Figure 10. Passive and active methods for stabilizing an emergent formation of four swimmers.

(A) In an inline school of four swimmers, the leading three swimmers flap inphase, but swimmer 4 actively controls its phase in response to the flow it perceives locally to match its phase to that of the local flow as proposed in Li et al., 2020. The phase controller stabilizes swimmer 4 in formation but at no hydrodynamic benefit. (B) Sequentially increasing the phase lag by a fixed amount $\mathrm{\Delta }\varphi =-{30}^o$ in an inline school of four swimmers stabilizes the trailing swimmer but at no hydrodynamic benefit. (C) Gradually tuning the phase lag $\mathrm{\Delta }\varphi$ in a school of four swimmers as done in (B). At moderate phase lags, the school stays cohesive (top plot) but swimmer 4 barely gets any power savings (bottom plot). (D) By laterally offsetting the swimmers, four swimmers, all flapping inphase, form cohesive schools with different patterns, e.g., with side-by-side pairing of two swimmers, staggered, and diamond patterns. The time evolution of separation distances is shown in Figure 10—figure supplement 1. Individual in each pattern receive a different amount of hydrodynamic benefit. Diamond formation provides the most power saving for the school as anticipated in Weihs, 1973, for a school in a regular infinite lattice. In (A, B, and D), %values indicate the additional saving or expenditure in cost of transport relative to solitary swimming.

Figure 10—figure supplement 1. Alternative formations of four flapping swimmers.

Vortex sheet simulation of four swimmers with alternative formations. On right hand side, we report pairwise spacing between them. The lateral distance is ${\displaystyle \ell =0.25L}$. (A) Two leading swimmers swim side by side. The third and forth swimmer collide to each other. (B) The same configuration as in (A) with larger initial distance between the third and forth swimmer. They form a stable school. The forth swimmer stays at the second equilibrium behind the third swimmer. (C) An additional swimmer is placed side by side to the second swimmer in a school of three inline swimmers. The second and third swimmer stays at the second equilibrium from the first swimmer, and the forth swimmer stays at the second equilibrium from the second and third swimmer. (D) An additional swimmer is placed side by side to the third swimmer in a school of three inline swimmers. The last two swimmers stay at the second equilibrium from the second swimmer. (E) Four inline inphase swimmers when initially placed close to the second equilibrium. Power saving per swimmer is reported in Figure 10.