Numerical investigation of schooling arrangement and phase differences on the hydrodynamic performance of fish swimming side-by-side behind a D-cylinder

As the fundamental unit in fish school studies, two fish provide a crucial theoretical foundation for understanding fluid dynamic interactions and energy-saving mechanisms. This study employs numerical simulations to examine the effects of fish positions relative to a D-cylinder and tail-beat phase on propulsion performance and energy-saving efficiency in the wake flow behind a D-cylinder. At a water speed of 4.5L/s, three typical arrangements of two fish swimming side-by-side behind a D-cylinder (Centered-Offset, Flush-Offset, and Flush-Flush) are compared and analyzed. By incorporating tail-beat phase variations (in-phase and anti-phase), the coupling mechanisms between fish school performance and wake dynamics are further elucidated. The results demonstrate that spatial arrangements of fish schools downstream of a D-cylinder are key to enhancing hydrodynamic efficiency. Specifically, the centered-offset arrangement significantly reduces drag on fishes through the shielding effect, the flush-offset arrangement enhances the utilization of local flow energy, and the flush–-flush arrangement achieves balanced optimization of lift and drag through symmetric alignment. Building on these findings, tail-beat phase modulation further regulates the intensity of flow interactions. In-phase tail beating increases flow field complexity due to enhanced synchronization of tail vortices, while anti-phase tail beating significantly improves propulsion efficiency and reduces drag fluctuations by mitigating wake interference. This study highlights the synergistic effects of spatial arrangement and tail-beat phases in enhancing the propulsion efficiency, minimizing drag fluctuations, and optimizing flow interactions of fish schooling. These findings provide valuable insights for the design of biomimetic fish schools and optimizing fishway designs in complex flow fields.