# The Aerodynamic Mechanisms of the Formation Flight of Migratory Birds: A Narrative Review
## 3 Results
### Numerical Studies
The flow around a Canadian goose wing geometry was studied using a CFD code that implements the solution of the time-averaged Navier-Stokes equations (RANS). From the velocity and pressure distributions on the wing, the researchers found that modifying the wing morphology reduced the induced drag and saved around 15% energy. Analysis of the flow in the wing wake revealed that a pair of beneficial alternating three-dimensional flapping vortices (FAVs) were generated. A goose positioned behind could save around 19% of its energy during the wing's downward phase and 14% during the upward phase. The researchers deduced an optimum depth of around 4 m from the wingtip and an optimum spacing between wingtips ranging from 0 to \(-0.40\) m. The differences in drag and lift coefficients between the different wing- flapping modes showed that the typical wing-flapping case was the most effective in terms of drag reduction, resulting in energy saving of around 16%, which is consistent with estimates  
Figure 8: Unlike CFD methods that numerically solve the conservation equations for macroscopic properties (i.e., mass, momentum, and energy), LBM models the fluid as fictitious particles, and these particles undergo successive processes of propagation and collision on a discrete lattice. from the most recent studies [13; 41; 42] that estimated these energy saving values to range between 11.4 and 26.5%. [PERSON] et al. [65] used the unsteady vortex lattice method (UVLM) to simulate the flight of flapping wings in formation. Their simulation considered the aerodynamic coupling between flapping wings and their interactions with wake vorticity. Motivated by field observations of migratory birds, the numerical analysis revealed that flying in a V-formation with optimal spacing led to a significant increase in lift and thrust, while saving energy costs. This improvement is explained by the inherent interaction between the following birds and the wake vorticity generated by the lead bird. The researchers also examined the effect of wake decay on the aerodynamic performance of flapping wings, reporting a limited influence on the generation of aerodynamic forces and power. It should be noted that the unsteady vortex lattice model (UVLM) does not account for viscous effects, and neglects wing deformability and fluid-structure interactions. This limitation limits its accuracy in modeling scenarios involving flow separation and high levels of interaction with the wing wake, introducing a potential risk of discrepancy between the results of this study and the true physical nature of this complex phenomenon. Vortex lattice methods (VLM) are often used in the early stages of aerodynamic studies because they calculate the flow around a wing with a rudimentary geometric definition. However, several authors have employed this method for advanced aerodynamic analyses. For example, [PERSON] et al. [66] used Ring and Horseshoe Vortex Lattice Methods to analyze the drag of wings arranged in a V-formation with different wingtip spacings. In addition, the authors 3D printed NACA 2412 airfoils and arranged them in a V-formation in a laminar flow water tunnel. They subsequently carried out Particle Image Velocimetry (PIV) measurements to study downwash and upwash flows. Theoretical investigations into drag reduction during formation flight were reviewed. It should be noted that in this study, analytical fluid dynamics was carried out with certain assumptions to model the aerodynamics of V-formation flight. In modeling the drag, the authors have assumed that the wings fly in a laminar flow regime (low Reynolds number). They also assumed that the flow pattern was incompressible and the air density constant. Furthermore, their study was carried out assuming that the downwash velocity of the wing was constant along the span, and that there was no turbulence in the wake. Despite these simplifying assumptions, the results of the numerical study are consistent with literature data showing that wing tip spacing (WTS) plays an important role in wing-induced drag.