Flag flapping instability

Settled in 2005 by Jun Zhang, Stephen Childress, Albert Libchaber, Michael Shelley (NYU/Chicago); Mederic Argentina, L. Mahadevan (Cornell).

Flags flapping in steady wind look chaotic but the underlying instability had no quantitative theory for a long time, because it requires solving the two-way coupling between a flexible sheet and the vortices it sheds. Zhang, Childress, Libchaber and Shelley built a clever analogue — flexible filaments in a flowing soap film — and in 2000 showed there are two genuinely stable states, a straight non-flapping state and a periodic flapping state, overturning the assumption that flags must always flap. In 2005 Argentina and Mahadevan supplied a full linear-stability theory including the added mass and drag of the surrounding fluid, correctly predicting the onset speed of flapping as a function of flag length and stiffness, which is now the standard explanation.

The rippling of a flag in the wind, one of the most common sights in nature, hides a textbook flutter instability shared with aircraft wings and heart valves.

Explanations that circulated

• A flag in steady wind is always aerodynamically unstable and must flap — the 2000 soap-film experiments showed a stretched-straight, non-flapping state is also stable, contradicting the old assumption that flapping is inevitable; which state occurs depends on flow speed and the filament's length and rigidity.

References

• Flexible filaments in a flowing soap film as a model for one-dimensional flags in a two-dimensional wind, J. Zhang, S. Childress, A. Libchaber, M. Shelley, Nature, 2000
• Fluid-flow-induced flutter of a flag, M. Argentina, L. Mahadevan, Proceedings of the National Academy of Sciences, 2005