Concept

Wagner function — where it appears

The fraction of its eventual circulation a wing has built, as a function of how far it has travelled since starting. It is about half at the first instant and reaches three quarters after four chords, because the starting vortex it shed is still nearby and still felt.

Named by 2 essays across 2 fields — each of them below, with the objects they name alongside it.

The lift does not arrive when the incidence does. Bound circulation against distance travelled, in units of the settled value. The wing starts far short of its final lift and takes tens of chords to collect the rest, because every scrap of circulation it takes has to be paid for with an opposite vortex shed behind it, and that vortex's own downwash holds the wing back until it is far away. Wagner's exact 1925 answer is drawn beside the model in the colour this site keeps for a borrowed claim: the shapes agree and the model is slower, by about a fifth at its worst.

The lift that arrives late

A wing set into motion does not have its lift. Every scrap of circulation it takes has to be paid for by shedding an equal and opposite vortex behind it, and until that debt is far downstream its own downwash holds the wing back — for tens of chords, not for an instant.

circulation · Unsteady lift $Where four insects have to turn round. Wagner's function — the fraction of its eventual circulation a wing has built after a stated distance of travel — with the half-stroke of each of four insects marked on it. Every one of them reverses while the curve is still climbing, so no insect wing ever reaches the circulation a steady calculation assigns it.$

A calculation with no memory in it

The bee calculation is famous for using the wrong velocity. Done with the right one it still falls short, and the reason is structural: a quasi-steady sum is a statement about a wing that has always been going, and an insect's wing travels between two and five chord lengths before it turns round.

misconceptions · Bumblebee

Named alongside it

The objects these essays reach for when they reach for this one.

Circulation Blade-element Bound vortex Downwash Flapping flight Induced velocity Kelvin's circulation theorem Kutta condition Leading edge vortex Memory kernel Model validity Quasi-steady

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