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 5 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

Named alongside it

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

MeasurementCorrelationDissipationFourierModel limitModel validityRegimeSpectrumStatisticsTurbulenceAdded massAeroelasticity

All concepts