Fluid Flow & Aerodynamics
why things fly, and why the usual explanation is wrong
Concept

Gust — where it appears

A change in the wind a wing meets, treated as an input to the wing's unsteady response rather than as a new steady condition. Its load is computed by convolving the gust's shape with Küssner's indicial function, which is not the function used for a change of incidence and differs from it by a factor of eight at one semichord.

Named by 3 essays across 2 fields — each of them below, with the objects they name alongside it.

The mean lift is not the lift at the mean angle. A finite wing's lift curve, with a gust distribution of standard deviation 3° about a mean angle of 10° drawn along the foot. The wing spends its time spread across that distribution, so what it averages is the average of the curve: 0.7951 against the 0.8368 the mean angle promises, a deficit of 5.0 per cent. Nothing has stalled, and no gust has taken the wing past the stall angle: the deficit comes entirely from the curve bending over, and it is there at every angle where the curve is not straight.

The lift at the mean angle

A wing in rough air flies at every angle in turn, so what it averages is the average of its lift curve rather than the lift at its average angle. Where the curve bends over near the stall the two differ by five per cent at three degrees of gust and by fourteen at five, and the drag goes the other way.

circulation · Gust Wagner's function and Küssner's, from one solver and two inputs. Lift as a fraction of its steady value, against distance travelled in semichords. The step in incidence and the sharp-edged gust are the same unsteady problem with two different right-hand sides, and Jones's exponential fits to both are drawn over the solve. From four semichords on they are nearly the same curve — which is why they get interchanged.

Two answers to one question

Unsteady aerofoil theory collapses a wing's whole history onto one function of one variable, and every quasi-steady gust calculation convolves something with it. There are two such functions, not one: a wing that is pitched changes its boundary condition everywhere at once, and a wing flying into a gust has not met most of the gust yet.

circulation · Unsteady lift A gust's lift keeps falling where a pitching wing's stops at a half. The magnitude of Sears' function, the lift a wing gets flying through a sinusoidal gust as a fraction of the quasi-steady value, against the reduced frequency on a logarithmic axis, beside Theodorsen's function for a wing that pitches or heaves. The two agree at low frequency. Above a reduced frequency of about a tenth they part: Theodorsen's levels off at one half, because a moving wing changes its whole boundary condition at once, while Sears' keeps falling as one over the square root of 2πk, because several wavelengths of gust lie along the chord and cancel.

The gusts that cancel along the chord

A wing that pitches keeps half its circulatory lift however fast it moves. A wing flying through a gust does not: once the gust is a few chords long, its ups and downs lie along the chord together and cancel, and the lift falls without limit. For an airliner that barely touches the root-mean-square gust load, and cuts the load spectrum at the wing's own torsion frequency to a quarter of the quasi-steady value.

regimes · Reduced frequency

Named alongside it

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

Unsteady liftLoad factorQuasi-steadyAdded massAveragingCirculationDrag polarDynamic pressureFlight envelopeIndicial responseKelvin's circulation theoremLift curve

All concepts