The summit forgets the forcing before the dissipation does

Forcing at the integral scale leaves the cascade alone put a forced flow and a decaying flow on one spectrum and compared their third moments. At a Taylor-scale Reynolds number of 200 the forced flow’s $-D_{LLL}/(\tfrac45\varepsilon r)$ climbed to 0.965 and the decaying flow’s turned over at 0.747. The difference was the source term: the energy that enters, or is released, at scales smaller than the separation does not have to be carried down through it, and a forcing confined to the largest scales releases almost none there, while a decay releases energy at every scale at once.

That essay ended on a question about time. A flow that is forced and then left alone has to pass from one curve to the other. The energy already in the inertial range was delivered by the forcing and takes a cascade time of its own scale to drain, so the guess it recorded was that the small separations would keep the forced value longest, the large ones would lose it first, and the summit would travel inwards as the flow forgot it had been forced. How long the forced plateau survives decides whether a decaying simulation — which almost always starts from a forced state — has a clean third moment for a while or loses it at once.

The answer is that it loses most of it at once. The direction of the guess is right: large separations go first. The time is wrong by a large factor, and the reason is worth more than the number.

A spectrum that can change #

A fixed spectrum cannot answer a question about a transient, so this essay evolves one. The energy spectrum obeys $\partial E/\partial t = -\partial\Pi/\partial k - 2\nu k^2E + F$, and the flux $\Pi$ through wavenumber $k$ is closed with Leith’s 1967 diffusion approximation,

$$\Pi(k) = -C\,k^{11/2}E^{1/2}\,\frac{d}{dk}\!\left(\frac{E}{k^2}\right).$$

It is the simplest closure with three properties this question needs. It carries a Kolmogorov inertial range: with $C = 3/(11C_K^{3/2})$ a spectrum $C_K\varepsilon^{2/3}k^{-5/3}$ carries a flux of exactly $\varepsilon$, and with $C_K = 1.5$ that is built in rather than fitted. It has a zero-flux state, $E \propto k^2$, which is equipartition among wavenumbers, so the scales larger than the forcing fill by backscatter as they do in a periodic box. And written in flux form it conserves energy to the last bit, so the energy lost through the transient is exactly the energy dissipated.

The flow is forced in a log-normal band centred at $k_f = 1$, half a unit wide in $\ln k$ — the band the previous essay used — with a total power of one, in a box whose smallest wavenumber is a quarter of the forcing’s. The viscosity is chosen to give a Taylor-scale Reynolds number of 199 once the flow has settled, which it does when its dissipation equals the forcing’s power to a part in ten thousand and its energy has stopped moving. Then the forcing is switched off, and the spectrum is followed.

The third moment is computed from the spectrum through the balance the previous essay used,

$$-\frac{D_{LLL}}{\tfrac45\varepsilon r} = 1 - V(r) - \int \frac{S(k)}{\varepsilon}\,\bigl[1 - 15\,c(kr)\bigr]\,dk,$$

where $V$ is the viscous term and $S$ the source: whatever the cascade does not have to carry down past $k$. In a stationary forced flow $S$ is the forcing. In a decaying flow it is the loss, $-\partial E/\partial t$. In between it is $\partial\Pi/\partial k + 2\nu k^2 E$, both at the same instant, and that is the quantity the transient changes.

The summit falls fast and then sits #

Forced, the closure’s third moment peaks at 0.940 near 67 Kolmogorov lengths. That is a little below the previous essay’s 0.965 on Pope’s model spectrum, because the closure’s forced spectrum is its own rather than Pope’s model, and the comparison within this essay is the closure against itself. Time is counted in large-eddy times of the forced flow, $T = K/\varepsilon$, 2.9 time units here.

A tenth of an eddy time after the forcing stops the summit is 0.899. At a fifth it is 0.838. At a half it is 0.759, and there it nearly stops: 0.742 at one eddy time and 0.721 at four, falling slowly as the decaying flow’s Reynolds number falls. Most of the transition from forced to decaying happens in the first fifth to third of a turnover. After that the curve is a decaying flow’s curve, and it is moving only because a decaying flow’s Reynolds number is moving.

The summit also travels. Forced it sits at 67 Kolmogorov lengths; by a third of an eddy time it has moved in to 33, and it stays within a few lengths of that for the next turnover; later it drifts out again, to 59 by four turnovers, because the separations here are counted in the forced flow’s Kolmogorov length and the decaying flow’s own Kolmogorov length grows as its dissipation falls. In the decaying flow’s own units the summit stays put.

Faster than the dissipation #

Here is the number that contradicts the expectation. The summit has made half of its whole fall, from 0.940 towards 0.721, by 0.22 eddy times. At that moment the dissipation is still 0.948 of its forced value. The dissipation does not reach half its forced value until 0.64 eddy times — three times later. The energy, drawn dashed, falls at once and steadily, because nothing is replacing what the dissipation removes.

