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straight vortex filaments whose strengths are determined by the bound circulation
distribution. Using the Biot–Savart law, the induced velocity components due to
these filaments are then computed and the initially assumed velocity field is modified
accordingly. The positions of the vortex elements, which ‘float’ with the fluid, are
modified in turn to conform with the velocity field. The induced velocity at the blade
and the bound-circulation distribution are also recalculated. The process is allowed to
iterate until satisfactory convergence has been achieved, indicating that the wake
geometry is consistent with the velocity field it induced. The calculations were
considerably simplified by defining a ‘far wake’ as that part of the wake beyond a
distance corresponding to two rotor revolutions and representing this far wake by a
stack of stepped vortex rings approximating to a helix whose spacing is determined
by the number of blades and mean local induced velocity.
The calculations of Clark and Leiper clearly identified the wake features found
experimentally by Landgrebe19, namely, that the outer trailing vortices roll up quickly
to form a strong tip vortex, while the inner vortices move downwards as a vortex
sheet which becomes progressively more inclined to the rotor plane. Their model
also showed that the initial position of the tip vortex depends strongly on the number
of blades; if the number of blades is high, the tip vortex remains roughly in the plane
of the rotor until it becomes close to the succeeding blade, when it is convected
downwards. As stated before, this feature had been noted by Landgrebe and is indicated
by the change of slope of ZT in Fig. 2.33.
The later model of Favier et al.26 was similar to that of Clark and Leiper in that a
rigid far wake was assumed (semi-infinite circular vortex cylinder beyond ψ = 10π/
b, and a constant pitch helix between ψ = 5π/b and 10π/b). A free wake was assumed
from the blade to ψ = 2π/b and a prescribed wake based on Landgrebe19 from here
to the start of the rigid wake. This confirmed the experimental observations previously
mentioned.
8°
6°
4°
2°
0
0.2 0.4 0.6 0.8 1.0
Root
cutout
Section angle of attack
θ0.75 = 12°
Radial co-ordinate, r/R
Goldstein–Lock analysis
Experimental prescribed wake
Fig. 2.36 Effect of experimental prescribed wake (Landgrebe) on incidence distribution
Rotor aerodynamics in axial flight 75
The free wake model of Brown and Fiddes27 develops a three turn free wake from
a constant pitch helical wake using a relaxation process, with the influence of the far
wake being accounted for. An interesting feature is the merging of adjacent vortices
according to a criterion in order to prevent mutual orbiting. The method was validated
against the experimental results of Carradonna and Tung28 on a constant chord,
untwisted, two-bladed rotor. The converged wake geometry is shown in Fig. 2.37. In
this case, a panel method was used on the blade itself, rather than a lifting line or
single bound vortex.
References
1. Houghton, E. L. and Carpenter P. W., Aerodynamics for engineering students, London, Edward
Arnold, 1993.
2. Durand, W. F. (ed)., Aerodynamic theory, vol. IV, section L, New York, Dover Publications,
1963.
3. Betz, A., Schraubenpropellers mit geringstem Energieverlust, Gottinger Nachrichten, 1919,
p. 193.
4. Goldstein, S., ‘On the vortex theory of screw propellers’, Proc. Roy. Soc., 123, 1929.
5. Theodorsen, T., Theory of propellers, New York, McGraw-Hill, 1948.
6. Bramwell, A. R. S., ‘A note on the static pressure in the wake of a hovering rotor’, Res. Memo.
City Univ. Lond. Aero. 73/3, 1973.
Fig. 2.37 Converged wake geometry for Caradonna and Tung28 rotor (from Brown and Fiddes27)
76 Bramwell’s Helicopter Dynamics
7. Brotherhood, P., ‘Flow through a helicopter rotor in vertical descent’, Aeronautical Research
Council R&M 2735, 1949.
8. Brotherhood, P., ‘Flight measurements of the stability and control of a Westland “Whirlwind”
helicopter in vertical descent’, RAE TR 68021, 1968.
9. Goorjian, P. M., ‘An invalid equation in the general momentum theory of the actuator disc’,
AIAAJ, 10(4) 1972.
10. Bramwell, A. R. S., ‘Some remarks on the induced velocity field of a lifting rotor and on
Glauert’s formula’, Aeronautical Research Council CP 1301, 1974.
11. Bailey, F. R., jnr, ‘A simplified theoretical method of determining the characteristics of a
lifting rotor in forward flight’, NACA Rep. 716, 1941.
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