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

Phase — where it appears

The relative timing of one oscillation against another, and the information a spectrum discards. Structure in a field lives in the phase relations between its scales, so two fields with identical spectra can look nothing alike.

Named by 9 essays across 5 fields — each of them below, with the objects they name alongside it.

One spectrum, two fields. Two one-dimensional fields built from the same amplitudes and different phases. Their energy spectra are identical to the last bit, their two-point correlations agree to 2e-15, their second-order structure functions are the same function — and no measurement of either kind can tell them apart. Everything that distinguishes them is in the phases, which is where the cascade lives.

The moment a spectrum cannot hold

A spectrum and a two-point correlation are the same object, transformed, so everything one of them records the other records too. Give a field the exact Kolmogorov amplitudes and independent phases and every second-order measurement comes out right — while the cascade, which lives in the phases, is not there at all.

turbulence · Spectrum What the phase reaches, and what it does not. The difference between the two records, as a fraction, for five quantities. The variance and the autocorrelation are the same to machine precision because they are the spectrum. A narrow-band linear oscillator answers its own frequency and almost nothing else, so it is nearly phase-blind too. Everything extremal — the crest, the peak drag load, the range of the running integral — is not.

The same statistics, and a different load

A wind or wave specification is written as a spectrum, and a spectrum discards the phases. Two records built from one spectrum agree in variance to thirteen figures and in peak drag load by twenty per cent — and with the phases lined up, the same spectrum is a single impulse thirty-one times worse.

misconceptions · Randomness 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 Morison's two terms over one cycle, at KC = 10. The drag term, in phase with the velocity and going as its square; the inertia term, in phase with the acceleration and ninety degrees ahead of it; and their sum, which is what a load cell records. The peak of the total is not the peak of either, and its position in the cycle is the only thing in the record that says how the two are divided.

Long enough to make a wake

A Reynolds number cannot ask whether an oscillating flow gets round a body before it turns and comes back, because it has no time in it. The Keulegan–Carpenter number can, and it decides which of Morison's two terms is the force. What it discards is the phase — and a peak force measurement cannot recover it.

regimes · Keulegan–Carpenter number Two flows with one mean profile. The time-averaged velocity of a plain shear and of the same shear carrying a zero-mean disturbance. There is one line on this plot: the largest difference anywhere across the channel is four parts in 10¹⁷.

What a mean profile cannot tell anybody

Two flows are built here with mean velocity profiles that agree to four parts in 10¹⁷. One of them carries momentum across the shear and dissipates forty per cent more energy; the other carries nothing. Everything that distinguishes them is second order in a disturbance the mean cannot see at all.

kinematics · Averaging The wake a flapping bird leaves is a wave in the air. Side view, the air at rest: the path the leader's wingtip traced, where its tip vortex now lies, over two wingbeats; lengths along the flight path in wake wavelengths — the distance flown in one beat — and heights in spans, for a tip swinging a fifth of a span each way. A follower three-tenths of a wavelength behind that beats three-tenths of a beat later traces the same path and flies along the leader's vortex all the way; one that beats half a beat off that traces the mirror image and meets the vortex only twice a beat.

The follower beats in time with the wake, not the bird

A gliding bird can sit in its neighbour's upwash and stay there. A flapping bird's wake is a wave left in the air — the path its wingtip traced, rising and falling with every beat — and a bird behind gains only if its own wing is where that wave is when it arrives. The best timing is a rule with no aerodynamics in it: lag the bird ahead by the time the wake took to come, so that each wingtip retraces the path of the one before. Directly behind, the rule flips by half a beat, and it buys a smaller loss rather than a gain.

circulation · Formation A follower can wander further fore and aft than sideways. A flapping follower's saving, averaged over a wander of its place with the standard deviation shown, as a fraction of its own induced drag: fore and aft, holding the phase that suits its average place, for tips swinging a tenth, a fifth and four-tenths of a span; and sideways, beating in phase. The saving in place is 0.837. Sideways it has halved at a wander of 0.185 spans; fore and aft, with a fifth-span swing, only at 0.804, a third of the wake's wavelength, and with the four-tenths a cruising bird swings, at 0.37. With a tenth-span swing it never halves.

A flapping follower can drift fore and aft, but not sideways

A bird in a flapping V has to be in the right place and beat at the right phase for that place, and no bird holds its place exactly. Drifting fore and aft costs it phase, at a full beat for every wavelength of the wake; drifting sideways takes it off the leader's tip vortex. The first is two to four times cheaper than the second, it can be bought back by re-timing the beat within about one beat of the drift, and the second cannot be bought back at all. So the precision a follower needs is sideways, and the attention it needs is on its timing.

circulation · Formation A lagged re-timing passes no error down the V. The variance of each bird's phase error, in radians squared, against its place in one arm of a V, for fore-and-aft wander of half a span correlated over four beats. Holding a fixed phase, every bird's error is its offset from the bird ahead, 3.16. Re-timing with a one-beat lag, every bird's error is 0.632 — the first follower's and the thirtieth's alike, to the last digit. Re-timing as smoothly but through two half-beat lags, the error grows from 0.819 at the first follower to 1.05 at the tenth and 1.08 at the thirtieth, and is still growing slowly there.

A wandering flock passes no error down the V

In a flapping V every bird re-times its beat to the wake of the bird ahead, whose own beat is imperfectly timed, so the errors ought to pile up along the arm. With the simplest way of re-timing they do not, at all: the thirtieth bird is off its phase by exactly as much as the first. A one-beat lag and its complement add to one at every frequency, and that identity telescopes the whole arm. Re-time more smoothly and the errors do accumulate — by about a third, and then they stop.

circulation · Formation Below separation Morison's drag coefficient is a Stokes layer's, and it grows without limit as the motion shrinks. Wang's drag coefficient for a circular cylinder in an attached oscillatory flow against the Keulegan–Carpenter number, for β of 10², 10³, 10⁴ and 10⁵, each drawn up to Hall's onset of the Honji instability. β = 10²: 2.77 at KC = 1, onset at KC 1.83; β = 10³: 0.845 at KC = 1, onset at KC 1.03; β = 10⁴: 0.264 at KC = 1, onset at KC 0.578; β = 10⁵: 0.0831 at KC = 1, onset at KC 0.325. At fixed β the coefficient falls as 1/KC.

Below separation a cylinder's drag is a Stokes layer, and KC stops deciding the force

Morison's equation splits the force on a cylinder in an oscillating flow into inertia and drag, and the Keulegan–Carpenter number says which dominates — once the flow separates and leaves a wake. Below that, nothing separates, and the drag term describes the thin oscillating boundary layer on the cylinder's surface. Its coefficient then grows without limit as the motion shrinks, as one over the Keulegan–Carpenter number, and that exactly cancels the number's power to decide anything: the drag's share of the force is fixed by the frequency parameter alone, a few per cent at most, until Honji's instability ends the regime.

regimes · Keulegan–Carpenter number

Named alongside it

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

Model limitWakeDissipationFlapping wingFormation flightInduced dragMeasurementRegimeUpwashAdded massAveragingCorrelation

All concepts