Vortex phase matching of a self-propelled model of fish with autonomous fin motion

It has been a long-standing problem how schooling fish optimize their motion by exploiting the vortices shed by the others. A recent experimental study showed that a pair of fish reduce energy consumption by matching the phases of their tailbeat according to their distance. In order to elucidate the dynamical mechanism by which fish control the motion of caudal fins via vortex-mediated hydrodynamic interactions, we introduce a new model of a self-propelled swimmer with an active flapping plate. The model incorporates the role of the central pattern generator network that generates rhythmic but noisy activity of the caudal muscle, in addition to hydrodynamic and elastic torques on the fin. For a solitary fish, the model reproduces a linear relation between the swimming speed and tailbeat frequency, as well as the distributions of the speed, tailbeat amplitude, and frequency. For a pair of fish, both the distribution function and energy dissipation rate exhibit periodic patterns as functions of the front-back distance and phase difference of the flapping motion. We show that a pair of fish spontaneously adjust their distance and phase difference via hydrodynamic interaction to reduce energy consumption.