# The Aerodynamic Mechanisms of the Formation Flight of Migratory Birds: A Narrative Review
## 3 Results
### Theoretical and Experimental Studies
Table 2 shows some of the WTS values estimated by various authors, together with an estimate of the associated energy savings. Figure 4: Geometric positioning of birds relative to each other within an organized formation flight. Many years after the pioneering study by [PERSON] and [PERSON] [28], theoretical methods aimed at deciphering the mechanisms contributing to energy savings during formation flight have evolved considerably. [PERSON] and [PERSON] [40] developed an optimal formation pattern aimed at reducing the power required during formation flight in birds by approximating the wing with a horseshoe vortex. The analysis was based on a rigid, rectangular wing without flapping, which the authors believed would be suitable for large birds flying in formation over long distances and flapping their wings slowly. Assuming strict uniformity between all birds, encompassing factors such as wingspan, weight, and aspect ratio, the authors used a method in which the vortex system is simplified. The aerodynamic profile is replaced by a bound vortex, while the trailing wake is represented by two tip vortices. As Figure 5 suggests, the wake of a flapping bird is different from that of a gliding bird. Various models of leader position change were also investigated, comparing the rates of power reduction required. The theoretical study of the aerodynamics of formation flight has revealed that V-formation is optimal for long-distance flights. This configuration offers a significant reduction in power requirements and an even distribution of drag among the birds, resulting in significant energy saving. [PERSON] and Speakman [41] used an innovative approach, drawing on both observation of formation flight in pink-footed geese (_Anser brachyynchus_) and aerodynamic theory. In their study, they photographed 54 birds from the ground to assess distances and angles, revealing that a significant number of birds did not correspond to the position theoretically predicted to maximize aerodynamic economies. They estimated that with an average wingtip spacing, the induced power saving would be 14%. A few years later, [PERSON] and [PERSON] [42] applied the same technique to photograph 25 formations of greylag geese (_Anser anser_). Their results revealed substantial variation in positioning, with only 17% of birds corresponding to the optimal position expected for aerodynamic saving. Using the assumptions of [PERSON] and [PERSON] [41], they proposed an average induced power saving of 26.5%, resulting in a 5-9% reduction in total flight costs. With the leitmotiv of understanding the aerodynamic implications leading to a reduction in the power required for formation flight, [PERSON] [43] demonstrated that for maximum power reduction, the trailing bird wing should be located as close as possible to the wake of the leading bird wing. He also suggested that the total reduction in flight power for the whole formation depends strongly on the number and lateral distance of the wings. \begin{table}
\begin{tabular}{c c c} \hline  & **WTS** & **Energy Savings** \\ \hline [PERSON] and Hainsworth [11] & \(-\)16 cm & 51\% \\ \hline [PERSON] [37] & \(-\)19.8 cm & 36\% \\ \hline [PERSON] [39] & \(-\)33.7 cm & 30\(-\)40\% \\ \hline \end{tabular}
\end{table} 
Table 2: Wing-tip spacing (WTS) of the follower bird compared to the leader bird (Canadian geese). In brackets, energy savings expressed as a percentage. Figure 5: Representation of the wake shape behind a migratory bird in flapping flight (**a**) and gliding flight (**b**) as determined by numerical simulation. Results obtained by numerical simulations were carried out by the authors of the present study. Observations of migrating cranes from a helicopter have shown that, in formations of this species, the bird at the top flies at the lowest position, and all the other birds are slightly offset in height. This subtle arrangement is necessary to achieve maximum energy saving in the case of a wake vortex.