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

Reduced frequency — where it appears

A frequency made dimensionless on the chord and the flight speed, being the number of chords the flow travels per radian of oscillation. It decides whether a flow may be treated as quasi-steady, and above about 0.05 it may not.

Named by 8 essays across 3 fields — each of them below, with the objects they name alongside it.

One calculation is about the animal; the other is about how fast it happens to be going. The mean lift coefficient required of each animal's wings, computed two ways, on a log scale. Treating the wings as fixed and flying them at the animal's forward speed gives answers spanning a factor of 29 — from 0.88 to 26 — because the number is governed by a speed that has nothing to do with how the animal makes its lift. Doing the flapping arithmetic gives answers spanning a factor of 1.90. And in a hover the fixed-wing calculation has no answer at all: there is no dynamic pressure, and no coefficient however large will do. That is the version of the famous claim that is actually true, and it is a statement about the calculation rather than about the bee.

The bee that cannot fly

The claim has a traceable origin and the calculation behind it was a real calculation done with the wrong velocity. Doing it with the right one gives a number an aerofoil might plausibly produce — and still leaves a gap, and the gap is what took another sixty years to close.

misconceptions · Bumblebee Quasi-steady stops being true a long way before one. The magnitude of Theodorsen's function, which is the factor a quasi-steady lift calculation is wrong by, and the phase the lift lags the motion. Quasi-steady means C = 1, and the amplitude is already one per cent low at k = 0.0061 and fifteen per cent low at k = 0.1 — a reduced frequency at which nobody hesitates to call a flow quasi-steady. The lag is worse: it reaches a degree at k = 0.003, and a flutter calculation is decided by phase rather than by amplitude.

Slow enough to be steady

A wing moving slowly enough is assumed to carry the lift its instantaneous angle asks for. The reduced frequency has two thresholds — one where the apparent-mass and circulatory lifts are equal, and one where the quasi-steady answer stops being right — and they are a hundred and seventy-eight apart.

regimes · Reduced frequency Two roots walking towards each other, and one of them crosses. The roots of the characteristic quartic in the complex plane as the airspeed is raised from nothing to 105 metres per second — growth rate across, frequency up. At rest the two sit on the imaginary axis at the uncoupled frequencies. As the speed rises, the aerodynamic coupling drags them towards each other in frequency while pushing one left and the other right, and at 80.8 metres per second the right-hand one crosses the axis. Everything about the failure is in this picture: the coalescence, the crossing, and the fact that the flutter frequency is neither of the two the structure started with.

The shake that is not resonance

A steady airstream contains no oscillation at any frequency, so nothing is driving anything. What happens instead is that the aerodynamic forces couple two structural motions that were independent, drag their frequencies together, and turn one damping negative — and the wing takes the energy out of the air itself.

circulation · Flutter The damping crosses zero once, and the crossing is the boundary. The least damping ratio of the four aeroelastic modes against speed. It falls through zero at 80.843 metres a second, and at that speed the crossing root's real part is four parts in 10¹⁸ — which is what an algebraic condition on a quartic with real coefficients looks like when it is solved numerically.

The speed where the damping is exactly zero

Flutter is an algebraic condition on a quartic: one root crosses the imaginary axis, at one speed, exactly. And the quantity that locates it is so nearly flat there that the standard way of finding it from flight test overshoots by a quarter.

circulation · Flutter The wake's memory, as a gain and a phase. Theodorsen's lift deficiency against reduced frequency. It is one at zero frequency — the quasi-steady limit, where the wake has had time to convect away — and falls to a half at high frequency, with a phase lag peaking near 15 degrees in between.

The lag that makes flutter possible

This site's own flutter model set the lift deficiency to one and recorded in its notes that doing so throws away the lag which stabilises the torsion mode. Putting the lag back moves the flutter speed from 80.8 metres a second to 131, and removes the need for the structural damping that was covering the artefact.

circulation · Flutter One incidence, two lifts. Lift coefficient against incidence for a wing pitched sinusoidally through the stall, with the static curve for comparison. The loop is traversed anticlockwise: at twelve degrees the wing carries 0.22 more lift going up than coming down.

Two lifts at one incidence

A wing pitched up and down through the stall does not retrace its own lift curve. At twelve degrees it carries 0.22 more lift going up than coming down, and the loop that opens between the two is the work the airstream does on it — which is where the energy for a stall flutter comes from.

circulation · Vortex lift Eight numbers, one construction. The dimensionless groups this collection has produced, placed on a logarithmic axis at a representative value. Each is a memory time divided by a process time, each was named separately in a different field, and each decides the same question: whether the past is still present.

Every memory number is one time over another

These essays produced eight dimensionless groups in eight different fields, named after eight different people, spanning a factor of four hundred in value. Written out, all eight are a memory time divided by a process time, and all eight govern the same curve.

regimes · Dimensional 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.

AeroelasticityFlutterQuasi-steadyUnsteady liftMeasurementMemory kernelModel validityWakeCirculationConvolutionDampingDivergence

All concepts