An energy analysis of a fluttering flag (Mathematical aspects of nonlinear waves and their applications)

Abstract

This work deals with the dynamics and stability of a flapping flag, with emphasis on the onset of flutter instability. The flag is considered as a thin plate. The fluid forces are evaluated by employing the theory of Theodorsen (1935). The aim of the work is to (i) obtain a physical understanding of the mechanism of the flutter oscillations, and (ii) to obtain a better understanding of dynamically varying fluid forces, expressed via the complex Theodorsen function C, as opposed to the quasisteady approximation, where C=1-i0. An energy balance analysis shows that (i) flutter can occur only in a 'dragging' sort of motion, in other words, in the form of downstream travelling wave motion, and (ii) a small imaginary part of the Theodorsen function, C=1-iepsilon, 0<epsilonll 1, has a destabilizing effect, in the sense that the critical flow speed is smaller than by the quasisteady approximation C=1-i0. These predictions have been verified by numerical eigenvalue analyses. The result (ii) is in opposition to previously reported results. Furthermore, it is found that certain terms in the equation of motion of the flag that previously have been discarded, on the assumption that they are associated with very slow changes across the flag, have a significant effect on the stability of the flag.