A theoretical analysis of formation flight as a nonlinear self-organizing phenomenon

This study analyses the existence, stability and self-organizationof formation flight utilized by migrant birds. Air is approximatedas an incompressible inviscid flow, while birds are modelledas elliptically loaded lifting-lines. Application of conventionalwing theory leads to newly derived, basic equations that describethe problem as a dynamical system of multiple wings interactingwith each other through induced flow field. Formation flightis defined as the steady-state solution of the basic equations,in particular the solution that all the birds fly at the samespeed. In the case of a prescribed thrust, constant transverseinterval between adjacent birds, and a flock of physically identicalbirds, analytical study of the basic equations reveals the factsthat (1) formation flight is self-organized and (2) this formationflight is stable. The new implication is that a configurationof formation emerges as a result of nonlinear dynamical interactionbetween many birds and that this nonlinear dynamical systemdoes not exhibit chaotic behaviour. Numerical calculation hasalso been done for cormorant-type birds with the same transverseinterval between flock members. The proposed numerical schemequickly converges to very accurate results owing to the recentlyderived, closed-form expression of induced velocity distributionaround an elliptically loaded lifting-line. Transverse intervalsbetween birds are found to be a more important factor than thenumber of birds. Configurations of formations are found to beinverted U rather than inverted V. In these formations everybird enjoys the same amount of drag reduction.