# The Aerodynamic Mechanisms of the Formation Flight of Migratory Birds: A Narrative Review
## 3 Results
### Numerical Studies
Their study shows that by increasing wingtip spacing over a certain distance, there is no change in drag coefficients, and maximum drag  
Figure 9: Example of numerical simulation result obtained using the RANS method. Visualization of the interaction of the vortex wake between birds in a V-formation using the volume rendering method. Results obtained by numerical simulations carried out by the authors of the present study. coefficients occur when the trailing wings are positioned in the downwash region of the leading wing. More recently, [PERSON] et al. [67] have used the UVLM to investigate the combined effects of active bending, twisting and group organization on the aerodynamic performance of flapping wings. The UVLM model was used to assess the aerodynamic forces generated on flapping wings in V-shaped formations of three to five members. The study explored the effects of morphing on lift, thrust and power coefficients, as well as propulsive efficiency, demonstrating improvements in thrust and power coefficients in solo flight and in V-formation. The model used by [PERSON] et al. employs active morphing in the UVLM-based solver. Unlike various numerical studies [61; 64; 68], the wing is not modeled as a rigid, non-deformable solid. Nevertheless, the study is based on the use of a rectangular-shaped wing, which generates a different wake from a complex-shaped bird's wing. In a recent paper, [PERSON] [69] used the open-source Ptera software (V2.0.0), which employs the unsteady vortex lattice method (UVLM) to analyze flapping wing V-formations. The author of the study validated the results with experimental data while demonstrating its ability to analyze simple ornithopters. A major limitation of this method is that the fluid is considered to be inviscid. In this scenario, the model does not take into account some aerodynamic phenomena such as flow separation, leading to some inaccuracies in the results. Nevertheless, this method, less commonly used to date, seems promising for simulating, in a simplified way, the organized flight of migratory bird flocks. Different numerical methods can be used to study the aerodynamics of migratory birds, each having its own advantages and disadvantages. Among the most commonly used methods are the Lattice Boltzmann method and the CFD method. A comparison of these models for aerodynamic purposes was conducted in a recently published study [70]. Among the authors who have used the CFD method, we can mention [PERSON] et al. [61]. They used CFD to determine the optimum position of gliding Canada geese within a V-shaped formation, taking into account the influence of wingtip spacing and depth (see Figure 6). A computational code based on the finite volume method was used to analyze the wake and three-dimensional vortex structures developing behind the birds. The results demonstrated that flying in formation could reduce energy expenditure by minimizing aerodynamic drag while improving lift. A significant finding suggested that wingtip spacing should be around \(-26\) cm to optimize aerodynamic efficiency. These results are quite close to previous estimates by [PERSON] and [PERSON] [11], who suggested that maximum energy saving was obtained for a wingtip spacing of \(-16\) cm, while another study by [PERSON] [37] experimentally determined a median wingtip spacing of \(-19.8\) cm, and yet another study [38] reported a wingtip spacing of \(-33.7\) cm. It should be noted that this optimal position requires a lateral overlap of the birds' wingtips (see Figure 6) to achieve the best aerodynamic efficiency [11]. [PERSON] et al. [71] have shown that the spacing between the wingtips of leader and follower birds is highly variable, suggesting that tracking the lateral position of the preceding bird is a difficult task. The study by [PERSON] et al. [61] also highlighted an imbalance of forces between the left and right wings of the second bird, which could be attributed to the interplay of wakes.