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

Fourier — where it appears

The representation of a field or a signal as a sum of sinusoids, one amplitude and one phase for each wavenumber or frequency. A spectrum keeps only the amplitudes, so everything that distinguishes two fields sharing a spectrum is in the phases.

Named by 2 essays across 2 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

Named alongside it

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

PhaseSpectrumStatisticsAutocorrelationCascadeCorrelationDesign loadExtreme-valueInertial rangeKolmogorov's theoryMeasurementModel validity

All concepts