On membrane-wing stability in laminar flow

Abstract

Abstract A computational study of the dynamic stability of a two-dimensional membrane wing in laminar steady flow is presented, with a focus on the role of membrane mass. The membrane is assumed to be linearly elastic, and no supports are used in the membrane model. The study focuses on small mass ratios of μ = ρ m h ∕ ρ c ≤ 1 , which are most relevant in today’s membrane-wing applications, and small angles of attack (AoAs), for which the massless membrane solution predicts a stable solution. For very small AoAs the membrane is stable, in accordance with the massless solution. As the AoA is increased, the membrane loses stability via limit-cycle oscillations (LCO). The instability threshold depends on the membrane mass-ratio such that any increase in the mass-ratio increases the AoA of LCO onset. Membrane oscillations improve the mean aerodynamic characteristics of the airfoil, presenting significantly higher lift-slope than stable membranes. Dynamic mode decomposition (DMD) analysis revealed that membrane-oscillations appear with a dominant mode-shape that is very similar to the second structural mode, and a frequency that is slightly lower than the structural frequency. A simple mathematical model is suggested for single-mode membrane oscillations from which a straight-forward criterion for predicting the membrane-stability is derived.