A relation with no turbulence in it

The fraction that is really four thirds ends by pointing at a relation it used and did not examine. Turning the mixed third moment into the longitudinal one needs isotropy and incompressibility and nothing else, and the same pair of assumptions applied at second order gives a relation between the longitudinal and transverse structure functions:

$$D_{NN}(r) = D_{LL}(r) + \frac{r}{2}\frac{\mathrm d D_{LL}}{\mathrm d r}.$$

There is no dynamics in that. No Navier–Stokes, no cascade, no Kolmogorov, no Reynolds number, no dissipation rate. It uses homogeneity, isotropy and incompressibility, and a flow that violates it has violated one of those three. Which makes it an unusually clean instrument, and it is almost never used as one. Beside the one exact result it is the only other statement about the interior of a turbulent flow with no model in it — and unlike that one it needs no dynamics at all.

What this essay does is divide it through. That step turns a relation into an identity, the identity collapses two of the most-quoted numbers in experimental turbulence into one number, and the one number turns out to be reproduced — without intermittency, without anomalous scaling, without anything but a smooth model spectrum — by the finite size of the Reynolds number.

The relation, divided through #

Dividing by $D_{LL}$ gives

$$\frac{D_{NN}}{D_{LL}} = 1 + \frac{n(r)}{2}, \qquad n(r) = \frac{\mathrm d\ln D_{LL}}{\mathrm d\ln r}.$$

The important word is local. $n(r)$ is the logarithmic slope at the separation $r$, not a fitted exponent over a window, so the identity holds at every separation including all the ones where nothing is a power law. That is worth insisting on, because the relation is usually quoted in its power-law form — “the ratio is 4/3 in the inertial range” — which is the identity evaluated at one value of $n$ and looks like a result about the inertial range when it is not.

Three values of $n$ are available without any turbulence theory at all. In a smooth field, which is what a fluid is below the dissipation scale, $D_{LL} \propto r^2$ and the ratio is exactly 2. Beyond the correlation length the two points are independent, both functions equal twice the variance, $n \to 0$ and the ratio is exactly 1. And where $n = 2/3$ the ratio is 4/3.

So the ratio is a single curve from 2 down to 1, and asking what it is “in the inertial range” is asking where on that descent it lingers. The answer is that it does not linger anywhere, and how nearly it does is exactly what a Reynolds number buys.

Where the relation comes from, and why incompressibility is the load-bearing part #

It is worth seeing what the three assumptions each contribute, because they are not equally strong and only one of them is likely to fail in a well-run experiment.

Homogeneity says the two-point statistics depend on the separation and not on where the pair is. That gives a function of a vector rather than of two positions. Isotropy says they depend on the separation’s length and not its direction, which reduces the general second-order two-point tensor to two scalar functions of $r$ — one for the component along the separation and one for the two perpendicular to it. Those two are $D_{LL}$ and $D_{NN}$, and at this stage they are independent: any pair of functions is allowed.

Incompressibility is what ties them together. The velocity field has zero divergence, so its correlation tensor has zero divergence in each index, and applying that to the isotropic form leaves exactly one constraint — which, written in terms of the structure functions rather than the correlations, is the relation above. One equation, and it removes one of the two free functions.

That ordering matters for reading a violation. Homogeneity is checkable directly, by measuring the same statistic at two stations, and a well-designed experiment establishes it before anything else. Isotropy is the assumption everyone worries about and the one the relation is usually invoked to test. Incompressibility is the one nobody worries about, and it is also the one that a measurement can break without the flow breaking it: Taylor’s frozen hypothesis converts a time series into a spatial increment by multiplying by a mean speed, and the resulting “field” is not the divergence-free one the relation is about. The error in that conversion goes as the turbulence intensity, so a flow with ten per cent fluctuations carries a percent-level violation that is an artefact of the instrument.

Which means a residual of a few per cent has at least three candidate owners, and separating them needs more than the relation itself. The relation’s value is that it produces a number at all — the alternative, comparing a measured exponent against a theoretical one, produces a number whose owner is even less clear.

