  In this paper, we study the fluid-structure interaction (FSI) of a
three-dimensional (3D) flexible membrane immersed in an unsteady separated flow
at moderate Reynolds numbers. We employ a body-conforming variational FSI
solver based on the recently developed partitioned iterative scheme for the
coupling of turbulent fluid flow with nonlinear structural dynamics. Of
particular interest is to understand the flow-excited instability of a 3D
flexible membrane as a function of the non-dimensional mass ratio, Reynolds
number and aeroelastic number. For a wide range of the parameters, we examine
two distinctive stability regimes of fluid-membrane interaction:
deformed-steady state (DSS) and dynamic balance state (DBS). We propose
stability phase diagrams to demarcate the DSS and DBS regimes for the parameter
space of mass ratio vs. Reynolds number and mass ratio vs. aeroelastic number.
Based on the aeroelastic mode analysis, we observe a frequency synchronization
between the vortex shedding frequency and the membrane vibration frequency
which leads to self-sustained vibrations in the dynamic balance state. To
characterize the origin of the frequency lock-in, we derive an approximate
analytical formula for the nonlinear natural frequency by considering the added
mass effect and employing a large deflection theory for a simply supported
rectangular membrane. Through our systematic high-fidelity numerical
investigation, we find that the onset of the membrane vibration and the mode
transition has a dependence on the frequency lock-in between the natural
frequency of the tensioned membrane and the vortex shedding frequency or its
harmonics. These findings on the fluid-elastic instability of membranes have
implications for the design and development of control strategies for membrane
wing-based unmanned systems and drones.