The dissipation lags because it is the far end of the cascade. Energy that was in the inertial range when the forcing stopped is still on its way down, and it keeps arriving at the Kolmogorov scale at the forced rate for about a cascade time — the delay a dissipation that lags its production measured from the other side, with the production stepped up rather than switched off. The previous essay’s guess was that the third moment would wait with it. It does not, and the reason is in what the third moment measures.

$-D_{LLL}$ at a separation $r$ measures the energy flux through the scale $r$. The four-fifths law says that flux is $\varepsilon$ — the statement where the energy goes put in words, that everything dissipated at the bottom was delivered from above — and the ratio plotted is flux over dissipation. In the forced flow the large scales are full and held full, so they hand down exactly what is dissipated. The moment the forcing stops the large scales start emptying, and an emptying scale hands down less than the scale below it is still dissipating, because what the scale below is dissipating was sent earlier. The flux through every inertial separation falls at once, while the dissipation at the bottom does not yet know. The ratio — flux now, over dissipation now — falls at once too.

Large separations lose first, and overshoot #

The direction of the old guess survives. At ten Kolmogorov lengths, deep in the viscous range, the ratio is 0.632 forced and still 0.631 a tenth of an eddy time later. At 200 lengths it drops from 0.799 to 0.657 in the same tenth. The large separations are where the source term moves into the inertial range first, so they are the first to stop counting the cascade’s full flux.

What the guess did not anticipate is the overshoot. The ratio at 200 lengths keeps falling past the value the decay will settle at, reaching 0.456 at 0.54 eddy times, and then climbs back — to 0.565 by four. At 100 lengths the same thing happens more gently. The cause is that the forced flow’s spectrum is not a decaying spectrum. It carries more energy near the forcing band than a flow that had always been decaying would, and once the forcing stops that excess is lost at a rate set by its own short turnover time. For half an eddy time the large scales are shedding energy faster than a self-similar decay would ask of them, the source at those scales is larger than in the eventual decay, and the ratio there dips below the decaying value until the spectrum has relaxed to a decaying shape.

The small separations behave differently and slowly. Once the large-scale transient is over, the ratio at ten and thirty lengths keeps falling for as long as the run is followed — from 0.63 to 0.26 at ten lengths over four turnovers — and this is not the forcing being forgotten. It is the Reynolds number falling, which the limit that is not the value showed is the one thing the dissipation itself is indifferent to: as the flow decays its Kolmogorov length grows, and a separation fixed in the forced flow’s units sinks deeper into the viscous range. That fall would happen in any decaying flow.

Where the source goes #

The mechanism can be seen directly. Forced, the source is the forcing band, all of it below $k\eta = 0.02$, and only 1.9 per cent of it above. A tenth of an eddy time after switch-off the source is the decay’s loss, and it has already spread: 16 per cent of it lies above $k\eta = 0.02$. By half an eddy time the share is 41 per cent. A decaying flow loses energy at every scale, because every scale’s turnover time is finite, and a source spread through the inertial range is exactly what the decay inside the four-fifths law found to cost the third moment a quarter of its value.

The speed of that spreading is the whole answer. A wavenumber in the inertial range has a turnover time much shorter than the large-eddy time — at $k\eta = 0.02$ it is about a seventh of it here — and it starts to lose energy as soon as the scale above it stops supplying the full flux. So the source spreads down through the inertial range on the inertial range’s own short clocks, not on the large eddies’ long one, and the third moment, which is the integral of that source against a kernel, follows it.

The inertial range drains from the top #

The compensated spectrum shows the same thing in the quantity measured most often. Forced, it holds a plateau at the closure’s Kolmogorov constant, 1.46 between $k\eta = 0.02$ and $0.06$ against the built-in 1.5. After the switch-off the energy-containing range empties first and the plateau falls from its large-wavenumber end inwards, while the dissipation range, scaled by the falling $\varepsilon$, keeps its shape longest. By one eddy time the same band reads 1.21. A measurement of the Kolmogorov constant made in this window would report a number nearly a fifth low, and the reason would not be intermittency, or the Reynolds number, but that the spectrum was not in equilibrium with the dissipation it was divided by — a dissipation still being paid for by energy sent before the forcing stopped.

That is the practical warning, and it has a size. A decaying simulation is normally started from a forced field and run for a few large-eddy times, and the first half-turnover of it is a transient in which the third moment, the compensated spectrum and the dissipation are each on a different clock. Discarding that half-turnover costs a sixth of a typical four-turnover run. Keeping it puts a third moment that is neither forced nor decaying into the average, and in the closure here that average over the first turnover is 0.79 — a value that belongs to no flow. The constant that travels found the decay’s own constants to depend on the initial state; here the transient from a forced state is a second source of the same kind of error, and it is largest exactly when a simulation’s decay phase has just begun.

