Schooling of tandem flapping wings*

We present the results of a theoretical investigation into the dynamics of tandem flapping wings, a model system for studying schooling swimmers in relatively fast flows. We develop a discrete dynamical system (iterated map) in which the swimmers shed point vortices during each flapping cycle, which in turn exert forces on the swimmers. Our model predictions exhibit good agreement with experimental data on the steadily-translating schooling states of tandem wings. If the wings execute identical flapping kinematics, we find that, as the flapping amplitude is progressively increased, the system undergoes a period-doubling cascade that ultimately leads to chaos. If the wings' flapping amplitudes or frequencies are distinct, we find that the wings may either separate, collide, or form unsteady bound states. Generally, our results indicate how hydrodynamics may influence the dynamics of schooling swimmers in biological contexts.