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

Abstract

We present new constrained and free-swimming experiments and simulations in the inertial regime, with Reynolds number $\mbox{Re} = O(10^4)$ , of a pair of two-dimensional and three-dimensional pitching hydrofoils interacting in a minimal school. The hydrofoils have an out-of-phase synchronisation, 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 two-dimensionally stable equilibrium point for a side-by-side formation. 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. Previously discovered one-dimensionally stable equilibria driven by wake vortex interactions are shown to be, in fact, two-dimensionally unstable, at least for an out-of-phase synchronisation. Additionally, it is discovered that a trailing-edge vortex mechanism provides the restorative force to stabilise 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 an 87 % increase in thrust. Freely swimming foils in a stable side-by-side formation show efficiency and speed enhancements 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.