Spectrum — where it appears
Named by 28 essays across 5 fields — each of them below, with the objects they name alongside it.
Where the energy goes
Energy enters a turbulent flow at the largest scale and leaves it at the smallest, and in between there is nothing for it to depend on but the rate at which it is passing through. Two quantities and one dimensional argument fix the shape of the spectrum, and the exponent is −5/3.
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.
The one exact result
Almost nothing in turbulence follows from the equations without a model in it. One thing does — Kolmogorov's four-fifths law, which fixes the third moment of the velocity differences at −(4/5)εr with no adjustable constant anywhere in it. The companion two-thirds law is not exact, and the 4.02 everybody quotes in it turns out to be a gamma function.
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.
What decay never forgets
Stir a box of fluid and stop. The turbulence decays, at a rate with no viscosity in it — so the rate cannot come from the fluid. It comes from an invariant of the very largest scales, fixed at the moment the stirring stops, and never revisited.
The range a real Reynolds number does not have
Kolmogorov's minus five thirds is a statement about a band of scales that has forgotten the forcing and does not feel the viscosity. Both conditions are about separation, and separation is exactly what a finite Reynolds number does not have much of.
The scalar has its own cascade
Below the Kolmogorov scale there is no turbulence left, and a dye stirred into the flow goes on cascading anyway — on a spectrum whose exponent is minus one and whose amplitude contains no velocity spectrum at all. Resolving it costs the three-halves power of the Schmidt number, which for dye in water is a factor of ninety thousand.
A dissipation correlated across every scale
The dissipation is supposed to be the most local quantity in turbulence — a thing happening at the smallest eddies, everywhere and independently. A multiplicative cascade makes its logarithm correlated over a distance that is a ratio of logarithms, so two points a thousand smallest scales apart still agree a quarter of the time.
Universal, and one of five
A turbulence that has run its Reynolds number down stops being turbulent, the equations go linear, and the decay picks up a new exponent. That exponent is quoted everywhere as 5/2 and as universal. It is neither: it is the same corner of the same spectrum deciding the answer a second time.
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.
A row that meets the row before it
A compressor blade is loaded and unloaded thirty times a revolution by geometry it does not have. The forcing sits at the blade count of the row in front of it and at multiples of that, and how far up the harmonics it reaches is decided by how far apart the two rows are.
A relation with no turbulence in it
Isotropy and incompressibility alone fix the transverse structure function from the longitudinal one. Divide the relation through and it says the ratio of the two is one plus half the local slope — so the exponent everybody measures as 0.70 and the ratio everybody measures as 1.35 are one measurement, and a model spectrum with no intermittency in it produces both.
The decay inside the four-fifths law
The four-fifths law gives the dissipation of turbulence from one measured moment with no constant in it, which makes it the obvious way to measure how the dissipation coefficient travels during a decay. But the law is exact only in a limit, and a decaying flow is not in it. The decay itself takes a quarter off the moment at the Reynolds numbers grids reach — and the bias moves as the flow decays, by a sixth, in the direction opposite to the effect being looked for.
The span takes away the infinity, not the gust
A two-dimensional wing in turbulence reverses its load infinitely often, because the gust spectrum never falls fast enough. A real wing averages the gust across its span, and the average keeps ninety-five per cent of the gust's strength and removes the infinity. The load's crossing rate settles at about two per turbulence scale flown and stops depending on where anyone cut the spectrum off.
A record is as long as its integral scales
A turbulence measurement of a million samples can hold less information than one of a thousand. What sets a record's worth is not how many numbers it contains but how many integral time scales it spans, and the integral scale — the unit every other error is counted in — is itself the hardest thing in the record to measure.
The best estimate of a scale assumes its shape
There are four common ways to read an integral time scale off a turbulence record, and on the right signal the best of them is four times more precise than the usual one. On a signal whose correlation has a different shape it is off by a factor that no length of record reveals — and one of them turns out to be measuring the sampling rate rather than the flow.
Forcing at the integral scale leaves the cascade alone
A decaying flow's third moment falls a quarter short of the four-fifths law at the Reynolds numbers a grid reaches. A flow forced at its largest scales, on the same spectrum at the same Reynolds number, falls short by three and a half per cent. The shortfall is not what a finite Reynolds number does to every flow; it is what the source does, and how fast it closes is set by how far the source reaches into the inertial range.
A snapshot counts areas, and a flow counts its edge
A record at one point holds one independent value for every two integral scales it lasts. A snapshot of a field holds one for every integral area it covers, so at the same number of samples it holds far fewer, and sampling it more finely adds nothing. Except for a velocity in a two-dimensional incompressible flow, whose mean over a window is fixed by the window's edge alone: it converges a whole power faster than its area allows, and a large enough snapshot of it beats the record.
