Amplitude Death in Aeroelastically-Driven Flapping Foil Motion

In two-dimensional simulations of aeroelastically-driven motion of an elastic foil in a plane parallel channel, studied earlier in an unbounded domain (Connell & Yue, J. Fluid Mech. 581, 33-67, 2007; Goza & Colonius, J. Comp. Phys. 336, 401-411, 2017), we considered Reynolds numbers (based on channel width) 160 ≤ Re ≤ 400, with a ratio of foil length to channel width of 2, "mass ratio" (product of foil density and foil thickness divided by fluid density and foil length) 0.1, and dimensionless bending rigidity (product of elastic modulus and second moment of cross-sectional area, divided by fluid density, square of characteristic velocity, and cube of foil length) in the range 10⁻⁵ ≤ K_B ≤ 4 × 10⁻⁴. Over these ranges, periodic solutions, with distinctly different flapping amplitudes, frequencies, and mode shapes, exist. For 190 ≤ Re ≤ 240, flapping is completely suppressed in an Re-dependent range of K_B. We attribute this to "amplitude death" found when two nonlinear oscillators with unequal natural frequencies are linearly coupled. Recognizing that the foil behaves like a rigid plate as K_B → ∞, we interpret K_B as a “coupling strength” between the mismatched-frequency modes. Dependence of the complete-suppression K_B range on frequency mismatch is consistent with predictions of a simple Stuart-Landau model. Qualitative analysis shows that increasing Re reduces dimensionless foil tension, which reduces frequency mismatch, and ultimately reduces the complete-suppression K_B range, consistent with the simulations.