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.

### Analysis

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.

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.

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.

### Optimization

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