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

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.

Nonlinear static response of low-aspect-ratio inverted flags subjected to a steady flow

“…As seen in the plot, the critical flow velocity decreases monotonically as the aspect ratio is increased. This is in good agreement with experimental observations and CFD model predictions in the literature [27,32]. For instance, the present theory predicts … P D 42:5 and … S D 8:9 for a slender inverted flag with A D 0:067, whereas it predicts … P D 4:6 and … S D 5:2 for a square one with A D 1.…”

“…In this case, it can be assumed that the flow reaches a steady-state and the flag loses its stability via a steady flow process. As such, steady and unsteady models are identical, and the system response is independent of time-dependent terms in the fluid forces [27]. For sufficiently large aspect ratios, on the other hand, the inverted flag displays a large-amplitude flapping motion.…”

“…Such applications have motivated a considerable amount of research [8,22,9,10]. Furthermore, many researchers have employed plate-interactingwith-flow models to explain trivial everyday phenomena such as flapping of flags or motion of tree leaves in a gentle breeze [31,27], as well as to analyse the in-flight or swimming motion of animals and explain the mechanism of snoring in humans [38,17,3].…”

“…In two remarkable studies, Sader et al [26,27] pursued the investigations of Kim et al [18] and conducted a set of experiments to explore the effect of the aspect ratio (i.e. the ratio of span to chord).…”

“…In addition to experiments, Sader and co-workers investigated the effect of the aspect ratio parameter separately in two asymptotic limits. In the first study [27], they employed a two-dimensional model based on a steady fluidelastic theory introduced by Kornecki et al [19]. Then, Prandtl's lifting-line theory and the vortex lattice method were utilized to account for flags of finite aspect ratio.…”