of curvature
of the sheets is large enough (especially for the important outer parts of the blade) for
the sheets to be regarded as straight. Actually this is the same assumption as the one
made by Prandtl in his method of calculating ‘tip loss’ in hovering flight, section
2.10.1.
Under ‘rectangularisation’, the trailing wakes, Fig. 6.9, are replaced by those
shown in Fig. 6.10. The wake attached to the reference blade is assumed to be a
straight sheet extending back to infinity, while those of the other blades are assumed
to be doubly infinite sheets placed so that the straight vortex lines are tangential to
the curved wakes where they pass under the reference blade. Part of the problem in
Willmer’s work was to determine the appropriate positions of the wakes of the
blades. Any number of parts of the original curved wake could be included, and the
choice made depended on the required accuracy. Once the numbers and positions of
the sheets had been chosen, the induced velocity at the reference blade could be
calculated by an extension of Glauert’s wing theory. It should be noted that this does
not imply that the vortex wake is regarded as a number of discrete vortices, as is
sometimes thought, but that the vortex wake is continuous and conditions are satisfied
at a number of points along the span (method of collocation). Unfortunately, because
the loading of a rotor blade is very different from the elliptic, or near elliptic, loading
of the conventional fixed wing, a larger number of spanwise points are needed for
acceptable accuracy. A comparison of the calculation of the blade thrust by Willmer’s
　
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