How wavelength affects hydrodynamic performance of two accelerating mirror-symmetric undulating hydrofoils

Fish schools are capable of simultaneous linear acceleration. To reveal the underlying hydrodynamic mechanism, we numerically investigate how Reynolds number Re = 1000-2000, Strouhal number St = 0.2-0.7, and wavelength lambda = 0.5-2 affect the mean net thrust and net propulsive efficiency of two side-by-side hydrofoils undulating in anti-phase. In total, 550 cases are simulated using immersed boundary method. The thrust increases significantly with the wavelength and the Strouhal number, yet only slightly with the Reynolds number. We apply a symbolic regression algorithm to formulate this relationship. Furthermore, we find that mirror-symmetric schooling can achieve a net thrust more than ten times that of a single swimmer, especially at low Reynolds numbers. The highest efficiency is obtained at St = 0.5 and lambda = 1.2, where St is consistent with that observed in the linear-accelerating natural swimmers, e.g., Crevalle jack. Six distinct flow structures are identified. The highest thrust corresponds to an asymmetric flow pattern, whereas the highest efficiency occurs when the flow is symmetric with converging vortex streets.