Numerical simulation of hydrodynamics and propulsion in dense bioinspired fish schools under oblique flows

Oblique inflow fundamentally reorganizes the wakes of dense fish schools, generating deflected reverse Kármán jets and multiscale vortex-bridging interactions that govern the propulsion and energetics at the group level. Using a Graphics Processing Unit (GPU)-accelerated ghost-cell immersed boundary solver, we conducted two-dimensional simulations of dense bioinspired fish schools, systematically varying the Strouhal number (St), Reynolds number (Re), and lateral spacing (ds). Force decomposition in body-tangential and normal coordinates, together with wake-topology analysis, revealed a unifying mechanism: propulsion efficiency peaks when the convective transit time of the shed vortices matches the undulation period, resulting in constructive phase locking among neighbors. This condition emerges within a moderate St window and at transitional Re, where coherent reverse Kármán jets remain compact. At higher St, resonance amplifies force fluctuations and dissipates power, whereas at higher Re, additional small-scale content disrupts phase coherence, reducing school-level gains. Spacing governs the coupling strength: an intermediate, dense but non-contact arrangement (ds = 0.6L) yields the broadest efficiency maximum and increases the school-averaged efficiency by approximately 12%–15% relative to both tighter and wider spacings. Furthermore, the benefits of interaction converge as the wakes diffuse. Energetic advantages are spatially selective across the formation: individuals directly exposed to the inflow incur greater drag, whereas shielded interior or trailing positions benefit from phase locking. These results provide a macroscale framework linking inflow obliquity, wake coherence, and array geometry, thereby offering design guidance for bioinspired multivehicle formations that adapt spacing and phasing to maximize collective propulsion while minimizing energy loss.