The forced motion of a flag

Abstract

The prevailing view of the dynamics of flapping flags is that the onset of motion is caused by temporal instability of the initial planar state. This view is re-examined by considering the linearized two-dimensional motion of a flag immersed in a high-Reynolds-number flow and taking account of forcing by a ‘street’ of vortices shed periodically from its cylindrical pole. The zone of nominal instability is determined by analysis of the self-induced motion in the absence of shed vorticity, including the balance between flag inertia, bending rigidity, varying tension and fluid loading. Forced motion is then investigated by separating the flag deflection into ‘vortex-induced’ and ‘self’ components. The former is related directly to the motion that would be generated by the shed vortices if the flag were absent. This component serves as an inhomogeneous forcing term in the equation satisfied by the ‘self’ motion. It is found that forced flapping is possible whenever the Reynolds number based on the pole diameter ReD ≳ 100, such that a wake of distinct vortex structures is established behind the pole. Such conditions typically prevail at mean flow velocities significantly lower than the critical threshold values predicted by the linear theory. It is therefore argued that analyses of the onset of flag motion that are based on ideal, homogeneous flag theory are incomplete and that consideration of the pole-induced fluid flow is essential at all relevant wind speeds.