Membrane flutter in three-dimensional inviscid flow

Abstract

We develop a model and numerical method to study the large-amplitude flutter of rectangular membranes (of zero bending rigidity) that shed a trailing vortex-sheet wake in a three-dimensional (3-D) inviscid fluid flow. We apply small initial perturbations and track their decay or growth to large-amplitude steady-state motions. For 12 combinations of boundary conditions at the membrane edges we compute the stability thresholds and the subsequent large-amplitude dynamics across the three-parameter space of membrane mass density, pretension and stretching rigidity. With free side edges we find good agreement with previous 2-D results that used different discretization methods. We find that the 3-D dynamics in the 12 cases naturally forms four groups based on the conditions at the leading and trailing edges. The deflection amplitudes and oscillation frequencies have scalings similar to those in the 2-D case. The conditions at the side edges, although generally less important, may have small or large qualitative effects on the membrane dynamics – e.g. steady vs unsteady, periodic vs chaotic or the variety of spanwise curvature distributions – depending on the group and the physical parameter values.

Large-amplitude membrane flutter in inviscid flow

“…Now, we build on that work and study in more detail the time-dependent spanwise variations in membrane deflection and how they depend on various material and geometric parameters. In [41] we compared dynamics in a 2-D flow with those in a 3-D flow in the quasi-2-D limit (large span, free side-edges) and found good agreement. To explore this point further we now increase the spanwise resolution by a factor of four over that used in [41].…”

“…In [41] we compared dynamics in a 2-D flow with those in a 3-D flow in the quasi-2-D limit (large span, free side-edges) and found good agreement. To explore this point further we now increase the spanwise resolution by a factor of four over that used in [41]. The main question that we answer in this work is: when are 3-D membrane flutter motions truly three-dimensional and when are they essentially 2-D?…”

“…In [41] we developed a mathematical model and numerical method to investigate the large-amplitude flutter of rectangular membranes that shed a trailing vortex-sheet wake in a three-dimensional (3-D) inviscid flow. There, we studied basic features of the membrane shapes and dynamics (steady, periodic, chaotic) and how they vary with the boundary conditions, fixed or free, at the membrane edges.…”

“…In the current work, we focus only on the FFFF and FRFR boundary conditions, where F stands for a fixed edge and R stands for a free edge. The first letter in each label is the leading-edge boundary condition type and the following letters are the boundary conditions moving clockwise around the rectangular membrane looking down from larger z values, as in [41]. Thus, the second and fourth letters are the side-edge boundary conditions and the third letter is the trailing-edge boundary condition.…”

“…Another way to simulate a free-end membrane condition, theoretically and experimentally, is by attaching the membrane edge to a thin rod or tether that can rotate freely about its other end at a pivot point [77,78]. These works motivated our choice of boundary conditions in [41] and in the present study.…”