Non-monotonic and non-reciprocal collective hydrodynamics in self-propelled foils under identical and mismatched kinematics

This study numerically investigates the hydrodynamics of two self-propelled undulating foils in an initial anti-phase, side-by-side configuration. The effects of lateral spacing, undulatory amplitude, frequency, and kinematic mismatches by independently varying amplitude or frequency, are systematically examined. When kinematics are identical, the foils maintain a symmetric formation, with forward speed varying non-monotonically with lateral spacing and power consumption always exceeding solitary swimming. A critical spacing exists where speeds match while energy efficiency remains lower. Under frequency differences, four modes emerge, separating (Se), side-by-side (Ss), bouncing side-by-side (Bs), and staggered (Sg), with the kinematically disadvantaged individual gaining speed and efficiency at the advantaged individual's energetic expense. Under amplitude differences, formations become more stable. The kinematically disadvantaged individual consistently gains speed with increased power consumption, while energy efficiency gains appear only when disparity is sufficiently large. Further analysis shows that reduced-order models for inline formations fail to capture the observed non-monotonic speed variation and critical spacing. Instead, collective performance in side-by-side configurations is governed by the coupling of full-body lateral velocity fields, rather than wake-vortex or leader's trailing edge-follower's leading edge interactions. These findings highlight fundamental differences between anti-phase, side-by-side and inline schooling, demonstrating that kinematic heterogeneity reshapes the distribution of hydrodynamic advantages among individuals.