The power requirements of flight at the size and speeds of birds have long been of interest to biologists, and are increasingly studied by engineers considering small Unmanned Air Vehicles. Techniques for calculating power are generally based on those that have proved successful for the much larger, faster machines that we regularly fly in. However, fluids (including air) behave quite differently at different sizes and speeds, so a new range of experimental validation, and perhaps also theoretical formulation, may be required for flight on smaller scales. Geoff Spedding and John McArthur from the University of Southern California report measurements on wings under conditions appropriate for bird flight, and highlight some troubling departures from the assumptions inherent in conventional drag – and so power – analyses.

Three components contribute to conventional drag calculations. The first form of drag is associated with pushing air downwards to provide lift upwards and is known as ‘induced’ drag. Even without air viscosity (‘stickiness’), this continuous acceleration of surrounding air requires power, and results in drag that is proportional to the square of the lift force. Well-designed wings do this efficiently, accelerating air as little as possible. However, this is never perfect, and the additional drag is accounted for in power calculations by a fudge factor, expressed either as a ‘span efficiency’ (<1) or its inverse, the ‘induced-power correction factor’ (>1) which, in the animal flight literature, is usually taken as close to 1.2.

Second, the duo begin to include the effects of viscosity. Even at zero lift, there is a small amount of drag. The exact value of this drag is often difficult to measure, even under extremely controlled windtunnel conditions; it can be a thousand times smaller than the other forces to be measured. In the bird flight literature, this drag, in coefficient form, tends to take the value 0.02.

The difficulty really starts with a third drag relationship. This drag is again due to viscosity but, confusingly, tends to rise in proportion to the square of lift (just like the ‘inviscid’ induced drag component). It is therefore often convenient to include this effect as an altered fudge factor in the induced drag component, as an adjustment to the ‘span efficiency’ (or its inverse). But this mixing up of aerodynamic mechanisms for mathematical convenience can cause considerable confusion.

Spedding and McArthur highlight the issues using their detailed measurements on a wing – the Eppler 387. They demonstrate that, at the sizes and speeds appropriate for much of animal flight, the relationship between drag and the square of lift is really very poor and highly dependent on speed. This is not just ‘stall’ – with increasing angle of attack, lift can suddenly increase at the same time as drag decreases. The derived induced power factor (including the viscous adjustment) comes out at 3.7 – more than triple that usually adopted in bird flight calculations.

The authors used a smooth Eppler 387 in part because it is known for its weird characteristics at low speeds; it may very well be that some aspect of animal wings returns aerodynamic performance to lower drag and easier theoretical tractability. But whether this is the case, and which aspect this turns out to be, awaits further experimentation. So the animal flight community looks destined to continue with the ‘1.2ish, 0.02ish’ rule of thumb, but be continually haunted by the possibility that the aerodynamic power calculations might not just be wrong, but very – and systematically – wrong.

G. R.
Span efficiencies of wings at low Reynolds numbers
J. Aircr.