Bird Migration V Formation - 2D Wake Vortex and Upwash Map

Why a V saves energy

A flying bird must produce lift L = W = m g, and doing so leaves behind a pair of wing-tip vortices that push air downward between them and upward outside them. With elliptic loading the circulation at the root is Γ0 = 4W / (π ρ U b), the vortices are a distance s = πb/4 apart, and the downwash at the wing is w0 = Γ0 / (2b). The induced drag is Di = W w0 / U = L² / (½ ρ U² π b²). A follower flying in the upwash outside the leader's wing-tip vortex sees a smaller effective downwash, w₀ minus the mean upwash, and so pays proportionally less induced drag.

The model

This is the far-wake (Lissaman and Shollenberger, 1970) idea: every bird upstream contributes the field of its vortex pair, with a small viscous core to avoid infinities, averaged across the follower's wingspan in closed form. The flock saving is the mean reduction of induced drag; the range gain converts it into total power using the share of flight power that is induced (about 40 to 50 percent for a goose at cruise).

Honest simplifications: vortices are straight and do not roll up, decay or sink; the follower's flapping phase and wing motion are ignored (real birds also stay in the upwash by timing their wingbeats); induced drag is floored at zero for a bird that would receive more upwash than it needs; only induced power changes, profile and parasite drag are the same for all birds.

• Choose Geese V and read the colours: the leader is red (full drag), followers turn green.
• Slide the sideways offset and watch the graph of upwash: the best position is just outside the leader's wing tip, not directly behind.
• Choose Too wide to see followers leave the upwash and lose the benefit.
• Add birds with the first slider and watch the range gain curve on the third graph climb and flatten.