Copyright © A. Filippone (1996-2005). All Rights Reserved.

Birds Formation Flight

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The formation flight of birds has been object of speculation and research for a long time. A considerable literature related to observations and theories is also available. However, the quantitative results are often in disagreement.

The problem has been analyzed with elementary aerodynamics of multi-body systems, which is an extension of Munk’s interference theory for two lifting lines. The problem consists in evaluating a drag matrix as shown in the sketch below. (data on CD-ROM)

The formation flight analyzed for some important configurations (convex vee, concave vee, planar, non-planar, etc.), then optimized by using heuristic techniques. This was required because the solution space is multimodal and strongly non-convex, and it changes unpredictably as more birds join the flock.

Heuristic strategies applied to this problem are (more info on CD-ROM):

  • downhill simplex method
  • simulated annealing
  • genetic algorithms


We define the ratio of the drag of the formation flight to that of the in-solo flight (denoted by the asterisk) Dr=(1 + D/D*).

Assume that the birds are flying line abreast. The induced drag depends only on the tip clearance, which is normalized with the wing semispan.

The induced drag increases from zero to a maximum when the birds are flying tip-to-tip. Such a limiting condition can not occur in practice due to the need to trim and control the flight. Therefore a characteristic tip clearance of s/b=1/4 has been considered in all the calculations in this analysis.

convex fee

It was found that if the system is unstaggered line abreast flight produces maximum induced lift at the central position.

The birds at the tip gain just half of these benefits. If the system is staggered, and the birds are flying in a convex vee formation, the birds at the tips gain most, due to the fully developed upwash field created by the leading birds. The bird at the apex withstands the hardest position.

concace vee

If the vee formation is a wedge, then the apex is critical. The results of the present simulation show clearly that the apex position is by far the hardest. For long flights it has to be constantly swapped, something that has been occasionally observed.


Starting from a first guess configuration the vee has been optimized using a heuristic approach. The GA has given the best results. It was assumed as objective function the the mean absolute deviation of the drag. The objective was to minimize this function.

It appears that the best configuration is a vee roughly parabolic, with a flatter nose. It is interesting to note that the flock does not have to be symmetric, and that one more bird can join the flock from either size of the apex.

Related Material

(on CD-ROM)
  • Wing Aspect-ratios of Some Birds
  • Aerodynamic Efficiency of Some Birds

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Copyright © A. Filippone (1996-2005). All Rights Reserved.