Collective locomotion of two unconstrained self-propelled plates with temporally asymmetric anti-phase flapping

Flow-mediated interactions can yield collective performance gains in animal groups without explicit communication, yet the role of temporal stroke asymmetry in formation selection and propulsive performance of fully unconstrained propulsors remains unclear. In the present study, numerical simulations are performed on two side-by-side self-propelled plates executing temporally asymmetric anti-phase flapping while remaining free to translate in both the horizontal and lateral directions. A regime map spanning stroke amplitude and temporal asymmetry over several actuation frequencies identifies four distinct modes: straight propulsion with a stable lateral spacing, deflected propulsion with persistent lateral drift, collision, and separation. Within the stable schooling regimes (straight and deflected propulsion), increasing asymmetry generally improves propulsive performance while reducing the equilibrium lateral gap distance. Propulsive performance within the stable modes is maximized when the trailing-edge outward stroke of the two plates is faster than the inward motion. Despite the slow inward motion, vortex-mediated interactions generated during the preceding rapid outward stroke strengthen a gap-directed flow that suppresses instroke-driven gap outflow, thereby sustaining a strong shared positive-pressure region. During the rapid outward motion, the plates further establish an intensified shared negative-pressure region. As a result, the combined enhancement of these pressures amplifies the pressure-driven normal forces, leading to improved propulsive performance. These results clarify how temporal asymmetry reorganizes wake-mediated interactions and formation selection in unconstrained side-by-side locomotion, and provide design guidelines for using asymmetric actuation to stabilize formations and enhance collective performance.