Dynamics of inverted flags: Experiments and comparison with theory

Abstract

The study of the dynamics of cantilevered thin flexible plates in reverse axial flow -also known as inverted flags -has become of significant interest, partly due to their energy harvesting potential. This paper presents fresh experimental results, aiming to enhance our understanding of the dynamics of inverted flags, particularly concerning stability and global dynamics sensitivity to various system parameters, such as the aspect ratio (i.e. height-to-length ratio) and flag material. This is achieved through testing flags of various dimensions, made from different materials. The dynamics of the system is presented in the form of bifurcation diagrams in which the flag tip angle is shown as a function of the dimensionless flow velocity. The frequency content of oscillatory motions is presented in the form of spectrograms. Also, time-histories, phase-plane portraits, as well as power spectral density and probability density function plots are presented for different flow regimes. Interesting dynamical features, such as small-and large-amplitude flapping, static divergence, large-amplitude buckling, and chaotic motions, have been observed experimentally, some for the first time. It is shown that for flags of very low aspect ratio the undeflected static equilibrium is stable prior to a subcritical pitchfork bifurcation. For flags of large aspect ratio, on the other hand, the undeflected static equilibrium becomes unstable via a supercritical pitchfork bifurcation leading to static divergence (buckling) at a sufficiently high flow velocity; at higher flow velocities, past the pitchfork bifurcation, a supercritical Hopf bifurcation materialises leading to flapping motion around the deflected static equilibrium; at even higher flow velocities, flapping motion becomes symmetric around the undeflected static equilibrium. Interestingly, it is found that heavy flags may exhibit large-amplitude flapping right after the initial static equilibrium, provided that they are subjected to a sufficiently large disturbance. In addition to experiments, nonlinear analytical models for the two asymptotic cases of zero and infinite aspect ratios are outlined, the results of which are used for comparison with the experimental results. Theory and experiments are found to be in reasonably good agreement.Keywords inverted flag, cantilevered plates in reverse axial flow, fluid-structure interactions, nonlinear dynamics, effect of aspect ratio, effect of flag material

Aspect ratio-dependent hysteresis response of a heavy inverted flag

“…Heavy inverted flags show consistent subcritical bifurcation for small to wide aspect ratios, similar to what is observed by Tavallaeinejad et al. (2021), as opposed to the results of the regular flag configuration in which the hysteresis is only present for greater than unity (Eloy et al. 2012).…”

“…The large-amplitude periodic fluttering only disappears at minimal aspect ratios () (Tavallaeinejad et al. 2021).…”

“…The heavy inverted flag, instead, exhibits hysteresis (subcritical Hopf and pitchfork bifurcations for and , respectively) over a wide range of parameters (Tavallaeinejad et al. 2021). Kim et al.…”

“…Tavallaeinejad et al. (2021) observed that flags with small have a larger hysteresis loop, distinguishing them from the regular flag case. Tang, Liu & Lu (2015) also observed a bistable response from the stationary to flapping modes in a computational study of three-dimensional flags with different aspect ratios, although the mechanism of bistability and the extension of the hysteresis loop were not explored.…”

Cross-Flow-Induced Vibration of an Elastic Plate

“…A number of studies considered axial flow over a cantilevered plate pinned at one end, which is sometimes referred to as the flag or the inverted flag [24][25][26]. It was shown recently that critical flow velocities for large amplitude divergence and flapping of the inverted flag can be predicted fairly well by a two-dimensional (2D) theoretical model when the ratio of the plate width to its length (i.e., the aspect ratio) is large [27]. Similarly, several studies considered flexible plates in 2D cross-flow, which can be rendered more readily to modeling and theoretical analysis [28][29][30][31].…”