The flapping-flag instability as a nonlinear eigenvalue problem

Abstract

We reconsider the classical problem of the instability of a flapping flag in an inviscid background flow with a vortex sheet wake, and reformulate it as a nonlinear eigenvalue problem. We solve the problem numerically for the 20 lowest wave number modes. We find that the lowest wave number mode is the first to become unstable, and has the fastest growth rate within 3.5 decades of the stability boundary in the parameter space of flag mass and flag rigidity. This and subsequent modes become unstable by merging into complex-conjugate pairs, which then lose conjugacy further into the region of instability. Eigenmodes exhibit sharp transitions in shape across the stability boundary. In the corresponding initial value problem, we show a correlation between the wave number of unstable modes in the small-amplitude (growth) regime and the large-amplitude (saturation) regime. Wave number increases with decreasing rigidity and shows a combination of discrete and continuous change in the shape of unstable modes. Using an infinite flag model we compute the parameters of the most unstable flag and show that a classical mechanism for the instability correlating pressure lows to flag amplitude peaks does not hold.