The dynamics of an inverted flag are investigated experimentally in order to find the conditions under which self-excited flapping can occur. In contrast to a typical flag with a fixed leading edge and a free trailing edge, the inverted flag of our study has a free leading edge and a fixed trailing edge. The behaviour of the inverted flag can be classified into three regimes based on its non-dimensional bending stiffness scaled by flow velocity and flag length. Two quasi-steady regimes, straight mode and fully deflected mode, are observed, and a limit-cycle flapping mode with large amplitude appears between the two quasi-steady regimes. Bistable states are found in both straight to flapping mode transition and flapping to deflected mode transition. The effect of mass ratio, relative magnitude of flag inertia and fluid inertia, on the non-dimensional bending stiffness range for flapping is negligible, unlike the instability of the typical flag. Because of the unsteady fluid force, a flapping sheet can produce elastic strain energy several times larger than a sheet of the deformed mode, improving the conversion of fluid kinetic energy to elastic strain energy. According to the analysis of the leading-edge vortex formation process, the time scale of optimal vortex formation correlates with efficient conversion to elastic strain energy during bending.

The dynamics of a cantilevered elastic sheet, with a uniform steady flow impinging on its clamped end, have been studied widely and provide insight into the stability of flags and biological phenomena. Recent measurements by Kim et al. (J. Fluid Mech., vol. 736, 2013, R1) show that reversing the sheet's orientation, with the flow impinging on its free edge, dramatically alters its dynamics. In contrast to the conventional flag, which exhibits (small-amplitude) flutter above a critical flow speed, the inverted flag displays large-amplitude flapping over a finite band of flow speeds. The physical mechanisms giving rise to this flapping phenomenon are currently unknown. In this article, we use a combination of mathematical theory, scaling analysis and measurement to establish that this large-amplitude flapping motion is a vortex-induced vibration. Onset of flapping is shown mathematically to be due to divergence instability, verifying previous speculation based on a two-point measurement. Reducing the sheet's aspect ratio (height/length) increases the critical flow speed for divergence and ultimately eliminates flapping. The flapping motion is associated with a separated flow -detailed measurements and scaling analysis show that it exhibits the required features of a vortex-induced vibration. Flapping is found to be periodic predominantly, with a transition to chaos as flow speed increases. Cessation of flapping occurs at higher speeds -increased damping reduces the flow speed range where flapping is observed, as required. These findings have implications for leaf motion and other biological processes, such as the dynamics of hair follicles, because they also can present an inverted-flag configuration.

Slender flexible structures such as plants are deformed by external flow. When the deformation is significant, this results in a reduction of drag. We give a theoretical value of the exponent that characterizes the drag law. This theoretical value is shown to compare well with experimental data on a very large variety of plants. It is found that reconfiguration affects more the local bending stress than the total drag. Moreover, a nonlinearity in the bending law does not affect significantly the mechanism.