A wandering forcing leaves the averaged four-fifths law alone

The summit forgets the forcing before the dissipation does switched off the forcing of a stationary flow and watched its third moment. The summit of $-D_{LLL}/(\tfrac45\varepsilon r)$ fell from the forced flow’s 0.940 to half its excess over the decaying value within a fifth of a large-eddy time, while the dissipation had hardly moved. It ended on the obvious worry. Simulations of forced turbulence are rarely forced steadily; most drive their large scales with a random process, an Ornstein–Uhlenbeck amplitude that wanders on a correlation time chosen for convenience. If the third moment responds in a fifth of an eddy time, a forcing that wanders on a longer time should keep tipping the flow towards the decaying state and back, and the four-fifths ratio such a simulation reports would depend on a parameter few of them report.

The calculation here does exactly that, and the worry turns out to be half right. The flow does swing between forced-like and decaying-like states, and its instantaneous four-fifths ratio swings with it, far further than either state’s value. But the number a simulation actually reports is an average, and the average is protected by a piece of arithmetic the switch-off essay did not need.

A forcing that wanders #

The forcing is the steady calculation’s band at the largest scales, with its amplitude multiplied by $a(t) = \exp(\sigma z - \sigma^2/2)$, where $z$ is an Ornstein–Uhlenbeck process of unit variance and correlation time $\tau$. The amplitude is always positive and its mean is one, so the mean rate of energy injection is the steady forcing’s; only its history differs. A log-spread $\sigma$ of 0.5 makes the injection wander by about half; $\sigma = 1$ makes it wander by a factor of e either way, occasionally falling to a few per cent of its mean for an eddy time. The spectrum is marched with Leith’s closure — the energy flux through each wavenumber a diffusion in wavenumber, with Kolmogorov’s constant built in — at a Taylor-scale Reynolds number of 200, on a grid of ten nodes per e-fold of wavenumber, where the steady summit is 0.94002 against 0.93996 on twenty.

Each run starts from the steady state, is given twenty eddy times to forget it, and is then averaged over 200 eddy times, or 400 for the longest correlation time, in four batches whose spread gives an error. A run of 200 eddy times costs about half a minute, so the twelve runs are stored, and a short check run — six eddy times at a stated seed — is re-marched on every check to the stored digits.

Averaged, the summit does not move #

The four-fifths ratio a simulation reports is built from averages: the third-order structure function averaged over the run, divided by $\tfrac45\langle\varepsilon\rangle r$. Computed that way, the summit lies between 0.937 and 0.944 across every correlation time from a thirtieth of an eddy time to ten, at both spreads, against the steady forcing’s 0.940. The batch errors are a few thousandths, and the departures are of the same size and of both signs. Within what 200 eddy times can resolve, the averaged four-fifths summit does not know how long its forcing remembers.

Why the average is protected #

The reason is that the quantity being averaged obeys a linear equation, and the average of its source is fixed. The Kármán–Howarth balance the steady calculation used,

$$-D_{LLL}(r) = \tfrac45\varepsilon r - 6\nu\frac{\mathrm{d}D_{LL}}{\mathrm{d}r} - \tfrac45 r\int S(k)\,[1 - 15c(kr)]\,\mathrm{d}k,$$

is linear in the spectrum, through $\varepsilon$ and $D_{LL}$, and linear in the source $S = F - \partial E/\partial t$. Average it over a stationary run and every term becomes the same term built from the averaged spectrum and the averaged source. And the averaged source is fixed by stationarity: the time derivative averages to the spectrum’s change between the run’s ends divided by its length, which vanishes as the run grows, so $\langle S\rangle = \langle F\rangle$, the mean forcing, whatever the forcing’s history. The averaged dissipation equals the mean injection for the same reason. So the source term of the averaged balance, normalised by the averaged dissipation, is exactly the steady forcing’s.

What is left to move the summit is the viscous term, built from the averaged spectrum’s dissipation range, and the averaged spectrum differs from the steady one only because the closure is nonlinear and a mixture of states is not a state. At the summit, 67 Kolmogorov lengths out, that term is a few hundredths of the whole, and its change is a fraction of that. The computed departures of a few thousandths are its size.

The contrast with the averaging every turbulence model starts from is exact. What averaging costs is that the momentum equation’s one nonlinear term does not average away, and leaves a Reynolds stress no averaged quantity determines. The Kármán–Howarth balance here has no such term in the place that matters: the third moment appears linearly, the closure’s nonlinearity is hidden inside the spectrum’s evolution, and the balance between the moments at one instant is linear in them. So the averaged balance is closed by the averaged moments, and the forcing’s history, which lives entirely in the nonlinear evolution, cannot reach it except through the averaged spectrum’s shape.