Two reported numbers that are one number #

The identity has an immediate consequence for how measurements are read, and it is the reason this relation deserves more attention than it gets.

Experiments report two quantities from the second-order structure function. One is the inertial-range exponent, which comes out at about 0.70 rather than 2/3 and is very widely taken as the second-order signature of intermittency. The other is the ratio $D_{NN}/D_{LL}$, which comes out between about 1.2 and 1.35 depending on the flow and the fitting window, and is usually discussed as a test of local isotropy. Both are read off a second-order statistic, which is everything a spectrum can hold and no more. Both are borrowed figures here rather than computed ones.

They cannot be independent. A measured ratio of 1.35 is a measured local slope of 0.70, exactly, and a measured slope of 0.70 is a measured ratio of 1.35. If an experiment reports a pair that does not satisfy $\text{ratio} = 1 + n/2$, it has not found a violation of Kolmogorov: it has found a violation of isotropy or of incompressibility, or an inconsistency between two windows.

Computing both sides separately — the transverse function from its own spectral kernel, the slope by differencing the longitudinal one — puts them on top of each other to $2\times10^{-3}$, and the residual is the numerical differencing rather than anything physical. Two routes to one number is the discipline this site applies to its own arithmetic; here the same discipline says something about everybody’s measurements.

Where 0.70 comes from, and it is not intermittency #

Which raises the obvious question: what does the local slope do in a real flow at a real Reynolds number?

Answering it needs a spectrum, and the spectrum here is a stated shape rather than a solution — Pope’s model form, two smooth factors multiplying $k^{-5/3}$, one cutting the energy-containing end and one the viscous end. What is not stated is its two shape constants. Those are solved: bisected until the spectrum integrates to the energy it was given and until twice the viscosity times its second moment returns the dissipation it was given. The two constraints come back satisfied to $6\times10^{-15}$.

There is a bump. It is the spectral bottleneck, it appears because the viscous factor and the energy-containing factor overlap rather than meeting cleanly, and nothing in the construction asked for it — it is a feature of the cascade’s two ends failing to meet in the middle. A bump in the compensated spectrum is a region where the spectrum is shallower than $-5/3$, which makes the structure function steeper than $r^{2/3}$ — and that is where the story is.

Why a smooth spectrum has a bump in it #

The bottleneck deserves a sentence of mechanism rather than being left as an artefact of a fitted shape, because it is a real feature of real spectra and its cause is understood.

In the inertial range the energy flux is carried by interactions between neighbouring scales, and the transfer is efficient because there is always a smaller scale waiting to receive. Approaching the dissipation range there is not: the scales below have been damped, so the modes just above them have fewer partners to hand energy to and the flux stalls slightly. Energy piles up in front of the viscous cut-off, the spectrum is locally shallower than $k^{-5/3}$, and the pile is the bottleneck. It is present in simulations, in measurements, and in any model spectrum whose two ends are joined smoothly rather than by a kink — which is why it appears here without being asked for.

What it does to a structure function is the mirror image. A spectrum shallower than $-5/3$ over a band gives a structure function steeper than $r^{2/3}$ over the corresponding separations, and those separations are just above the dissipation scale — which is exactly the region an experiment includes when it extends its fitting window downward to get another half-decade of range. The part of the record an experimenter is most tempted to use is the part where the bottleneck is largest, and using it steepens the fitted exponent in the direction that looks like intermittency.

The flattest point of the slope curve — the nearest thing to a plateau any of these Reynolds numbers provides — sits at 0.7093 at $Re_\lambda = 200$, 0.7042 at 500, 0.6957 at 1,000, 0.6826 at 3,000 and 0.6737 at 10,000. The corresponding ratios are 1.3547, 1.3521, 1.3479, 1.3413 and 1.3368.

Those are the measured numbers. 0.70 at the Reynolds numbers of laboratory grid and jet experiments is the value reported from those experiments for fifty years, and this calculation contains no intermittency, no anomalous scaling, no multifractal model and no fitted exponent anywhere. It contains a spectrum with a bottleneck in it and the exact isotropic relation.

