The Design of UAV Systems
3.3 Rotary-wing Aerodynamics
Aerodynamics and Airframe Configurations 29
0 50 100 150 200 250 300
0 500 1000 1500 2000 2500 3000
Wing Loading [N/m2]
Minimum Flight Speed [m/s]
20,000 m
15,000 m
10,000 m 5,000 m Sea Level Global Hawk
Predator B
Figure 3.3 The variation of Vminwith aircraft wing loading
where w is the aircraft wing loading in N/m2,ρ0is the air density at sea-level standard conditions andσ is the relative air density at altitude.
The variation of Vmin with aircraft wing loading at several altitudes is shown in Figure 3.3 which covers wing loadings appropriate to the close-range, medium-range and MALE and HALE UAV. Figure 3.4 covers the lower values of Vmin, resulting from the wing loading relevant to the smaller mini and micro UAV.
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0 4 8 12 16 20
0 50 100 150 200
Wing Loading [N/m2]
Minimum Flying Speed [m/s]
MAV
MINI UAV
Figure 3.4 The variation of Vminwith aircraft wing loading (lower values)
At each element of the rotary wing, the aftward-inclined vector of the lift force produces a drag at the element which translates into a torque demand at the rotor hub. For the same reason as with the fixed wing, the rotary wing must induce a large mass of air for efficiency. Therefore the larger the diameter of the circle (or disc) traced out by the rotary wing, the more efficient it is.
Thus the disc-loading p N/m2, i.e. lift produced divided by the disc area, may be seen as the equivalent of span loading for the fixed wing. The difference, however, is that in the case of the fixed-wing lift, the lift is produced by the air being deflected over a sensibly single linear vortex so that the induced downwash and drag would theoretically increase to infinity at zero speed if the aircraft were able to fly at that condition. For the rotary wing, lift is produced by the air being deflected downwards over a large area covered by a number of rotating vortices so that the induced velocity and drag of the rotor remains at a finite value at zero airspeed. The velocity induced at the helicopter rotor in the hover is given by:
vi= kn( p/2ρ)1/2 (3.7)
where knis a correction factor to account for the efficiency of the lift distribution and the strength of the tip vortices generated at the rotor blade tips. In practice this can vary between about 1.05 to 1.2, with 1.1 usually being appropriate to rotors of moderate disc loading p.
The induced power in hover flight is then given by Pi= kn.Tvi,where T is the thrust produced by the helicopter rotor.
In forward flight the rotor is able to entrain a greater mass of air and so, as with the fixed wing, it becomes more lift-efficient and the induced power rapidly reduces with increasing speed.
At a forward speed of about 70 km/hr, the flow pattern through the rotor becomes similar to that passing a fixed wing. Therefore at that speed and above, the same expression may be used to calculate the helicopter-induced power and drag as that used for the fixed-wing i.e. Equation (3.1).
Aerodynamics and Airframe Configurations 31
1.0
0.8
0.6
0.4
0.2
00 1 2 3 4 5 6 7 8 9 10
k [ND]
V
√p
Figure 3.5 The ‘Hafner’ induced power factor
For airspeeds between zero and 70 km/hr, the airflow through a rotor is more complex and an empirical solution, credited to Raoul Hafner, is normally used. This takes the form of a curve whereby an induced factor k is plotted against a function of aircraft forward speed and disc loading (V÷√
p), The induced power calculated for the hover is multiplied by the factor k to obtain the induced power obtaining at the intermediate airspeeds. The ‘Hafner curve’ is shown in Figure 3.5.
3.3.2 Parasitic Drag
Again, the same elements, form drag, friction drag, momentum drag, etc., as discussed for the estimate of fixed-wing aircraft performance, apply to the rotary-wing aircraft. There is a small difference, however, in the accounting.
Due to the more complex ‘flight path’ of a rotor, its drag is accounted for separately under the heading of
‘profile drag’ or ‘profile power’. The drag of the remaining elements, i.e. fuselage, undercarriage, cooling drag, etc., then comprise the parasitic drag which is calculated in the same way as for a fixed-wing aircraft.
3.3.3 Profile Drag or Power
In hover flight, the profile power required to turn the rotor against the profile drag is computer-calculated by the summation along the length of the rotor blade of the drag of the blade split into elemental sections and multiplied by the local element velocity and radius from the hub. This is referred to as blade element theory. However, for simpler analysis, an average value of drag coefficientδ is used and mathematical integration produces the following expression;
Profile power Po= 1
8ρ δ AVT3 (3.8)
where A is the total blade area and VTis the speed of the rotor tip.
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In forward flight, each blade element no longer describes a circular orbit, but a longer, spiral path which becomes increasingly asymmetric as the forward speed increases. The result is an increase of the profile power compared with that of the hover. Power is expended in translating the rotor as well as rotating it.
An exact expression of the multiplier to obtain this power is a power series equation inµ, the ‘advance ratio’ – i.e. the ratio of the forward speed to the rotor tip speed. This multiplier is usually simplified to the close approximation: (1+ 4.73 µ2), so that the profile power in forward flight is calculable as:
Poµ= Po(1+ 4.73 µ2). (3.9)