Heavy Flags Undergo Spontaneous Oscillations in Flowing Water

Abstract

By immersing a compliant yet self-supporting sheet into flowing water, we study a heavy, streamlined, and elastic body interacting with a fluid. We find that above a critical flow velocity a sheet aligned with the flow begins to flap with a Strouhal frequency consistent with animal locomotion. This transition is subcritical. Our results agree qualitatively with a simple fluid dynamical model that predicts linear instability at a critical flow speed. Both experiment and theory emphasize the importance of body inertia in overcoming the stabilizing effects of finite rigidity and fluid drag.

Effects of body elasticity on stability of underwater locomotion

“…This problem is somewhat reminiscent to that of the fluttering flag instability. The latter has been the focus of a large number of experimental ( [21]), computational ( [1,18]), and theoretical ( [2]) investigations. Of course, the flag is held stationary at one end in an incoming uniform flow field whereas the fish is freely moving.…”

Response of a flexible filament in a flowing soap film subject to a forced vibration

“…The wake instability of a cylinder exists even when the cylinder is fixed. However, in the onset problem of a flag flapping, the study by Shelley, Vandenberghe, and Zhang 7 shows that the instability of a flag can be considered as a local phenomenon without considering the effect of vortex shedding. In the forced vibration of a cylinder, the cylinder is rigid.…”

“…Their experimental observations showed that an initiallystationaryfilamentstarteditsflappingmotionwhenbothitslengthandtheexternalflowspeed exceeded some threshold values. A further study by Shelley, Vandenberghe, and Zhang 7 showed that the instability of a flag also relied on the density ratio of the flag to fluid. They employed a temporal linear instability analysis to study the onset of the flag flapping.…”

“…11 Their theory predicted the critical speed for the onset of the flapping as well as the frequency of the flapping. Both the analyses of Shelley, Vandenberghe, and Zhang 7 and Argentina and Mahadevan 10 are based on linearized theories. The onset of flag flapping is in the small amplitude regime, where a linear analysis is applicable.…”

“…Experimental studies on the flapping behavior of a flag have been conducted in soap films, water tunnels, and low-speed wind tunnels. 1,7,8 By placing flexible filaments in a flowing soap film, Zhang et al 1 studied the model of onedimensional flags in a two-dimensional wind. Their experimental observations showed that an initiallystationaryfilamentstarteditsflappingmotionwhenbothitslengthandtheexternalflowspeed exceeded some threshold values.…”