So the second-order departure from 2/3 is not evidence of intermittency. It is consistent with intermittency, and intermittency is real and is measured convincingly at higher orders, where the exponents genuinely stop being thirds by amounts no finite-Reynolds-number effect explains. At second order the two explanations are the same size, and the finite-Reynolds one is already there in a calculation with nothing in it.

There is a further reason to be careful with the second-order exponent specifically, which is that it is the order at which the two candidate explanations are closest in size. Intermittency corrections grow with the order of the moment — that is their defining feature, and it is why the sixth and eighth moments are where the evidence for them is convincing. At second order the correction predicted by every intermittency model is a few hundredths, which is precisely the size of the finite-Reynolds-number effect computed above. Two effects of the same size, in the same direction, at the one order where the measurement is easiest and therefore the most often quoted: that is a situation in which a number can be reported consistently for fifty years and mean something different from what it is taken to mean.

There is a further reason to be careful with the second-order exponent specifically, and it is that both candidate explanations act on a statistic that carries no transfer in it: a spectrum and a second-order structure function are the same object, and neither can see the cascade whose intermittency is being inferred.

How far 10,000 still is from infinity #

The rate at which the departure closes is the discouraging part.

From $Re_\lambda = 200$ to 10,000 — a factor of fifty, which is the whole distance from a laboratory grid to a large wind tunnel — the slope falls from 0.7093 to 0.6737 and the ratio from 1.3547 to 1.3368. The excess over the asymptotic value falls from 0.0213 to 0.0035, roughly as the reciprocal square root of the Reynolds number, so closing the remaining gap by another factor of ten needs a hundredfold rise. The atmospheric surface layer reaches $Re_\lambda$ of a few thousand and is the largest routinely measured turbulence there is.

This is the same arithmetic that the range a real Reynolds number does not have records for the inertial range’s width, and it produces the same conclusion from the other side. There the finding is that the range in which an exponent could be measured cleanly is never long enough; here it is that the exponent measured in the range that exists is systematically wrong by an amount comparable to the effect being looked for.

One detail in that picture is worth extracting because it bites in practice. The two functions are not flattest at the same separation. The transverse one is the longitudinal one plus half its derivative, and adding a derivative shifts a maximum, so a window chosen to be “the inertial range” by looking at one function is not centred on the other’s flattest region. Fitting both over one window — which is what an experiment does, because the window is chosen once — introduces a systematic difference between the two fitted exponents that has nothing to do with the physics of either.

Why the relation is used as a shortcut instead of as a test #

There is a reason the residual is rarely measured, and it is not carelessness. The relation is much more useful as a substitution than as a check, and having been used once as a substitution it cannot be used as a check.

A single hot wire in a mean flow gives one velocity component. Getting $D_{NN}$ from it requires either a second wire displaced perpendicular to the flow — a real experiment with a real calibration problem, since the separation has to be known to better than the smallest separation being measured — or an X-wire, which measures two components at one point and gives transverse increments only along the mean flow direction, which is a different quantity again. Faced with that, the standard move is to compute $D_{NN}$ from the measured $D_{LL}$ through the relation, quote it, and proceed.

That is a perfectly sound thing to do and it is exactly what the relation is for. What it forecloses is the test: a $D_{NN}$ obtained from $D_{LL}$ through the relation satisfies the relation identically, and comparing them measures nothing at all. Any dataset in which the transverse function was derived rather than measured carries a residual of zero by construction, and a reader cannot tell from the plot which it was.

So the instrument exists, costs one extra probe, and is bypassed by the same relation that makes it possible. The measurements that do carry an independently obtained $D_{NN}$ are the ones worth going back to, and the residual is sitting in them unread.

The size of the simulation that would settle it is worth stating, because it is the reason the question is still open. Separating a finite-Reynolds-number effect of a few hundredths from an intermittency correction of a few hundredths needs a decade more inertial range than exists in any laboratory flow, and the grid that would resolve one is the calculation nobody can run.

What a genuine violation would look like #

None of the above is a test of isotropy. Everything computed here assumes isotropy, so it cannot detect a failure of it; what it establishes is that the departures usually cited need no such failure.

