How do birds flock? Uncovering aerodynamic mysteries through mathematics

Findings have potential applications for transportation and energy.


How does a flock of birds fly in a coordinated and seemingly effortless manner?

A new study from a team of mathematicians at New York University suggests that the answer lies in precise and previously unknown aerodynamic interactions. Aerodynamics’ impact depends on the size of the flying group. It is beneficial for small groups and disrupts large ones.

In small bird flocks, the aerodynamic interactions enable each member to maintain a particular position relative to their leading neighbor, but in larger groupings, an effect causes members to be moved from their positions and may result in collisions.

Previously, scientists determined how birds move in groups. In this new study, scientists conducted experiments mimicking the interactions of two birds.

The researchers developed mechanical flappers that mimic the columnar formations of birds, in which the birds line up one behind the other. The plastic wings, which were 3D printed and used motors to flap in water, were designed to mimic the airflow around a bird’s wings during flight. A video of the experiment showed how this “mock flock” moved through the water and could freely arrange itself in a line or queue.

Depending on the group size, the flows had varying effects on the group’s organization.

The scientists found an effect in which each flyer in small groups of up to four receives assistance from the aerodynamic interactions in maintaining its position in relation to its neighbors.

Leif Ristroph, an associate professor at New York University’s Courant Institute of Mathematical Sciences, said, “If a flyer is displaced from its position, the vortices or swirls of flow left by the leading neighbor help to push the follower back into place and hold it there. This means the flyers can automatically assemble into an orderly queue of regular spacing with no extra effort since physics does all the work.”

“For larger groups, however, these flow interactions cause later members to be jostled around and thrown out of position, typically causing a breakdown of the flock due to collisions among members. This means that the very long groups seen in some types of birds are not easy to form, and the later members likely have to constantly work to hold their positions and avoid crashing into their neighbors.”

Then, the scientists used mathematical modeling to comprehend the fundamental forces guiding the experimental data. They concluded that similar to how springs keep train cars together, flow-mediated interactions between neighbors serve as the actual springs holding each member in place.

However, because these “springs” only operate in one direction—a lead bird can push a follower, but not vice versa—later members tend to reverberate or oscillate erratically due to this non-reciprocal interaction.

Joel Newbolt, an NYU graduate student in physics at the time of the research, said, “The oscillations look like waves that jiggle the members forward and backward and which travel down the group and increase in intensity, causing later members to crash together.”

Scientists named these new types of waves “floors,” based on the similar concept of phonons, which refer to vibrational waves in systems of masses linked by springs and are used to model the motions of atoms or molecules in crystals or other materials.

Newbolt adds“Our findings, therefore, raise some interesting connections to material physics in which birds in an orderly flock are analogous to atoms in a regular crystal.”

Journal Reference:

  1. Newbolt, J.W., Lewis, N., Bleu, M. et al. Flow interactions lead to self-organized flight formations disrupted by self-amplifying waves. Nat Commun 15, 3462 (2024). DOI: 10.1038/s41467-024-47525-9


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