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.

A cyber–physical underwater vehicle is equipped with bio-inspired flapping fins positioned on the sides of the vehicle’s main body. The proposed control surfaces are inspired by fish pectoral fins, generating forces and moments that can potentially be harnessed for maneuvering, hovering and station keeping. The streamwise and cross-stream forces produced by the fins are characterized for a range of reduced frequencies and Strouhal numbers. The streamwise forces are shown to be predominantly a function of the fin’s projected frontal area, while the lateral forces also depend on the Strouhal number. When operated simultaneously, different flapping synchronizations can be employed for specific goals; a symmetric motion suppresses the lateral forces, while an anti-symmetric motion decreases the peaks of the streamwise force produced. The cyber–physical vehicle demonstrates how the pair of fins can successfully maneuver the vehicle in the lateral direction.

New free-swimming experiments and simulations are conducted on a pair of three-dimensional, bio-robotic swimmers composed of a body and tail section based on Yellowfin tuna, Thunnus albacares. It is discovered that the pair converges spontaneously to a side-by-side schooling formation that is stable to perturbations in the swimming direction at a fixed lateral spacing. We reveal that for close lateral spacings of 43% of the body length and thick, tuna-like bodies with a 22% thickness-to-length ratio, the flow between the swimmers is accelerated in a "channeling effect" due to flow constriction. Consequently, this creates a low-pressure zone that is the primary mechanism generating a fluid-mediated restorative force, thereby making the side-by-side formation hydrodynamically stable. This quasi-steady mechanism makes the stability of the formation insensitive to the phase synchronization between the bio-robots in contrast to previous results for schooling foils. Moreover, in the side-by-side formation tunalike swimmers are seen to have only a small reduction in their swimming speed and a concurrent small rise in their cost of transport. By leveraging this channeling effect, bio-robotic schools may be able to maintain a schooling formation with little or no control. This flow mechanism may also be present in biological schools of tuna-like fish where it may sculpt the formations observed in nature.