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We establish through numerical simulation conditions for optimal undulatorypropulsion for a single fish, and for a pair of hydrodynamically interactingfish, accounting for linear and angular recoil. We first employ systematic 2Dsimulations to identify conditions for minimal propulsive power of aself-propelled fish, and continue with targeted 3D simulations for a danio-likefish. We find that the Strouhal number, phase angle between heave and pitch atthe trailing edge, and angle of attack are principal parameters. Angular recoilhas significant impact on efficiency, while optimized body bending requiresmaximum bending amplitude upstream of the trailing edge. For 2D simulations,imposing a deformation based on measured displacement for carangiform swimmingprovides 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 Reynoldsnumber 5000.  Next, we turn to 2D simulation of two hydrodynamically interacting fish. Wefind that the upstream fish benefits energetically only for small distances. Incontrast, the downstream fish can benefit at any position that allowsinteraction with the upstream wake, provided its body motion is timedappropriately with respect to the oncoming vortices. For an in-lineconfiguration, one body length apart, the optimal efficiency of the downstreamfish can increase to 66%; for an offset arrangement it can reach 81%. Thisproves that in groups of fish, energy savings can be achieved for downstreamfish through interaction with oncoming vortices, even when the downstream fishlies directly inside the jet-like flow of an upstream fish.

Optimal undulating swimming for a single fish-like body and for a pair of interacting swimmers