The summit forgets the forcing before the dissipation does
A flow forced at its largest scales carries the four-fifths law closer to exact than a decaying one. Switch the forcing off and the third moment has to pass from one value to the other. It does not wait for the cascade to drain: half the forced summit is gone in a fifth of an eddy turnover, while the dissipation has hardly moved, and the largest separations overshoot the decaying value before they settle on it.
A wing averages a gust through its own loading
A finite span averages a turbulent gust and so tames the load it causes, but it does not average uniformly. The reverse-flow theorem says exactly how it weights the span: by the loading the wing makes at a uniform incidence. An elliptic wing therefore filters the gust the way a round aperture diffracts light, passes eight per cent more of its short scales than a uniform strip, and crosses its mean load up to four per cent more often.
A foil in a random sea follows it up to its peak
In a regular wave a foiling boat chooses between holding its height and following the surface. A real sea is a spectrum, and a wand that filters its signal can do both at once: follow the long waves and fly level through the short. The best place to divide them is close to the frequency of the sea's own peak, the divided control beats either pure strategy, and in a rough sea it is the only one of the three that keeps the foil a safe distance under the surface and the drag near its calm-water value.
At higher orders the bottleneck hides intermittency
At second order the spectral bottleneck makes a field with no intermittency look intermittent. At higher orders it does the opposite. Every moment's local slope is steepened by the same finite-Reynolds-number excess, multiplied by half the order, while the intermittency correction pulls the other way. At a laboratory Reynolds number a fitted sixth-order exponent therefore shows less than two-thirds of the anomaly the field carries, and a fourth-order one less than a quarter.
A wandering forcing leaves the averaged four-fifths law alone
A forced flow's third moment loses half its excess within a fifth of an eddy time of the forcing stopping, so a forcing that wanders ought to drag a simulation's four-fifths ratio about with it. Instantaneously it does: the summit swings from 0.8 to 1.4. Averaged it does not — the third moment averaged and then divided stays at the steady forcing's 0.94 for every correlation time from a thirtieth of an eddy time to ten. The reason is that the balance is linear and stationarity fixes its source; what the forcing's memory changes is only the answer to a different question, the average of a ratio.
The camera needs the window's memory, not the probe's
A run of particle-image frames stops gaining independent values for its mean once frames are closer than the flow's memory — but the memory that counts is the window mean's, and when large eddies live longer than small ones it is longer than the integral time a single probe measures. Give each scale its own memory and the frame rate that stops buying moves by up to a quarter where the flow forgets before it is swept, by nothing where the sweep sets it, and by however much longer the largest eddies live. Fed the window's own memory, the single-memory rule gives it back to two per cent.
One wing root feels the rolling gust its lift cannot
The reverse-flow theorem says the gust weight of any load is the loading the wing would carry at an incidence shaped like that load's arm. For the lift that is the familiar uniform-incidence loading. For the bending moment at one root it is the loading of a wing washed in towards one tip only — a weight that is not symmetric, and so passes the rolling half of a gust that the lift and the two roots together cancel. That half is a sliver of the variance and adds up to a fifth to how often the root's moment crosses its mean.
A wand that lags its sea makes flying level the better bet
A foiling dinghy whose height control divides the sea sharply follows the long waves, flies level through the short and beats both pure strategies. Give the division the lag a real wand or filter has and the gain disappears in a moderate sea; in a rough one no crossover both keeps the foil under and spares it from pulling down. A second-order filter, which cuts more sharply, does worse rather than better, because it flies through more of the sea near its crossover than the sea contains.
Sweep slows a wing root's gust cycles and leaves their excess over the lift
On a straight wing every station meets a gust at the same instant, and one wing root's bending moment crosses its mean nearly a fifth more often than the lift, because it reads the rolling half of the gust. Sweep the wing back and the outer stations meet the gust later, by their distance aft. That delay averages the gust along the flight path as a second, softer chord would: at 35° every load crosses its mean about a tenth less often, the root slightly more than the lift since its weight sits further out, and the variance hardly changes at all. The rolling excess survives almost whole.
A camera zoomed into the cascade needs frames faster than its window remembers
A wide camera window averages away the small eddies, so its mean remembers what the large eddies remember, and a run of frames is worth as many independent values as that memory allows. Zoom the window into the inertial range and the opposite is not what happens. A small window keeps nearly all the variance and almost exactly a single probe's memory — every eddy larger than itself still fills it — but much of what it keeps is the cascade's eddies, which forget far faster than the largest ones. Its frames then stop buying at an interval a third shorter than its own memory says, and a velocity gradient in the same window needs frames thirty times faster.
Named alongside it
The objects these essays reach for when they reach for this one.
MeasurementModel limitDissipationInertial rangeStructure functionAveragingIntegral scaleCascadeReynolds numberTurbulenceCorrelationFrequency response