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Now consider the points P and P′ just above and just below one of the sheets
(x < 0), Fig. 2.20. The required circulation, as explained earlier, is equal to the
difference of potential between these points. Then taking, for simplicity, the sheet on
the x axis, on which ψ = 0, the potential difference across the sheet at a distance a
from the edge of the sheet is easily found from eqn 2.62 to be
φ φ π
π
P P
–1
–
– = = 2 cos e p
′ Γ ws
a
s
= wsk
where k
a
s = 2 cos –1 e
–
p
π
π
* φ in eqn 2.62 is the standard symbol for velocity potential and should not be confused with the
inflow angle.
Rotor aerodynamics in axial flight 63
Hence,
w = Γ/spk
Now, if we can take the induced velocity at the blade to be half this value, i.e. if
we suppose vi = 12
w, we have, on substituting from eqn 2.63 for sp
vi = bΓ/4πrk sin φ (2.64)
This is precisely the same expression as eqn 2.61 except for the factor k, which can
be regarded as a correction factor for the number of blades. Since k is always less
than unity, the induced velocity for a given circulation is always larger the fewer the
number of blades. Put in another way, there is a loss of circulation near the blade tips
when the number of blades is finite.
If a is interpreted as the distance R – r from the blade tips and s is based on the
value at the blade tip, k can be written as
k
b x
= 2 cos –1 e
– (1 – )
sin
π
φ
= 2 cos –1 e
π
– f (2.65)
where f b x = (1 – )/sin 12
φ . This relationship is shown plotted in Fig. 2.22.
Goldstein’s more exact analysis for the helical vortex sheets resulted in an equation
identical with eqn 2.64 but k did not have the simple form of eqn 2.65. Goldstein’s
values of k, which are a function of the number of blades, the radial position of the
element in question, and the inflow angle φ, are given in Fig. 2.23.
Proceeding with the rotor analysis, we see from Fig. 2.21, that
tan ρ = vi/W
= bcCL/8πrk sin φ
from eqns 2.60 and 2.64, or
tan ρ = σ CL/8k sin φ (2.66)
1.0
0.8
0.6
0.4
0.2
0 0.5 1.0 1.5 2.0 2.5 3.0
f
Fig. 2.22 Variation of circulation factor as function of f
k
64 Bramwell’s Helicopter Dynamics
In hovering flight ρ = φ and, since CL = a(θ – φ), eqn 2.66 becomes
8kφ2 = σa(θ – φ) (2.67)
which is precisely the same as eqn 2.52 except for the factor k.
The value of k in eqn 2.64 has been derived on the assumption that the ideal wake
conditions exist, and this, in turn, implies a certain range of blade loadings depending
on the inflow angle and number of blades. Hence, any calculations using these values
of k are strictly valid only for these particular loadings. Lock16, however, widened the
range of application by assuming that the values of k would apply with reasonable
accuracy to any practical load distributions.
The rigid wake analysis of Goldstein implies a certain variation of the inflow
angle φ with respect to the radius; Lock’s assumption allows us to use φ freely, just
as we did for the calculations shown in Table 2.1.
0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0
1.0
0.8
0.6
0.4
0.2
b = 2
0.3
0.1
0.2
1.0
0.8
0.6
0.4
0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0
0.1
0.2
0.3
0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0
1.0
0.8
0.6
0.4
k
k
0.05
r/R
φ = 0.05
0.3
0.2
0.1
r/R
φ = 0.05
Fig. 2.23 Goldstein circulation factors
r/R
b = 3
b = 4
k
Rotor aerodynamics in axial flight 65
Let us apply the Goldstein–Lock analysis to the rotor whose induced velocity and
local lift coefficient were calculated in section 2.5. This time we solve eqn 2.67 with
the appropriate value of k instead of taking it as unity. Actually, as φ is unknown, the
correct value of k cannot be found immediately, but let us first calculate φ with k =
1 (corresponding to an infinite number of blades) and then find k from Fig. 2.22; if
necessary we can use the new value of φ to obtain a better value of k, and so on. The
convergence is usually very rapid. For the case under consideration, k becomes worth
considering only for radial distances greater than 0.9 R. The results of the calculations
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Bramwell’s Helicopter Dynamics(37)
