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

Quasi-steady — where it appears

Treating an unsteady flow as a succession of steady ones, which is valid while the reduced frequency is small. It is the assumption behind most engineering estimates of an oscillating body's forces, and its failure is what unsteady aerodynamics is about.

Named by 11 essays across 4 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 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 Where a profile stops being a parabola. Two measures of how far Womersley's solution has left the quasi-steady parabola, against the Womersley number, both logarithmic. The amplitude deficit reaches a hundredth at α = 0.28 and the phase lag reaches a hundredth of a right angle at α = 0.22 — both well below α = 1, which is where the unsteady and viscous terms are equal and is the value the number is named for. By α = 1 itself the flow is already a tenth short and eight degrees late.

Where the parabola goes

A pipe carrying a steady flow has a parabolic profile, and the Womersley number is supposed to say when an oscillating one still does. For the flow rate it is very nearly honest — one per cent at α = 0.91 — and for the phase it is out by a factor of three, because a lag is second order in the number and an amplitude deficit is fourth.

regimes · Womersley 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 A tube opened at the foot of a reservoir. Speed in a two-metre tube fed by a one-metre head, from the instant it is opened. The quasi-steady answer — Torricelli's √(2gh), which is what dropping the time derivative from Bernoulli's equation gives — is reached at the instant of opening, before any fluid has moved. The true answer is a hyperbolic tangent with a time constant of 2L/√(2gh), and it takes 2.4 seconds to come within one per cent. The whole of that transient is one term.

The pressure that depends on the past

Bernoulli's equation for an unsteady flow has a term nobody writes down and a right-hand side that is a function of time rather than a constant. The term is exactly zero once a flow has started and is the whole of the flow while it is starting, which is why it never appears in an answer and is never negligible in getting to one.

inviscid · Unsteady bernoulli 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 A bottom panel overshoots its static stress, and a lighter hull's panel is loaded less. The largest bending stress anywhere on the panel against time from first contact, for a 10° wedge entering at 5 m/s at constant speed and as free hulls of the full width's added mass and three tenths of it; solid, the panel's own response, dashed, the static stress the same pressure would give a massless panel. Constant speed: 361 MPa against 287 static; Hull = full width's added mass: 245 MPa against 195 static; Hull = 0.3 of it: 132 MPa against 106 static. The water reaches the panel at 4.5 ms and covers it at 11.2 ms at constant speed; the wetted panel's period is 7.84 ms.

A hull bottom feels the slam it has time to answer

The slamming pressure on a V-bottom is a narrow ridge at the spray root, a megapascal high, that sweeps across each panel of plating in a few milliseconds — about the panel's own period. So the panel neither feels the ridge's height nor ignores its speed: it overshoots the static stress by up to half, a rule that designs for the peak pressure asks for half as much plating again as it needs, and a light hull's slower, softer slam is gentler on its plating twice over.

inviscid · Unsteady bernoulli Over a swell the wing's lift swings further above its calm value than below it. The wing's lift coefficient through one encounter with a 50 m swell, heading into it at 50 m/s, for swells of amplitude 0.25, 0.5 and 0.7 m, with the calm sea's value. Phase zero is a crest under the leading edge. At 0.25 m it runs from 0.6 to 0.803, averaging 0.694. At 0.5 m it runs from 0.534 to 1, averaging 0.729. At 0.7 m it runs from 0.489 to 1.34, averaging 0.795. Calm, it is 0.684. One encounter takes 0.85 s.

A swell lifts a ground-effect wing more at its crests than it takes in its troughs

A wing skimming the sea gains its lift from the sea's reflection of it, and the reflection grows sharply as the gap closes. Over a swell the gap opens and closes with every wave, so the lift swings — further up at each crest than down at each trough, which raises the mean lift as well as shaking it. Climbing out of the swing is possible, but the height that calms the lift spends the cushion the craft was built for, and in a sea about two metres high it has spent nearly all of it.

misconceptions · Ground cushion On a flat bottom the plating's own give slows the water climbing it and takes a seventh off the slam; on a deep V, almost nothing. At the impact speed that loads the one-way panel to 200 MPa at each deadrise: left, the coupled panel's peak stress over the one-way panel's, against deadrise; right, the same against the panel's largest deflection over its rise across its width, w/(ℓ tan β). 4°: 2.27 m/s, ratio 0.847, deflection 3.99 mm; 6°: 2.9 m/s, ratio 0.894, deflection 4.24 mm; 8°: 3.44 m/s, ratio 0.934, deflection 4.44 mm; 10°: 3.94 m/s, ratio 0.972, deflection 4.61 mm; 15°: 4.99 m/s, ratio 1, deflection 4.81 mm; 20°: 5.8 m/s, ratio 0.983, deflection 4.69 mm.

Plating that gives way slows the water that slams it, but only on a flat bottom

The slamming pressure on a hull's bottom is a ridge that rides at the edge of the wetted width, and how fast that edge moves depends on how far the water has to climb. A panel of plating that bends inward under the load gives the water further to climb, and while it is moving away it slows the climb itself. Fed back into Wagner's condition, that takes a seventh off the peak stress of a panel on a four-degree bottom loaded to the edge of what aluminium can bear, and almost nothing on a deep V — because what matters is the plating's deflection against the bottom's own rise across the panel.

inviscid · Unsteady bernoulli

Named alongside it

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

Model limitAdded massReduced frequencyCirculationPhase lagUnsteady liftAeroelasticityBlade-elementDimensionlessFlapping flightFlutterFree surface

All concepts