Concept

Final period — where it appears

The last stage of a decaying turbulence, where the Reynolds number has fallen far enough that the nonlinear term no longer matters. Every Fourier mode then decays on its own and the energy follows an exact power law, whose exponent is fixed by the spectrum at scales larger than any eddy.

Named by 2 essays across one field — each of them below, with the objects they name alongside it.

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.

turbulence · Decay

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.

turbulence · Decay

Named alongside it

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

Decay Initial condition Integral scale Invariant Model limit Power law Reynolds number Self-similarity Spectrum Dissipation Energy cascade Measurement

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