How long a run has to be for the protection to hold is a number the identity gives directly. The averaged source differs from the mean forcing by the change in the spectrum’s energy between the run’s ends, divided by its length. If the energy wanders by up to its own steady value — the trace below shows swings of about that size at a log-spread of one — the error in the normalised source is about the steady energy over the run’s length: one large-eddy time divided by the number of eddy times averaged. A run of a hundred eddy times leaves a source error near one per cent and a summit error of a few thousandths, which is the scatter the stored runs show. A run of ten would not.

This is the result’s surprising connection, and it runs through the decay inside the four-fifths law: the four-fifths law is a statement about a balance, its corrections are terms in that balance, and averaging a linear balance cannot create a correction its averaged terms do not contain. The switch-off essay’s decaying flow had a source that was the energy loss; a stationary flow with a wandering forcing has, on average, a source that is its mean forcing. The dramatic response of the summit to a sudden switch-off is real, and in the average it is paid back exactly by the overshoot when the forcing returns.

Three readings of one flow #

The protection belongs to one order of operations. A simulation that computes the ratio at every snapshot — each third moment divided by that snapshot’s own dissipation — and then averages, reads a different number. Averaged at the steady summit’s separation, that reading is 0.953 at a correlation time of a tenth of an eddy time and 0.933 at ten. Averaged as the summit of each snapshot’s curve, it is 0.991 at a tenth and 0.940 at ten. These readings do depend on the forcing’s memory, by up to five per cent, and the dependence is the one the switch-off essay feared.

Two things make them differ from the ratio of averages, and neither is turbulence. Dividing by the instantaneous dissipation weights each snapshot by the inverse of its dissipation, and the dissipation lags the forcing, so the snapshots taken while the forcing has just risen — third moment already up, dissipation still low — count for more than their share. And taking each snapshot’s maximum over separation before averaging picks out the fluctuation’s peaks wherever they fall, which biases the average upward by an amount that grows with the fluctuation. A paper that reports “the four-fifths ratio” without saying which reading it took has reported a number that can differ from its neighbour’s by the size of the correction it was looking for.

How far the instantaneous summit swings #

The spread of the instantaneous summit has a maximum, and it sits where the switch-off essay’s timescale says it should. That timescale has a twin one field over: a dissipation that lags its production found the dissipation of a sheared patch trailing a sudden change of strain by about a turnover, out of balance by a factor of six in the meantime. The summit here is the cascade’s middle, and it responds faster than the dissipation at the cascade’s end and slower than the forcing at its start. With a log-spread of one it peaks at 0.265 near a correlation time of 0.3 eddy times and falls to 0.069 at ten. A forcing that wanders much faster than the summit can respond is averaged by the flow before it is felt, and the third moment sees a nearly steady source. A forcing that wanders much slower is followed: the flow passes through a sequence of nearly stationary states, each with its own summit near 0.94, and the snapshot ratio wanders only a little. Between them, where the forcing changes on the summit’s own response time of a fifth to a third of an eddy time, the flow is caught out of balance most often and furthest.

A trace shows what out of balance looks like. Over twelve eddy times the amplitude wanders between 0.06 and 1.8 of its mean, the dissipation between 0.2 and 1.3 of its forced value — following, with a lag — and the instantaneous summit between 0.78 and 1.41. A summit above one is a third moment larger than the dissipation can account for: the forcing has just risen, energy is pouring down the cascade, and the dissipation at the bottom has not yet heard. It is the mirror image of the switch-off, and averaged over a run the two cancel in the balance even though they do not cancel in the ratio.

The whole curve, not only the summit #

The argument covers every separation, not only the summit, and the curves confirm it. Over sixty eddy times with a log-spread of one, the averaged ratio at correlation times of a tenth, one and ten eddy times lies on the steady forcing’s curve across the inertial range, with summits of 0.9405, 0.9400 and 0.9417. At the largest separations, where the source term carries most of the balance, the curves agree because the averaged source is the steady source; at the smallest, where the viscous term carries it, they agree because the averaged spectrum’s dissipation range is close to the steady one’s.

