Eigenmode analysis of membrane stability in inviscid flow

TL;DR

This paper investigates the stability of a thin membrane in inviscid flow, analyzing eigenmodes and instabilities across various boundary conditions and membrane properties, with implications for fluid-structure interaction understanding.

Contribution

It provides a comprehensive eigenmode analysis of membrane stability in inviscid flow, exploring different boundary conditions and membrane parameters, and compares linear predictions with nonlinear simulations.

Findings

Instability occurs via divergence or flutter across membrane densities.

Eigenmodes become more complex ('wavier') at lower mass density and tension.

Good agreement with small-amplitude unsteady simulations, but only qualitative with large-amplitude steady states.

Abstract

We study the instability of a thin membrane (of zero bending rigidity) to out-of-plane deflections, when the membrane is immersed in an inviscid fluid flow and sheds a trailing vortex-sheet wake. We solve the nonlinear eigenvalue problem iteratively with large ensembles of initial guesses, for three canonical boundary conditions---both ends fixed, one end fixed and one free, and both free. Over several orders of magnitude of membrane mass density, we find instability by divergence or flutter (particularly at large mass density, or with one or both ends free). The most unstable eigenmodes generally become "wavier" at smaller mass density and smaller tension, but with regions of nonmonotonic behavior. We find good quantitative agreement with unsteady time-stepping simulations at small amplitude, but only qualitative similarities with the eventual steady-state large-amplitude motions.