The test that would detect one is different, and almost nobody makes it. Measure $D_{LL}$, put it through the operator, and compare the result with a separately measured $D_{NN}$. The residual is a direct measure of how far the flow is from satisfying homogeneity, isotropy and incompressibility at those separations, with no model, no fitted exponent and no assumption about a range.

One version of that test needs no apparatus at all, only consistency. Suppose an experiment reports, as experiments do, that the longitudinal and transverse functions are both power laws over a common range with different exponents — a longitudinal 0.70 and a transverse 0.66, say. The relation forbids it. Its right-hand side is $D_{LL}$ plus a constant multiple of $D_{LL}$, so it carries the longitudinal exponent whatever the transverse function does: a transverse power law with a different exponent cannot satisfy it anywhere except at the single separation where the two are matched. Over a decade and a half the two part by 4.7 per cent at one end and 4.5 at the other, and the control with equal exponents parts by nothing.

A reported pair of unequal second-order exponents is therefore already a statement that one of the three assumptions has failed — before any question of intermittency is raised, and independently of what the values are. That is a much stronger reading of a routine result than the one usually given, and it is available for free from data that has already been published.

What is assumed here, and what is not measured #

The spectrum is a shape, not a solution. Everything above is computed from Pope’s model form, which was constructed to look like measured spectra and fitted here only to its own two integral constraints. Its bottleneck is a property of how its two factors overlap. Real spectra have a bottleneck too, measured and not in doubt, but the size of the one in this model is a property of the model, so the 0.709 at $Re_\lambda = 200$ should be read as “an effect of this size arises with no intermittency” rather than as a prediction of what an experiment will find.

Isotropy is assumed everywhere, including in the two spectral kernels, so nothing here can detect the thing the last section says the relation is good for. That test needs data.

The relation assumes incompressibility as well, and that is not a formality in a hot-wire measurement: the increments are obtained from a time series through Taylor’s frozen hypothesis, which is itself exact only in a limit, and a compressibility error and a frozen-hypothesis error enter the relation the same way.

And nothing here is an argument against intermittency. The higher-order exponents depart from their dimensional values by amounts that grow with order, in a way no smooth spectrum reproduces, and the correlated dissipation field that produces them is measured rather than modelled. The claim is narrower and is about the second order alone, where the effect is small enough that a finite Reynolds number accounts for it.

The arithmetic is checked the usual way. The fitted spectrum returns the energy and dissipation it was given to $6\times10^{-15}$. The ratio from its own kernel and $1 + n/2$ from the other agree to $2\times10^{-3}$. The smooth-field limit comes out at 1.9999939 against 2 and the uncorrelated limit at 0.9999976 against 1, neither of which was put in anywhere. And the calculation refuses a Reynolds number no inertial range could survive, and a structure function at zero separation.

Still open: what the residual measures when it is not zero #

The relation’s residual is an isotropy measurement that costs nothing beyond data already taken, and the question nobody has answered is what a non-zero value of it means quantitatively.

A residual of five per cent says the flow is not isotropic at those separations. It does not say by how much, in what direction, or whether the anisotropy is in the large scales leaking down or in the measurement geometry. Turning a residual into a number needs the relation’s generalisation to an axisymmetric field, where the two structure functions are replaced by a family indexed by the angle between the separation and the symmetry axis, and the residual becomes a projection onto the first anisotropic sector. That calculation exists in the literature as a formalism and has rarely been carried through to a number; doing it for a shear flow, and asking how large a shear produces a five per cent residual, would turn a consistency check into an instrument.

Beside it is the question this essay’s central result raises about its own subject. If the second-order departure from 2/3 is a finite-Reynolds-number effect of the size computed here, the same effect is present at every order, and the exponents at fourth and sixth order carry it too. How much of the measured anomaly at those orders is the bottleneck rather than intermittency — computed from the same model spectrum, with the higher-order structure functions obtained from a stated joint distribution rather than from a spectrum — is the calculation that would say how much of the evidence for anomalous scaling survives being corrected for the Reynolds number the evidence was taken at.