On the mechanism of flutter of a flag

This paper is concerned with the dynamics and stability of a flapping flag, with emphasis on the onset of flutter instability. The mathematical model is based on the one derived in a paper by Argentina and Mahadevan (Proc Nat Acad Sci 102:1829–1834, 2005). In that paper, it is reported that the effect of vortex shedding from the trailing edge of the flag, represented by the complex Theodorsen function C, has a stabilizing effect, in the sense that the critical flow speed (where flutter is initiated) is increased significantly when vortex shedding is included. The numerical eigenvalue analyses of the present paper display the opposite effect: the critical flow speed is decreased when the Theodorsen function (i.e., vortex shedding) is included. These predictions are verified by an analytical energy balance analysis, where it is proved that a small imaginary part of the Theodorsen function, \(C = 1 - \mathrm {i} \, \epsilon \), \(0 < \epsilon \ll 1\), has a destabilizing effect, i.e., the critical flow speed is smaller than by the so-called quasi-steady approximation \(C = 1 - \mathrm {i} \, 0\). Furthermore, order-of-magnitude considerations show that Coriolis and centrifugal force terms in the equation of motion, previously discarded on the assumption that they are associated with very slow changes across the flag, have to be retained. Numerical results show that these terms have a significant effect on the stability of the flag; specifically, the said destabilizing effect of the vortex shedding is significantly reduced when these terms are retained. The mentioned energy balance analysis illuminates the nature of the flutter oscillations and the ‘competition’, at the flutter threshold, between the different types of fluid forces acting on the flag.