Optimal undulating swimming for a single fish-like body and for a pair   of interacting swimmers
  We establish through numerical simulation conditions for optimal undulatory propulsion for a single fish, and for a pair of hydrodynamically interacting fish, accounting for linear and angular recoil. We first employ systematic 2D simulations to identify conditions for minimal propulsive power of a self-propelled fish, and continue with targeted 3D simulations for a danio-like fish. We find that the Strouhal number, phase angle between heave and pitch at the trailing edge, and angle of attack are principal parameters. Angular recoil has significant impact on efficiency, while optimized body bending requires maximum bending amplitude upstream of the trailing edge. For 2D simulations, imposing a deformation based on measured displacement for carangiform swimming provides efficiency of 40%, which increases for an optimized profile to 57%; for a 3D fish, the corresponding increase is from 22% to 35%; all at Reynolds number 5000.   Next, we turn to 2D simulation of two hydrodynamically interacting fish. We find that the upstream fish benefits energetically only for small distances. In contrast, the downstream fish can benefit at any position that allows interaction with the upstream wake, provided its body motion is timed appropriately with respect to the oncoming vortices. For an in-line configuration, one body length apart, the optimal efficiency of the downstream fish can increase to 66%; for an offset arrangement it can reach 81%. This proves that in groups of fish, energy savings can be achieved for downstream fish through interaction with oncoming vortices, even when the downstream fish lies directly inside the jet-like flow of an upstream fish. 
Audrey P. Maertens, Amy Gao, Michael S. Triantafyllou
physics.flu-dyn