MMSC case study in mathematical modelling
Flapping flyers in formation
Report © 2026 George Hildick-Smith, shared under CC BY-NC-ND 4.0. Third party images and quotations remain the property of their owners.
Overview
We often notice that birds fly in flocks but why? Experiments with mechanically flapped foils show that objects interacting only through the fluid can settle into orderly formations, which suggests that fluid dynamics may drive some of the behaviour seen in flocks of birds. This project builds a one dimensional model of of flyers (e.g. birds). Each is pushed forward by its own flapping and disturbed by the wake of the flyer in front, which gives a system of delay differential equations that we solved numerically.
Starting from a single flyer, the report adds a background wake, then several flyers, and analyses which formations are stable, how a follower should flap to hold a fixed gap, and where each flyer should sit to minimise its effort, comparing selfish and cooperative strategies.
What I found
- A steady background wake (constant updraft) always raises the average speed, even though it might be expected to slow the flyer, because thrust depends on the square of the relative velocity.
- A follower stays with the leader when its amplitude times frequency matches the leader's. Flap less and it drifts away, flap more and it collides. Some stable cases exist on the other side of that line, where two distinct flapping periods occur.
- With flyers spaced at their most efficient distance, a cooperative strategy cuts the group's flapping effort by 24.3% compared with flying alone, and is more stable than each flyer acting selfishly.
