Large-amplitude flapping of an inverted flag in a uniform steady flow – a vortex-induced vibration
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.
© 2016 Cambridge University Press. Received 29 August 2015; revised 21 December 2015; accepted 15 February 2016; first published online 18 March 2016. This research was supported by a grant of the Gordon and Betty Moore Foundation, the Human Resources Program in Energy Technology of the Korea Institute of Energy Technology Evaluation and Planning (KETEP), granted financial resource from the Ministry of Trade, Industry and Energy, Republic of Korea (no. 20134030200300), Caltech's Kavli Nanoscience Institute and the Australian Research Council grants scheme.
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