Two-dimensionally stable self-organization arises in simple schooling swimmers through hydrodynamic interactions

Abstract

We present new experiments and free-swimming simulations of a pair of pitching hydrofoils interacting in a simple school. The hydrofoils have an out-of-phase synchronization and their arrangement is varied from in-line to side-by-side arrangements through a series of staggered arrangements representing the two-dimensional interaction plane. It is discovered that there is a two-dimensionally stable equilibrium point for a side-by-side arrangement. In fact, this arrangement is super-stable meaning that hydrodynamic forces will passively maintain this arrangement even under external perturbations and the school as a whole has no net forces acting on, drifting it to one side or the other. Moreover, previously discovered onedimensionally stable equilibria driven by wake vortex interactions are shown to be, in fact, two-dimensionally unstable, at least for an out-of-phase synchronization. Additionally, the stable equilibrium arrangement is verified for freely-swimming foils undergoing dynamic recoil motions. When constrained, the swimmers experience a collective thrust and efficiency increase up to 100% and 40%, respectively, in a side-by-side arrangement. However, in a staggered arrangement where there is direct vortex impingement on a follower, an even higher efficiency improvement of 87% is observed, which is coupled with a 94% increase in the thrust. For freely-swimming foils, the recoil motion attenuates the performance improvements showing a more modest speed and efficiency enhancement of up to 9% and 6%, respectively, when the swimmers are at their stable equilibrium. These newfound schooling performance and stability characteristics suggest that fluid-mediated equilibria may play a role in the control strategies of schooling fish and fish-inspired robots.collective locomotion | hydrodynamic interactions | fish schooling | pattern formation | collective performance M. K. helped design the study, gathered and processed experimental measurements, and drafted the manuscript. A. M. helped design the study, gathered and processed the numerical data, and helped revise the manuscript. K. M. helped design the study, and helped revise the manuscript.

Generalization of waving-plate theory to multiple interacting swimmers

“…Furthermore, for the case of anti-phase pitching (Fig. 5.8c), our prediction of a single stable inline equilibrium is consistent with recent experiments on pitching foils in a water tank [36].…”

“…Our predictions in Fig. 5.8c are qualitatively consistent with recent experiments on pitching hydrofoils in a water tank with an imposed free stream flow [36], in which a single in-line equilibrium with closely-spaced foils was observed.…”

“…Most previous laboratory experiments constrained the vertical positions of swimmers, and thus only observed the inline equilibria, but this constraint could be removed to search for staggered arrangements (e.g. [36]). For biological systems, the vertical constraint is completely artificial, so staggered arrangements would be expected to be just as likely as inline ones.…”

Intermittent swimming of two self-propelled flexible fins with laterally constrained heaving motions in a side-by-side configuration

“…However, it should be noted that the propulsive performance in this case is underestimated due to the dynamic recoil motion (i.e. opposite lateral motion of the leading and trailing edges) that occurs (Kurt et al 2019(Kurt et al , 2021Lin et al 2021). Based on these observations, it is reasonable to infer that the occurrence of equilibrium lateral states is a common phenomenon for two side-by-side anti-phase flapping fins and a single fin near the ground during continuous swimming regardless of the conditions of the model set-up, such as flexibility, moving or stationary conditions and types of active motions.…”

“…Furthermore, the presence of equilibrium states for two side-by-side anti-phase flapping rigid fins and a single rigid fin near the ground during continuous swimming has been proved in theoretical studies with mathematical expressions for the hydrodynamic forces acting on rigid fin(s) with constrained lateral motions in tethered (Baddoo et al 2020(Baddoo et al , 2021 and self-propulsion systems (Oza, Ristroph & Shelley 2019). Although the models using the constrained lateral motions above are unable to move toward an equilibrium lateral position while reacting to the surrounding fluid, an idealized pure pitching motion of two side-by-side anti-phase flapping rigid fins and a single rigid fin under a ground effect to move freely in the y-direction has shown flow-mediated organization (at equilibrium positions) in the y-direction (Kurt et al 2019) and in the x-and y-directions (Kurt et al 2021;Lin et al 2021). However, it should be noted that the propulsive performance in this case is underestimated due to the dynamic recoil motion (i.e.…”