Dynamic Orbital Formations and Energetics of Free-Swimming Foil Schools

Recent research has shown that fish schooling can lead to significant energy savings when individuals maintain fixed positions relative to one another. However, empirical observations indicate that real fish often move dynamically and frequently change positions, transitioning between inline, staggered, and side-by-side arrangements rather than holding such fixed formations. This work aims to investigate whether simple, self-organized dynamic formations can remain stable without relying on closed-loop control, and whether such formations can offer energetic benefits. To this end, we introduce a two-dimensional computational framework in which pitching hydrofoils are free to swim in both dimensions and interact solely through passive hydrodynamic forces. We examine the emergence of a stable, orbital schooling behavior resulting from these interactions. We further examine the energetic benefits of these interactions. These findings offer a new perspective on schooling dynamics and may help bridge the gap between theoretical models and the unstructured behaviors observed in natural fish schools.