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behind the wing the induced velocity becomes independent of the rearward distance,
and such a flow can be produced by a two-dimensional strip whose width is equal to
the span and which moves perpendicular to its plane with velocity w, Fig. 2.17. The
circulation about the wing, corresponding to this optimum case, can be evaluated by
60 Bramwell’s Helicopter Dynamics
calculating the line integral C1 joining the points P and P′ which are, respectively,
just above and just below the wing, Fig. 2.17. Now, since the flow is irrotational, the
line integral C1 is equal to the line integral C2, and this integral is equal to the
difference of potential between the points P and P′, which are equidistant from the tip
of the strip, Fig. 2.18. The velocity potential, and hence circulation Γ, for a twodimensional
strip moving normal to itself is found to be elliptical with a maximum
value Γ0 = 2ws at the wing centre, 2s being the width of the sheet (and the wing span).
For the wing itself, however, the maximum circulation should be given by Γ0 = 4ws,
since the wing and its trailing wake represent only half the assumed two-dimensional
strip, and the induced velocity for a given circulation is half the two-dimensional
value.
The corresponding problem for the rotor is to find the velocity potential for a
series of helical vortex sheets, Fig. 2.19, one for each blade, moving with constant
velocity w relative to the otherwise undisturbed air. Once this problem has been
solved, the effect of the number and relative spacing of the sheets on the blade
circulation can be investigated. It is also possible to determine the relationship between
the induced velocity at the blade and the mean velocity between the sheets.
The formidable problem presented by this motion was eventually solved by Goldstein4
in 1929, but here we shall merely describe a simpler method due to Prandtl15, especially
as Prandtl’s result has been used to calculate the rotor blade ‘tip loss’. Prandtl’s
method, which contains all the essentials of Goldstein’s problem, was to replace the
curved sheets of the helical surface by a series of two-dimensional sheets on the
C2
P′
P
C1
Fig. 2.17 Induced velocity distribution behind elliptically loaded wing
w
C2
P′
Fig. 2.18 Circulation about wing
w
P
Rotor aerodynamics in axial flight 61
w
Fig. 2.19 Vortex sheet arrangement near rotor blade
•P′
•P
y
x
w
Fig. 2.20 Two-dimensional flow about vortex sheets
d T d L
W
vi
Vc
Ωr φ
Fig. 2.21 Velocity and force components at a blade element
assumption that the radius of curvature of the outer parts of the sheets is so large that
they can be considered as doubly infinite straight strips, Fig. 2.20.
Before considering the flow about these sheets, let us rewrite the thrust equations
for an element of a blade in terms of the local circulation about the element. Let the
induced velocity at the blade element be vi, the axial velocity be Vc, and the overall
resultant velocity be W, Fig. 2.21.
For an infinite number of blades the thrust on an annulus of radius r and width dr
is, from momentum considerations,
dT = 2πrρ(Vc + vi cos φ) 2vi cos φ dr (2.58)
where we have assumed that the induced velocity in the far wake is twice that at the
disc.
From blade element theory we also have
d = d cos = cos d 12
T L φ bW 2ρCLc φr (2.59)
α
ρ
62 Bramwell’s Helicopter Dynamics
where dL is the lift on the element, b is the number of blades, and CL is the local lift
coefficient.
Now, from the Kutta–Zhukowsky theorem,
12
ρW2CLc = ρWΓ (2.60)
where Γ is the circulation about the element. Then from eqns 2.58, 2.59, and 2.60 we
obtain
dT = ρbWΓ cos φ dr
But from Fig. 2.21 we see that
W sin φ = Vc + vi cos φ
so that on eliminating dT we finally get
vi = bΓ/4πr sin φ (2.61)
which gives a relationship between the local induced velocity and the circulation
when the number of blades is infinite.
Returning now to Prandtl’s representation of the vortex sheets, Fig. 2.20, let the
infinite array of sheets move relative to the surrounding air with velocity w. The
complex potential of such a flow is known* to be
φ ψ π
π
+ i = ws cos–1 e p
z
s (2.62)
where sp is the spacing of the sheets, which is easily seen to be given by
sp = (2πr/b) sin φ (2.63)
It can be verified that eqns 2.62 and 2.63 satisfy the requirements of the problem.
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