# The Aerodynamic Mechanisms of the Formation Flight of Migratory Birds: A Narrative Review
## 3 Results
### Theoretical and Experimental Studies
To mitigate these potential problems, the authors have incorporated the fundamental effects of viscosity using the concept of zero drag. Formation flight was defined as the steady-state solution to the basic equations, particularly the solution in which all birds fly at the same speed. Still on a purely theoretical approach, [PERSON] [50] analyzed the potential reduction in power when flying in formation by considering the wing as a horseshoe vortex. Various patterns of leader change were investigated, comparing the rates of power reduction required, including V-formations. Using an approach similar to that of [PERSON] and [PERSON] [40], a rigid, rectangular, non-flapping wing was considered. [PERSON]'s theoretical study revealed that the U-formation (see Figure 6) would be optimal for long-distance flying. The U-formation would provide a significant reduction in power requirements compared to solo flight, and significant energy savings on extended flights. To our knowledge, [PERSON]'s study [50] is the only one that argues in favor of U-shaped formation. The theoretical results of this study need to be compared with experimental data to evaluate the potential energy savings of U-shaped formation compared to other types of formations. The constant development of algorithms has led [PERSON] et al. [51] to use a new approach called metaheuristics, which mimics the behavior of migratory birds. This so-called \"Migratory Bird Optimization\" (MBO) algorithm is inspired by the V-shaped formation of migratory birds. Parameters such as bird number, speed, wingtip spacing (WTS), and wing flapping were incorporated into the optimization model. Nevertheless, crucial details on bird species or size, which play a fundamental role in energy savings, are not disclosed. One of the studies that can be considered a major contribution to the scientific literature in the field of migratory bird aerodynamics was published in 2014. In this study, [PERSON] et al. [52] investigated the V-formation flight of bald ibises (_Geronticus eemita_). The findings of their study indicate that, in V-formation flight, ibises occupy positions mathematically predicted by fixed-wing aerodynamics, as described notably by [PERSON] and [PERSON] [28]. In-flight measurements showed that the birds flew precisely as predicted by theoretical simulations, about one meter behind the lead bird and another meter to the side. Some ibises showed a preference for flying on the right or left side of the V, while others favored the center or edges. However, the overall pattern indicated frequent exchanges of positions, and the group lacked a consistent leader. They also suggested that the wake of flapping birds (in their study, ibises spent 97% of their time flapping) is likely to be much more complex than predicted by theoretical studies. In addition, they showed that the birds exhibited coherence in wingtip trajectories when flying in V formation, beating their wings spatially in phase to maximize upwash capture throughout the wingbeat cycle (Figure 7). In contrast, when flying directly behind another bird, they adopted a spatially antiphase wingbeat, potentially to mitigate the detrimental effects of the downwash. [PERSON] et al. also suggested that energy saving could be increased by 20% when wing flapping is performed optimally in a spatial phase compared to out-of-phase wing flapping. These aerodynamic achievements, which might be considered impossible for birds due to the complexity of their flight dynamics, suggest an awareness of the spatial structures generated by the wakes of neighboring birds. These results provide an in-depth understanding of the complex aerodynamic interactions in birds' V-formation flight, demonstrating their remarkable ability to anticipate or detect these dynamic wakes. The investigation carried out by [PERSON] and colleagues, incorpo  
Figure 6: Example of an inverted V-formation (**a**) and an inverted U-formation (**b**).