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method with that of experiment and of the theory using constant induced velocity for
Fig. 6.9 Plan view of successive vortex sheets
Fig. 6.10 ‘Rectangularisation’ of vortex sheet
204 Bramwell’s Helicopter Dynamics
μ = 0.08 is shown in Fig. 6.11. Although the agreement with experiment can only be
described as fair, it is clear that the new model provided a great improvement on the
theory using constant induced velocity and went far to explain the large thrust variations
which often lead to large vibration at low values of μ.
Two important defects of the work were that it took no account of shed vorticity
and, related to this, it assumed that the strength of the trailing vortex elements were
constant along their length.
6.2.2 The shed vorticity
The question of shed vorticity and investigations into suitable representations of the
vortex wake have been considered by Miller2,4. Since Miller and others have given
considerable attention to the effect of the shed wakes, it may be useful to give a short
account of the methods of calculation here.
Consider an aerofoil, Fig. 6.12, whose incidence is changing. Any change of
circulation about the aerofoil must be accompanied by a corresponding vortex or
vorticity of the opposite sense in the wake, since Kelvin’s theorem requires that the
total circulation in a circuit containing the aerofoil and the wake must remain constant
with time. Thus, an aerofoil whose incidence is continually changing must deposit a
sheet of vorticity in the wake whose local strength is related to the time history of rate
of change of circulation. Now, the presence of vorticity in the wake plays a part in
satisfying the Kutta–Zhukowsky condition at the trailing edge of the aerofoil, so that
the circulation about the aerofoil is different from what would have been the case had
the vorticity been absent. Also, since the motion we are discussing is unsteady, there
1.75
1.50
1.25
1.0
0.75
0.50
Experimental
Willmer
Constant induced velocity
90° 180° 270° 360°
Lift
Mean lift
Fig. 6.11 Variation of blade loading with azimuth angle
Γ
Fig. 6.12 Vortices shed by aerofoil changing its incidence
Rotor aerodynamics in forward flight 205
is an extra pressure, or ‘virtual mass’, term which relates to the acceleration of the air
particles in the flow about the aerofoil.
The most important case of unsteady motion is a sinusoidal variation of incidence.
The first complete theory was given by Theodorsen5, who showed that for an aerofoil
whose incidence is changing by performing a sinusoidal vertical (heaving) motion,
the lift L as a fraction of the steady lift L0 at the instantaneous incidence is given by
L/L0 = C(k) + i(k)
12
(6.10)
where C(k) is Theodorsen’s function and k is the frequency parameter nc/2V, n being
the frequency of oscillation and c the aerofoil chord. Theodorsen’s function is defined
by
C(k) = K1(ik)/[K0(ik) + K1(ik)]
where K0(ik) and K1(ik) are Bessel functions of the second kind.
For an aerofoil oscillating about its mid-chord, the corresponding result is
L/L0 = [i + (i + 2/k)C(k)]
12
(6.11)
These results can be expressed in vector form as shown in Figs 6.13 and 6.14.
The results eqns 6.10 and 6.11 can be obtained in a number of ways6 but, since the
‘method of vortices’ is closely related to the ideas being discussed in this chapter, it
will be described briefly below.
Corresponding to changes of circulation dΓ about the aerofoil, vortex elements of
strength γ bdξ, equal and opposite to the circulation, are deposited in the wake, γ
being the vorticity and ξ the distance of the element from the centre of the aerofoil,
non-dimensionalised in terms of the semi-chord b, Fig. 6.15.
1.0
0.9
0.8
0.7
0.6
0.5
0.4
0.3
0.2
0.1
0
–0.1
–0.2
–0.3
–0.4
Component in quadrature with w
0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0
0.1
1.6
1.2
k = 2
1.0
0.8
0.6
0.4 Component in phase with w
1.8
1.2
0.4 0.2 0.1
0.04
L
2πρVw
M
πρVw
0.20.10.04
Fig. 6.13 Amplitude and phase of lift and moment of aerofoil oscillating in heave
206 Bramwell’s Helicopter Dynamics
1.0
0.9
0.8
0.7
0.6
0.5
0.4
0.3
0.2
0.1
0
–0.1
–0.2
–0.3
–0.4
–0.5
0.1 0.2 0.3 0.4 0.5 0.6
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