Fluid Flow & Aerodynamics
why things fly, and why the usual explanation is wrong
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Decay — the series

7 essays on one idea, from the one that introduces it to the one that assumes the rest.
  1. Two decay laws from two invariants, and nothing in the equations to choose. The energy of a decaying turbulence against time, integrated from dK/dt = −A K^(3/2)/l with the large scales conserving u² l³ in one case and u² l⁵ in the other. The exponents come out at 1.1997 and 1.4282 against the closed forms 6/5 and 10/7. Which invariant holds is decided by the shape of the spectrum at the very largest scales, at the moment the stirring stops.

    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.

    part 1 · turbulence
  2. Five spectra, five exponents, one linear equation. The energy of a decaying turbulence after the nonlinear term has stopped mattering, computed by integrating the exact modal solution E(k,0)exp(−2 nu k² t) at five different shapes of the spectrum at the origin. Each is a straight line on these axes and no two have the same slope: the exponent is (m+1)/2, where k^m is the spectrum's behaviour at wavenumbers smaller than any eddy. The 5/2 that is quoted as the final period's universal exponent is the m = 4 line and one of five.

    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.

    part 2 · turbulence
  3. The coefficient that was a constant, against the number it is said not to depend on. The dissipation coefficient Cε = eps·l/u³ along two decays, plotted against the Taylor-scale Reynolds number they pass through. One is flat because it was put in flat; the other falls as the reciprocal of the Reynolds number, which is what is measured in the near field of a grid. Neither line is a derivation. What is exact is the relation between them, Cε = 15(ℓ/λ)/Reλ, which is a rearrangement of two definitions and holds along both curves to 5·10⁻¹⁶.

    The constant that travels

    Every decay law in the subject rests on the dissipation being some constant times u³ over a length. The constant is not one. Letting it move the way grid measurements say it moves changes the decay exponent by a quarter — and lands one of the answers five per cent from another that is entirely different physics.

    part 3 · turbulence
  4. The plateau everybody looks for is a summit, and a low one. −Dₗₗₗ/((4/5)εr) against separation in decaying turbulence at five Taylor-scale Reynolds numbers. None has a plateau at one. Each rises through the viscous range and turns over, peaking at 0.49, 0.63, 0.75, 0.85, 0.90 for Reλ = 50, 100, 200, 500, 1000. A measurement of ε that takes the largest value of this curve as four-fifths reads each of those shortfalls as a smaller dissipation.

    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.

    part 4 · turbulence
  5. Forced at the integral scale, the third moment nearly reaches four-fifths. The Kármán–Howarth balance at a Taylor-scale Reynolds number of 200, on the same model spectrum, for a flow forced in a band at the spectrum's peak and for one decaying. Each term is divided by (4/5)εr and plotted against separation in Kolmogorov lengths. The viscous term is the same for both. The forcing term is negligible until the separation approaches the integral scale, and −Dₗₗₗ/((4/5)εr) for the forced flow peaks at 0.965 at 90 Kolmogorov lengths, where the decaying flow's peaks at 0.747 at 32.

    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.

    part 5 · turbulence
  6. Half the forced summit is gone in a fifth of an eddy time. The third moment of the velocity differences as a fraction of four-fifths εr, against separation in Kolmogorov lengths of the forced flow, at the instant the forcing is switched off and 0.1, 0.2, 0.5, 1 and 4 large-eddy times later. Forced, the curve peaks at 0.94 near 67 η. A tenth of an eddy time later the summit is 0.899, at a fifth 0.838, at a half 0.759, and after that it hardly moves: 0.742 at one and 0.72 at four, where the flow is simply decaying. The large separations fall first and farthest.

    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.

    part 6 · turbulence
  7. However long its forcing remembers, a stationary flow's averaged four-fifths summit stays at 0.94. The summit of −Dₗₗₗ/((4/5)εr) computed from the time-averaged spectrum and source — the third moment averaged, then divided by the averaged dissipation — against the forcing's correlation time in large-eddy times, for a forcing amplitude whose logarithm wanders with standard deviation 0.5 and 1, over runs of 200 to 400 eddy times. The steady forcing's summit is 0.94; across correlation times from 0.03 to 10 eddy times and both spreads the averaged summit lies between 0.9371 and 0.9438, within its batch errors of the steady value.

    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.

    part 7 · turbulence

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