Falling leaves and tumbling sheets of paper can momentarily float upward against gravity as they flutter through still air. Now, a new analysis explains the curious effect and points to novel aerodynamic mechanics for producing lift.
The fluttering, looping, and tumbling motions of falling sheets of paper and other flat objects have fascinated thinkers since the 19th century, when famed Scottish physicist James Clerk Maxwell studied falling cards. But previous analyses have not realistically explained how a tumbling sheet rises against gravity as it flips over. Researchers have assumed that the sheet acts much like an airplane wing, tilted slightly upward as it slices through the air. That produces a "lift" force pushing the sheet up that is proportional to the square of the speed with which it moves through the air. In principle, this lift can make the paper rise as it noses up and flips over. But in reality, this explanation has a problem: The paper rises only if the air has no viscosity--which certainly isn't the case for crisp, fall days.
There is another way to produce aerodynamic lift, report Z. Jane Wang and Umberto Pesavento of Cornell University in Ithaca, New York. Using a computer to model the flow of air around a falling sheet, they found that the sheet also experiences a lift force proportional to the speed of the sheet times the rate at which it is turning. This contribution to the lift remains high as the sheet flips because it is then rotating rapidly. In fact, it provides enough lift to make the sheet move upward against gravity, as the researchers report in the 1 October issue of Physical Review Letters. Moreover, the novel lift mechanism overwhelms the airfoil effect even between flips, when the sheet moves through the air quickly but rotates only slowly.Other researchers had developed more simplistic ad hoc simulations that generated upward motion, but the new work derives from fundamental "Navier-Stokes equations" that govern aerodynamics, says Elisha Moses, an experimental physicist at the Weizmann Institute of Science in Rehovot, Israel. Such detailed simulations should also allow researchers to study the whirlwinds or "vortices" produced by the tumbling sheet, Moses says, which is "what you want to see."