Phase-mediated interactions on collective locomotion of two unconstrained self-propelled flexible fins

In fish schooling, unrestricted movement enables the emergence of complex mutual interactions and adaptive dynamics, which are observed in the realistic collective behavior of fish. In the present study, the effects of phase difference (ϕ/π) on the collective locomotion and schooling performance of two unconstrained self-propelled flexible fins are numerically investigated. Two distinct collective formations emerge spontaneously depending on ϕ/π: side-by-side and diagonal configurations. The trends of flapping kinematics and propulsive performance of the upper fin are symmetric to those of the lower fin with respect to ϕ/π = 1.0. Regardless of ϕ/π, the cruising speed of the two schooling fins is consistently lower than that of a single fin due to reduced jet velocity strength and jet deflection. Although the leading fin consumes more energy to assist the following fin, the energy consumption of the following fin is significantly reduced due to shared fluid environments. At specific ϕ/π values, the average efficiency of the two fins improves due to a significant reduction in the time-averaged input power of the following fin. This reduction is attributed to a combination of decreased peak-to-peak flapping amplitude, shorter energy-consuming stages, and compensatory motion of the following fin induced by shared pressure fields. Vortical structures and corresponding jet patterns are analyzed to explain the reduced cruising speed caused by the collective formation of the two fins. Additionally, the effects of initial horizontal and lateral gap distances on the emergence of schooling formations are examined.