The same fraction at three Reynolds numbers #

The obvious objection is that a fifth of an eddy time is a Reynolds-number effect: at a higher Reynolds number the inertial range is longer, the cascade time from the top to the bottom is longer, and perhaps the summit would last. The closure says not. At forced Taylor-scale Reynolds numbers of 113, 199 and 491, half the summit’s fall is complete by 0.20, 0.22 and 0.22 eddy times. The gap between forced and decaying shrinks as the Reynolds number rises — the forced summits are 0.892, 0.940 and 0.978, the decaying ones they fall to 0.655, 0.734 and 0.832 — but the time it takes to close half the gap is fixed by the large eddies alone.

That follows from the mechanism. The summit lies near the top of the inertial range, at separations a modest fraction of the integral scale, and the source reaches those separations in a turnover time of those scales, which is a fixed fraction of the large-eddy time whatever the Reynolds number. The length of the range below does not enter.

Checks on the closure and the balance #

Each property used above is tested where it can fail. A pure $k^{-5/3}$ spectrum at the Kolmogorov constant carries a flux of $\varepsilon$ to 0.13 per cent on the grid used — the discretisation’s error, falling with the grid spacing. An equipartition spectrum carries no flux to rounding. The forced state dissipates its forcing’s power to two parts in ten million, and at a fiftieth of a Kolmogorov length the viscous term is one to a part in a hundred thousand, which is the statement that $D_{LL} = \varepsilon r^2/15\nu$ there and is a check on the kernel’s normalisation. Through the first eddy time of the transient the energy lost equals the energy dissipated to five parts in a million, the residual being the trapezoid rule applied to a dissipation sampled every hundredth of an eddy time. The tests also refuse a viscosity of zero and a negative one.

What a closure cannot say #

Nonlocal transfer. Leith’s closure moves energy only between neighbouring wavenumbers, and a closure with no memory at all is the reminder of what a closure that forgets its own history gives up. The real equations move it through triads that can span a decade of scales, and a large-scale strain acting on small eddies is one of them. A closure with that coupling, such as the eddy-damped quasi-normal Markovian approximation, would let the small scales feel the forcing’s removal sooner still — so the fifth of an eddy time is more likely an overestimate than an underestimate, but the closure here cannot show it.

Intermittency and the bottleneck. What decay never forgets and universal, and one of five are about the large scales; the small ones carry their own corrections. The spectrum here is smooth and the closure has no bottleneck bump at the viscous end. Both would move the forced and decaying numbers by a few points, as the previous essay noted for its fixed spectrum.

A real forcing. The forcing here is steady and then gone. A simulation’s forcing is usually random in time with a correlation time of its own, and a wind tunnel’s grid stops “forcing” as the flow moves downstream past it rather than at an instant. Both are smoother switch-offs, and the transient would be stretched to their time scale.

Anisotropy. The Kármán–Howarth equation assumes isotropy at every instant, including the instant the forcing stops.

The convention: whose eddy time #

Time is measured in large-eddy times of the forced flow, $T = K/\varepsilon$ at the moment of switch-off, and separations in the forced flow’s Kolmogorov length. Both are fixed through the transient, so the curves are on one axis; in the decaying flow’s own units the Kolmogorov length grows and the late drift of the summit disappears. The ratio is the third moment over four-fifths of the instantaneous dissipation times the separation, which is one when the four-fifths law holds and is what the one exact result is a statement about. Kolmogorov’s constant is set to 1.5 and the forcing band is the one forcing at the integral scale used.

Leith’s closure, and where the question came from #

Leith proposed the diffusion approximation in 1967 for two-dimensional turbulence and it has been used in three dimensions since as the simplest closure carrying both a cascade and equipartition. The source-term form of the Kármán–Howarth balance is the one the earlier essays on decay derived from Kolmogorov’s 1941 equation and the large-scale terms added to it by Danaila and colleagues in 1999. That a decaying simulation started from a forced field carries a transient in its third moment is known from direct simulations, where the forced state is the usual initial condition; its duration in large-eddy times, and that it is shorter than the dissipation’s lag, is what this calculation adds.

Still open: a forcing with a memory of its own #

The switch-off here is instantaneous. The next calculation gives the forcing a correlation time — an Ornstein–Uhlenbeck amplitude as most simulations use — and asks how the third moment’s summit in a stationary flow fluctuates as the forcing wanders, and whether its time-averaged value is the steady forcing’s 0.940 or something nearer the decaying one. If the summit responds in a fifth of an eddy time, a forcing whose correlation time is longer than that keeps flipping the flow between forced and decaying states, and the measured four-fifths ratio of a forced simulation would then depend on its forcing’s correlation time — a number that most simulations choose for convenience and few report. Beside it is the version in a wind tunnel, where the forcing is a grid and its removal is a distance, and the question is how many mesh lengths downstream the third moment is a decaying flow’s.