Inverted flags – clamped–free elastic thin plates subjected to a fluid flowing axially and directed from the free end towards the clamped end – have been observed experimentally and computationally to exhibit large-amplitude flapping beyond a critical flow velocity. The motivation for further research on the dynamics of this system is partly due to its presence in some engineering and biological systems, and partly because of the very rich dynamics it displays. In the present paper, the goal is to develop a nonlinear analytical model for the dynamics and stability of high aspect ratio (i.e. height to length ratio) flags. The inviscid fluid flow is modelled via the quasi-steady version of Theodorsen’s unsteady aerodynamic theory, and the Polhamus leading-edge suction analogy is utilized to model flow separation effects from the free end (leading edge) at moderate angles of attack. Gear’s backward differentiation formula and a pseudo-arclength continuation technique are employed to solve the governing equations. Numerical results suggest that fluidelastic instability may be the underlying mechanism for the flapping motion of high aspect ratio heavy inverted flags. In other words, flapping may be viewed as a self-excited vibration. It was found from numerical results that the undeflected static equilibrium of the inverted flag is stable at low flow velocities, prior to the occurrence of a supercritical pitchfork bifurcation. The pitchfork bifurcation is associated with static divergence (buckling) of the flag. At higher flow velocities, past the pitchfork bifurcation, a supercritical Hopf bifurcation materializes, generating a flapping motion around the deflected static equilibrium. At even higher flow velocities, flapping motion becomes symmetric around the undeflected static equilibrium. Interestingly, it was also 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. Moreover, inverted flags with a non-zero initial angle of attack were found to be less stable than their perfectly flow-aligned counterparts.

中文翻译：

---

轴流中二维倒旗的不稳定性和后临界行为

倒置的旗子——受约束的自由弹性薄板受到轴向流动并从自由端朝向夹紧端的流体的作用——已经通过实验和计算观察到，表现出超过临界流速的大振幅拍动。对该系统动力学进行进一步研究的动机部分是由于它存在于一些工程和生物系统中，部分是因为它显示了非常丰富的动力学。在本文中，目标是为高纵横比（即高长比）旗帜的动力学和稳定性开发非线性分析模型。无粘性流体流动是通过 Theodorsen 非定常空气动力学理论的准定常版本建模的，Polhamus 前缘吸力类比用于模拟中等迎角下自由端（前缘）的流动分离效应。齿轮的后向微分公式和伪弧长延拓技术被用来求解控制方程。数值结果表明，流体弹性不稳定性可能是高纵横比重倒旗扑动运动的潜在机制。换句话说，扑动可以被视为一种自激振动。从数值结果中发现，在超临界干草叉分叉发生之前，倒旗的未偏转静态平衡在低流速下是稳定的。干草叉分叉与旗帜的静态发散（屈曲）有关。在更高的流速下，经过干草叉分叉处，超临界 Hopf 分岔出现，在偏转的静态平衡周围产生拍打运动。在更高的流速下，扑动运动围绕未偏转的静态平衡对称。有趣的是，还发现重旗可能会在初始静态平衡后立即表现出大幅度的摆动，前提是它们受到足够大的干扰。此外，发现具有非零初始迎角的倒旗比其完美的流动对齐的旗更不稳定。还发现重旗可能会在初始静态平衡后立即表现出大幅度的摆动，前提是它们受到足够大的干扰。此外，发现具有非零初始迎角的倒旗比其完美的流动对齐的旗更不稳定。还发现重旗可能会在初始静态平衡后立即表现出大幅度的摆动，前提是它们受到足够大的干扰。此外，发现具有非零初始迎角的倒旗比其完美的流动对齐的旗更不稳定。