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

We present new constrained and free-swimming experiments and simulations of a pair of pitching hydrofoils interacting in a minimal school. The hydrofoils have an out-of-phase synchronization and they are varied through in-line, staggered, and side-by-side formations within the two-dimensional interaction plane. It is discovered that there is a \textit{two-dimensionally} stable equilibrium point for a side-by-side formation. In fact, this formation is super-stable, meaning that hydrodynamic forces will passively maintain this formation even under external perturbations and the school as a whole has no net forces acting on it that cause it to drift to one side or the other. Moreover, previously discovered \textit{one-dimensionally} stable equilibria driven by wake vortex interactions are shown to be, in fact, two-dimensionally \textit{unstable}, at least for an out-of-phase synchronization. Additionally, it is discovered that a trailing-edge vortex mechanism provides the restorative force to stabilize a side-by-side formation. The stable equilibrium is further verified by experiments and simulations for freely-swimming foils where dynamic recoil motions are present. When constrained, swimmers in compact side-by-side formations experience collective efficiency and thrust increases up to 40\% and 100\%, respectively, whereas slightly staggered formations output an even higher efficiency improvement of 84\% with a 87\% increase in thrust. Freely-swimming foils in a stable side-by-side formation show an efficiency and speed enhancement of up to 9\% and 15\%, respectively. 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.