Miniature flying machines

A NEW mathematical theory of insect flight could help engineers copy flies' tricks to design new miniature flying machines. It is the first concrete explanation of how some insects can manipulate the flow of air around them so as to switch in an instant between darting motions and hovering stillness.

There is an old myth that conventional aerodynamics indicates that the flight of the bumble-bee is impossible. This is true only if one treats a bee like an aeroplane, which of course is absurd: insects, unlike aircraft, stay aloft by flapping their wings very rapidly. So air, instead of flowing smoothly over the wing, is whipped into a frenzy of curling vortices.

But a beating wing does not produce chaos. Insects coordinate their wing movements with exquisite timing to generate a lift force. They key is how wings shed vortices from their edges. Vortices are moving parcels of air which carry away momentum, so like the air streams from a propeller they can generate a 'back' force on the object that sheds them.

Previous studies of insect flight have suggested that rotation of the wings during flapping is a crucial part of the mechanism by which an insect controls lift forces and alters direction during flight. During the downstroke, an insect's wing is oriented mostly parallel to the ground. At the end of the stroke, it rotates the wing to lie roughly perpendicular to the ground: a kind of figure-of-eight motion. Z. Jane Wang of Cornell University has developed a theory of how this rotating motion creates vortices that let an insect hover, as she now reports in Physical Review Letters .

All the subtleties of fluid flow - whether it be water travelling down a river channel or air blowing in the treetops - are captured in a single mathematical equation deduced in the nineteenth century: the `Navier-Stokes equation'. It is easy to write this equation down - but in all but a handful of simple cases, impossible to solve it with pen and paper.

Wang has done so using a computer to divide up the airflow around the wing into a fine grid and calculate the flow at each point on the grid. Performing the calculations for three-dimensional space would be too computer-intensive. Wang made the number-crunching possible by gambling that the flow in two-dimensional slices of space perpendicular to the wing would be the key consideration. By looking at the airflow patterns she was able to gain some insight into how an insect orchestrates the vortices.

During the downstroke, the wing creates a pair of counter- rotating vortices: one at the leading edge and one at the trailing edge. The rotation at the end of the stroke then combines these into a `dipole', a coherent jet-like flow structure in which the vortex pair moves together in a single direction. If the timing is right, the dipole moves downwards, generating a lift force on the wing while stripping it free of vortices so as to be ready for another pair in the following stroke.

Wang calculated how much lift this generates for a typical flapping cycle. For a dragonfly, it is more than adequate to support the insect's weight. And the force rises to a steady value after just a few wing strokes, enabling the insect to take off rapidly. Maybe now we will be able to work out how it is that some insects can take off backwards, fly sideways and land upside down.

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