Energetic benefits in coordinated circular swimming motion of two swimmers

The coordinated movement of multiple swimmers is a crucial component of fish schools. Fish swimming in different formations, such as tandem, side-by-side, diamond, and phalanx, can achieve significant energetic advantages. However, the energetic benefits of nonstraight swimming behaviors, such as the collective motion of a milling pattern, are not well understood. To fill in this gap, we consider two swimmers in circular tracks, controlled by a PID approach to reach stable configurations. Our study finds that the optimal phase is affected by circumferential effects, and that substantial energy savings can result from both propulsion and turning. We also explore the radial effect in terms of energetic benefits. In a milling pattern, the inner swimmers can easily gain a certain energetic benefit ($\ensuremath{-}8%$), while their peers on the outside must be close enough to the inner swimmer with a proper phase to gain the energetic benefit ($\ensuremath{-}14%$). When the radial spacing becomes larger or is in an unmatched phase, the swimming of the outer swimmers becomes more laborious ($+16%$). Our results indicate that swimmers who maintain a matched phase and minimum radial effect obtain the highest energetic benefits ($\ensuremath{-}26%$). These findings highlight the energetic benefits of swimmers, even in a milling pattern, where the position difference dominates the extent of benefit.