What a simulation should report #

The practical conclusion is a recommendation about arithmetic. Average the structure function and the dissipation separately over the run, then divide, and the forcing’s correlation time does not matter; divide snapshot by snapshot and then average, and it matters by up to five per cent at the correlation times most simulations use, which are of order an eddy time. The two readings coincide only for a steady forcing. A comparison between two simulations’ four-fifths ratios is a comparison of their forcings unless both took the first reading. The forcing’s correlation time, which most papers choose for convenience and few report, then genuinely does not matter; the averaging recipe, which almost none report, does.

The same caution applies to the constant that travels between experiments and simulations: any ratio of two fluctuating quantities is a nonlinear functional of the flow, and its average carries the flow’s fluctuation statistics along with the physics it was meant to isolate.

And it applies to experiments in a form that is easy to miss. A wind-tunnel measurement of the third moment usually divides by a dissipation estimated from the same record’s second-order statistics, which is a ratio of averages; a measurement that conditions on the large-scale energy — binning records by how energetic the flow was — and then compares the bins is taking the snapshot reading, and will find a dependence on the bin that the averaged law does not have. It is the kind of dependence on the large scales that universal, and one of five found hiding in a decaying flow’s final exponent, there set by the spectrum’s large-scale corner and here by which records were kept; in both cases the small scales are being asked to report on a choice made at the large ones.

The convention: whose average, whose eddy time #

All times are in the steady forced state’s large-eddy time $K/\varepsilon$, 2.88 in the closure’s units. Separations are in the steady state’s Kolmogorov length. The spread $\sigma$ is the standard deviation of the forcing amplitude’s logarithm; the amplitude’s mean is one by construction, so every run injects energy at the steady rate on average. “Summit” is the maximum over separation of a ratio curve; “ratio of averages” divides the run-averaged third moment by four-fifths of the run-averaged dissipation times $r$. The batch error is the spread of the four batch summits divided by $\sqrt3$.

What the closure cannot say #

Real triads. Leith’s flux is local in wavenumber, so a burst of energy injected at the large scales diffuses down the spectrum rather than jumping; a real cascade’s response to a sudden rise in forcing has non-local transfers that could change how far the instantaneous summit overshoots, though not the average, which the linearity argument protects in any closure. A flux with no lag of its own. Leith’s flux responds to the spectrum’s present slope at each wavenumber, so the cascade’s delay is carried only by the time energy takes to diffuse down the spectrum; a real cascade’s transfer also lags its own drive, the response with a lag in it that an eddy viscosity’s zero-frequency limit hides. A closure with memory in its flux would make the instantaneous summit’s swings later and probably larger, and would leave the averaged balance exactly where it is, because the argument for the average never asked how fast anything responds. Intermittency. The closure has none, and the instantaneous summit’s distribution in a simulation would carry the dissipation’s own fluctuations as well as the forcing’s. Forcing in phase. Real Ornstein–Uhlenbeck forcings act on individual Fourier modes, and their injection rate can be negative for a while; the log-normal amplitude here is always positive. And finite runs, which is the one caveat on the average itself: the identity $\langle S\rangle = \langle F\rangle$ holds only as the run grows, and a run of a few eddy times carries the difference in its end states.

And the record a measurement can afford. A probe in a wind tunnel or a simulation’s time series averages over a record of finite length, and the argument above says what that record has to cover: many correlation times of the forcing, not merely many eddy times. Where the stirring itself wanders over tens of eddy times — a fan’s speed hunting, the weather over an atmospheric surface layer — a record of a few hundred eddy times holds only a handful of independent forcing states, and its averaged summit carries their scatter.

Where the question came from #

The Ornstein–Uhlenbeck forcing of the large scales is Eswaran and Pope’s, from 1988, and it became the default for spectral simulations of isotropic turbulence because it keeps the forcing smooth in time without making it deterministic. Leith’s closure is from 1967. The concern that a forcing’s statistics leak into the small-scale statistics a simulation is meant to measure is as old as the method; the switch-off calculation put a timescale on it, and the calculation here separates the part of that concern the averaged balance absorbs from the part a choice of arithmetic creates.

Still open: the forcing a wind tunnel applies #

A wind tunnel’s turbulence is forced by a grid and then left alone, and its forcing’s removal is a distance rather than a time. Downstream of the grid the flow decays, and the four-fifths ratio at a probe is the switch-off’s decaying value — unless the grid’s wake is still feeding the large scales. An active grid, whose paddles flap at random, is a wandering forcing in space: the next calculation converts the switch-off transient and the wandering forcing into distances in mesh lengths, and asks how far downstream of an active grid the averaged ratio stays at a forced flow’s value before it falls to a decaying one’s.