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Large-amplitude flapping of an inverted flag in a uniform steady flow – a vortex-induced vibration

Published online by Cambridge University Press:  18 March 2016

John E. Sader ,
Julia Cossé ,
Daegyoum Kim ,
Boyu Fan  and
Morteza Gharib
John E. Sader*
Affiliation:
School of Mathematics and Statistics, The University of Melbourne, Victoria 3010, Australia Kavli Nanoscience Institute and Department of Physics, California Institute of Technology, Pasadena, CA 91125, USA
Julia Cossé
Affiliation:
Division of Engineering and Applied Science, California Institute of Technology, Pasadena, CA 91125, USA
Daegyoum Kim
Affiliation:
Department of Mechanical Engineering, KAIST, Daejeon 34141, Republic of Korea
Boyu Fan
Affiliation:
Division of Engineering and Applied Science, California Institute of Technology, Pasadena, CA 91125, USA
Morteza Gharib
Affiliation:
Division of Engineering and Applied Science, California Institute of Technology, Pasadena, CA 91125, USA
*
†Email address for correspondence: jsader@unimelb.edu.au
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Abstract

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.

JFM classification

Aerodynamics: Aerodynamics Vortex Flows: Vortex shedding

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Papers
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Journal of Fluid Mechanics , Volume 793 , 25 April 2016 , pp. 524 - 555
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© 2016 Cambridge University Press 

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Sader et al. supplementary movie

Measurement of flapping motion of the inverted-flag showing the effect of the pole (air speed is 3.5 m/s).

Download Sader et al. supplementary movie(Video)
Video 1.5 MB