Modeling of flapping-fin propulsion with Stuart-Landau oscillator equation

Recently, the lowest order thrust measurements in an abstracted twisting and flapping fin have been modeled using a van der Pol-like oscillator (\textit{JFM }\textbf{702,} 298-331). A Stuart-Landau oscillator is used here as a higher order model of the interaction of the low aspect ratio flapping fin with its downstream thrust-producing reverse Karman vortex street. ``Quasi-steady'' equations for the forces produced on flapping fins or wings by the surrounding fluid assume that the lift and drag coefficients are based on `a$_{g}$(t)', a time-variable angle of attack based on the fin's instantaneous position and velocity. In this work, a wake-modified angle of attack `a(t)' is used, such that `a = a$_{g}$ + a$_{w}$' where `a$_{w}$(t)' is based on the circulation in the wake. This modification of the geometric angle of attack `a$_{g}$' is justified generally by the conservation of circulation within the fin-wake system, and we argue that a Stuart-Landau oscillator represents a good approximation of the circulation within the wake. Results of this modeling are compared with experimental data taken on the abstracted penguin wing planform; a strong quantitative agreement exists between the experimental and modeled systems. We also model the effects of Reynolds number and the dependence of system oscillation lock-in on initial condition.