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

Abstract

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 arrangements within the two-dimensional interaction plane. It is discovered that there is a \\textit{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 it, causing 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 arrangement and the stable equilibrium is further verified for freely-swimming foils where dynamic recoil motions are present. When constrained, the swimmers experience a collective thrust and efficiency increase up to 100\\% and 40\\%, respectively, in a side-by-side arrangement, whereas the staggered arrangements output an even higher efficiency improvement of 87\\% with a 94\\% increase in thrust. For freely-swimming foils, the recoil motion attenuates the improvements at the stable equilibrium, showing a more modest speed and efficiency enhancement of up to 9\\% and 6\\%, 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.