# The Aerodynamic Mechanisms of the Formation Flight of Migratory Birds: A Narrative Review
## 3 Results
### Numerical Studies  
Numerical studies of the aerodynamics and organization of birds in formation use computational models to simulate and comprehend the complex phenomena associated with formation flight. These models help researchers handle a range of variables and conditions to analyze how birds interact aerodynamically and organize their formation. These numerical studies help deepen our understanding of the aerodynamic mechanisms and organization of birds in formation flight, providing valuable information for the biology of flight and the design of nature-inspired drone systems. Among the methods used are the standard two-dimensional lattice Boltzmann method (LBM), computational fluid dynamics (CFD), including the vortex lattice method, and tools such as Matlab and Ptera software. As shown in Figure 8, the system of equations depends on the scale of modeling (microscopic, mesoscopic, or macroscopic). [PERSON] et al. [60] used the two-dimensional lattice Boltzmann method (LBM) to solve the Navier-Stokes equations. The authors introduced a model for investigating the flow over cylinders placed in a V-formation, and drew a basic comparison with birds in flight. The impact of formation angle, spacing in the flow direction and number of cylinders on parameters such as the drag coefficient, lateral force coefficient and Strouhal number was investigated. The results indicated that reducing the formation angle resulted in lower drag, with the leading cylinder experiencing the lowest drag. This counter-intuitive finding contradicts the observations of other researchers, as the lead bird is generally the one that expends the most effort [7; 37; 61]. This increase in drag for the leader of a formation is generally accompanied by a decrease for athletes in the leader's wake. This is what is observed in sports, where the principle of drafting stipulates that an athlete can benefit from the aerodynamic wake created by the athlete in front of him, while protecting himself against air resistance [30; 62]. The simplicity of this study provided a basic understanding of some aspects of birds in flight in V-formation, revealing a trade-off between energy conservation and lateral forces. The results were in line with the positioning rules proposedby previous studies for artificial birds in V-formation. Overall, the study offered insights into the forces exerted on birds in V-formation and their implications in energy conservation. Some studies are based on mathematical modeling and numerical simulation methods. This is the case of [PERSON] and [PERSON]'s study [63], which presents an approach to animating flocks of birds flying according to line formation models. The method used distinguishes between the behavior of these flocks during initiation and their behavior during stable flight. A data-driven approach using an energy-saving model was proposed for animating stable flight. The authors showed that this problem could be formulated as a system of nonlinear equations and solved. The authors calculated the power reduction value for each bird in the flock. Using a mathematical model [36] which assumes a two-dimensional flock, with equal speeds and non-flapping wings. Although interesting, the initiation algorithm has several limitations, such as the inability to handle inverted shapes (such as inverted U's or V's) that exist in nature. The method, which is inspired by the actual position of birds within formations, offers interesting prospects, although it cannot, for the moment, model the flapping of bird wings. Among the numerical approaches used to study the unsteady dynamics of flapping or fixed wings, computational fluid dynamics (CFD) is one of the most widely used. An example of the results obtained with the CFD method is shown in Figure 9. [PERSON] et al. [64] used this method to study the flight mechanics of the Canadian goose. A two-joint arm model was used to assess unsteady aerodynamic performance and estimate potential energy savings for geese during migration.