Membrane Flutter in Inviscid Flow
Mavroyiakoumou, Christiana
2022
Abstract
Despite the advantages associated with extensible membranes in biological and engineering applications, the majority of previous studies have focused on the motion of bending-dominated flexible bodies through a fluid flow. In this thesis, we develop a variety of mathematical models and numerical methods to analyze the small- and large-amplitude dynamics of thin membranes (of zero bending rigidity) with vortex-sheet wakes in two- and three-dimensional inviscid flows. In chapter II, we study the dynamics of membranes initially aligned with a uniform background flow. This is a benchmark fluid-structure interaction that has previously been studied mainly in the small-deflection limit, where the flat state may be unstable. Here we study the initial instability and large-amplitude dynamics with respect to three parameters: membrane mass density, stretching rigidity, and pretension. With both membrane ends fixed, we find that all membranes become unstable by divergence below a critical pretension close to the value identified in previous studies, and converge to steady deflected shapes. With the leading edge fixed and trailing edge free, divergence and/or flutter occurs, and a variety of periodic and aperiodic oscillations are found. With both edges free, the membrane may also translate transverse to the flow, with steady, periodic, or aperiodic trajectories. In chapter III, we investigate the instability of membranes in terms of growth rates, angular frequencies, and eigenmode shapes, by solving a nonlinear eigenvalue problem iteratively. When both membrane ends are fixed, the stability boundary is fairly simple: light membranes become unstable by divergence and heavy membranes lose stability by flutter and divergence, which occurs for a pretension value that increases with the mass. With the leading edge fixed and trailing edge free, or both edges free, the membrane eigenmode shapes become more complicated and eigenmodes transition in shape across the stability boundary. We also compare our results against the simulations of the corresponding initial value problem in the growth regime and find excellent agreement. In chapter IV, we consider membranes that are held by freely-rotating tethers and find that the tethered boundary condition allows a variety of unsteady large-amplitude motions---both periodic and chaotic. We characterize the oscillations over ranges of: membrane mass density, stretching stiffness, pretension, and tether length and determine the region of instability and small-amplitude behavior by solving a nonlinear eigenvalue problem. We additionally consider a simplified model: an infinite periodic membrane, which yields a regular eigenvalue problem, analytical results, and asymptotic scaling laws. We find qualitative similarities among all three models in terms of the oscillation frequencies and membrane shapes at small and large values of membrane mass, pretension, and tether length/stiffness. In chapter V, we develop a model and numerical method to study the large-amplitude flutter of rectangular membranes that shed a trailing vortex-sheet wake in a 3D inviscid fluid flow. For all 12 combinations of boundary conditions at the membrane edges we compute the stability thresholds and the subsequent large-amplitude dynamics across the same three-parameter space as before. We find that 3D dynamics in the 12 cases naturally form four groups based on the conditions at the leading and trailing edges. The conditions at the side edges, though generally less important, may have qualitative effects on the membrane dynamics---e.g. steady versus unsteady, periodic versus chaotic, or the variety of spanwise curvature distributions---depending on the group and the physical parameter values.Deep Blue DOI
Subjects
extensible membranes flutter fluid-structure interactions
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Thesis
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