Stable schooling formations emerge from attractive/repulsive interactions between two self-propelled undulatory foils with an initial staggered configuration

Biological schooling exhibits diverse configurations with varying hydrodynamic performances, which emerge through flow-mediated interactions under different initial conditions. A numerical study was performed to explore the effects of initial spatial spacings on the schooling performance of two self-propelled foils in anti-phase. Our results indicate that compared to solitary swimming, individuals in a side-by-side schooling configuration generally show no evident hydrodynamic benefits, as any improvement in forward speed is associated with an evident increase in power consumption and a decrease in energy efficiency. In contrast, a staggered formation, generated by strong interaction between the leader's wake and the follower, may be more preferable when considering enhancements in both forward speed and energy efficiency. To demonstrate the formation of different schooling configurations, we propose a conceptual model where the competition and balance between attractive effect, originating from the stability of side-by-side configuration, and repulsive effect, arising from the leader's wake, constitute the primary mechanism that leads to stable schooling in either side-by-side or staggered formations. In addition, we suggest that rapid attainment of stability may be one of the important reasons affecting the formation of stable schooling configurations.