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Instability and the post-critical behaviour of two-dimensional inverted flags in axial flow

Published online by Cambridge University Press:  12 March 2020

Mohammad Tavallaeinejad Open the ORCID record for Mohammad Tavallaeinejad [Opens in a new window] ,
Michael P. Païdoussis Open the ORCID record for Michael P. Païdoussis [Opens in a new window] ,
Mathias Legrand Open the ORCID record for Mathias Legrand [Opens in a new window]  and
Mojtaba Kheiri
Mohammad Tavallaeinejad
Affiliation:
Department of Mechanical Engineering, McGill University, Montréal, Québec, H3A 0C3, Canada
Michael P. Païdoussis*
Affiliation:
Department of Mechanical Engineering, McGill University, Montréal, Québec, H3A 0C3, Canada
Mathias Legrand
Affiliation:
Department of Mechanical Engineering, McGill University, Montréal, Québec, H3A 0C3, Canada
Mojtaba Kheiri
Affiliation:
Department of Mechanical, Industrial and Aerospace Engineering, Concordia University, Montréal, Québec, H3G 1M8, Canada
*
†Email address for correspondence: [email protected]
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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.

Type
JFM Papers
Information
Journal of Fluid Mechanics , Volume 890 , 10 May 2020 , A14
Copyright
© The Author(s), 2020. Published by Cambridge University Press

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