Collective locomotion of two uncoordinated undulatory self-propelled foils

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Fish schooling with stable configurations is intriguing. How individuals benefit from hydrodynamic interactions is still an open question. Here, fish are modeled as undulatory self-propelled foils, which is more realistic. The collective locomotion of two foils in a tandem configuration with different amplitude ratios Ar and frequency ratios Fr is considered. Depending on Ar and Fr, the two foils without lateral or yaw motion may spontaneously form stable configurations, separate, or collide with each other. The phase diagram of the locomotion modes in the (Fr, Ar) plane is obtained, which is significantly different from that in Newbolt et al. [“Flow interactions between uncoordinated flapping swimmers give rise to group cohesion,” Proc. Natl. Acad. Sci. U. S. A. 116, 2419 (2019)]. For stable configurations, the gap spacing may be almost constant [stable position (SP) mode] or change dynamically and periodically [stable cycle (SC) mode]. In our diagram, the fast SP mode is found. Besides, the border between the separation and SP/SC modes is more realistic. In the fast SP cases, analyses of hydrodynamic force show the phenomenon of inverted drafting, in which the leader achieves hydrodynamic advantages. For the SC mode, the cruising speed increases piecewise linearly with FrAr. When Ar < 1, the linear slope is identical to that of the isolated leader, and the follower-control mechanism is revealed. Our result sheds some light on fish schooling and predating.