Abstract
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.
Highlights
Exploring the dynamics of a thin, flexible plate-like structure subjected to axial flow has been of constant interest to researchers as it can enhance our understanding of natural processes, such as the reconfiguration of plants in wind (Gosselin et al 2010), snoring in humans (Huang 1995a; 1995b) and flight of insects (Sane 2003)
A summary and taxonomy of previous studies is presented in Table 1. The aim of the present paper is to investigate analytically the nonlinear dynamics and postcritical behaviour of inverted flags in axial flow
The main concern of the present paper is the mechanism for flapping of infinitely high aspect ratio heavy inverted flags in axial flow
Summary
Exploring the dynamics of a thin, flexible plate-like structure subjected to axial flow has been of constant interest to researchers as it can enhance our understanding of natural processes, such as the reconfiguration of plants in wind (Gosselin et al 2010), snoring in humans (Huang 1995a; 1995b) and flight of insects (Sane 2003). Adopting steady-state aerodynamic theory in the equations obtained by Kornecki et al (1976), they showed that the onset of flapping is due to a static divergence instability and is independent of the mass ratio They showed analytically (using lifting-line theory and the vortex-lattice method) and experimentally that the critical dimensionless flow velocity at which the zero-deflection equilibrium loses stability decreases as the aspect ratio is increased. At a critical flow velocity, a saddle-node bifurcation occurs, from which a new deflected equilibrium state emanates and grows in amplitude as the flow velocity is increased Their experimental measurements for low aspect ratio (down to A D 0:033) inverted flags agree well with their theoretical results, both qualitatively and quantitatively. In ÷ 5, the effectiveness of the model is demonstrated by comparing the obtained results with those available in the literature, highlighting that the proposed model captures both the qualitative behaviour and the detailed dynamics observed in previous studies
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