# Dependence on aspect ratio of symmetry breaking for oscillating foils: implications for flapping flight

Article

Using two-dimensional direct numerical simulations, we investigate the flow in a fluid of kinematic viscosity ν and density ρ around elliptical foils of density ρ_s with major axis c and minor axis b for three different aspect ratios: AR = b/c = 1 (a circle); AR = 0.5; and AR = 0.1. The vertical location of these foils y_s(t) = A sin(2πf₀t) oscillates with amplitude A and frequency f₀ in two distinct ways: ‘pure’ oscillation, where the foils are constrained to remain in place; and ‘flying’ oscillation, where horizontal motion is allowed. We simulate the flow for a range of the two appropriate control parameters, the nondimensional amplitude or Keulegan-Carpenter number KC = 2πA/c and the nondimensional frequency or Stokes number β = f₀c²/ν. We observe three distinct patterns of asymmetry, labelled ‘S-type’ for synchronous asymmetry, ‘QP_H-type’ and ‘QP_L-type’ for quasi-periodic asymmetry at sufficiently high and sufficiently low (i.e. AR = 0.1) aspect ratios respectively. These patterns are separated at the critical locus in KC − β space by a ‘freezing point’ where the two incommensurate frequencies of the QP-type flows combine, and we show that this freezing point tends to occur at smaller values of KC as AR decreases. We find for the smallest aspect ratio case (AR = 0.1) that the transition to asymmetry, for all values of KC, occurs for a critical value of an ‘amplitude’ Stokes number βA = β(KC)² = 4π² f0A²/ν ≃ 3. The QP_L-type asymmetry for AR = 0.1 is qualitatively different in physical and mathematical structure from the QP_H-type asymmetry at higher aspect ratio. The flow at the two ends of the ellipse become essentially decoupled from each other for the QP_L -type asymmetry, the two frequencies in the horizontal force signature being close to the primary frequency, rather than twice the primary frequency as in the QP_H-type asymmetry. Furthermore, the associated coefficients arising from a Floquet stability analysis close to the critical threshold are profoundly different for low aspect ratio foils. Freedom to move slightly suppresses the transition to S-type asymmetry, and for certain parameters, if a purely oscillating foil subject to S-type asymmetry is released to move, flow symmetry is rapidly recovered due to the negative feedback of small horizontal foil motion. Conversely, for the ‘higher’ aspect ratios, the transition to QP_H-type asymmetry is encouraged when the foil is allowed to move, with strong positive feedback occurring between the shed vortices from successive oscillation cycles. For AR = 0.1, freedom to move significantly encourages the onset of asymmetry, but the newly observed ‘primary’ QP_L-type asymmetry found for pure oscillation no longer occurs when the foil flies, with S-type asymmetry leading ultimately to locomotion at a constant speed occurring all along the transition boundary for all values of KC and β.