The one exact result

Turbulence is the part of classical physics where the equations are known, complete and useless. Everything computed about it in practice descends from a model — a mixing length, an eddy viscosity, two transport equations with five constants — and every one of those constants was measured. The field’s own honest summary of itself, made repeatedly by the people who built it, is that there is no theory in the sense that the rest of fluid mechanics has one.

There is one exception, and it is worth knowing exactly what it says.

$$\langle [u(x+r) - u(x)]^3 \rangle = -\tfrac{4}{5}\,\varepsilon\,r$$

No adjustable constant appears in that. It is a consequence of the Navier–Stokes equations, homogeneity, isotropy and the limit of large Reynolds number, and nothing else — no closure, no dimensional argument with a fitted coefficient, no appeal to experiment.

What has to be assumed, and what does not #

The route to the result is the Kármán–Howarth equation, which is the two-point correlation’s own evolution equation, obtained by writing the momentum equation at two points and multiplying them together. It is exact, and it is not closed — the time derivative of a second moment involves a third moment, which is the closure problem arriving in its original form.

Kolmogorov’s move was to stop trying to close it and to look at what it says when the flow is statistically steady: the second moment is then not changing, so the third moment’s divergence has to balance the dissipation. Carried through with isotropy, the balance reads

$$S_3(r) = -\tfrac{4}{5}\varepsilon r + 6\nu\frac{dS_2}{dr},$$

and in the inertial range — separations far above the dissipation scale — the viscous term is negligible and the four-fifths law is what is left.

The assumptions are worth listing plainly, because each of them is a real restriction:

• homogeneity, so the statistics do not depend on position — false near any wall;
• isotropy, so they do not depend on direction — false in any shear;
• stationarity, so the energy fed in equals the energy dissipated — false in decaying turbulence, and repairable there with an extra term;
• and the limit $\mathrm{Re}\to\infty$ with $\varepsilon$ held fixed, which is the assumption the next essay in this field is entirely about.

The third of those is worth a sentence more, because it is the one that is quietly relaxed most often. In decaying grid turbulence the energy is falling rather than being supplied, so the balance acquires a term in $\partial S_2/\partial t$ and the law becomes $S_3 = -(4/5)\varepsilon r$ plus a correction that is small when the decay is slow compared with the eddy turnover. Measurements in grid turbulence are made in that regime and the correction is estimated and subtracted; it is not ignored, and a law quoted without it in a rapidly decaying flow is being quoted outside its hypothesis.

What is not assumed is anything about the mechanism. No eddy is modelled, nothing cascades in the derivation, and the word “cascade” appears nowhere in it. The law is a conservation statement about a flux, and the flux is what a cascade would be if there were one.

The constant in the companion law is a gamma function #

The four-fifths law’s famous relative is the two-thirds law, $S_2(r) = C_2(\varepsilon r)^{2/3}$, which is not exact: it follows from dimensional analysis rather than from the equations, and $C_2$ is measured. That is the standard account and it is half right. The dimensional argument fixes the exponent and leaves a number; what is less often said is that the number is not independent of the spectral constant, and the relation between the two is exact.

With a one-dimensional spectrum $E(k) = C\varepsilon^{2/3}k^{-5/3}$ held over an unbounded range,

$$S_2(r) = 2\int_0^\infty E(k)\,[1 - \cos kr]\;dk = 2C(\varepsilon r)^{2/3}\int_0^\infty x^{-5/3}(1-\cos x)\,dx,$$

and that integral has a closed form. It converges at both ends — as $x^{1/3}$ at the origin, where $1-\cos x$ vanishes as $x^2$, and as $x^{-5/3}$ at infinity — and its value is $-\Gamma(-2/3)\cos(\pi/3)$.

So the whole constant is $-\Gamma(-2/3) = 4.018408$, and the only empirical quantity left is $C$ itself, the Kolmogorov constant, which experiment puts near 0.5 for the one-dimensional spectrum. The useful consequence is that a measurement of $C_2$ and a measurement of $C$ are the same measurement, and any pair of published values whose ratio is not 4.0184 contains an error somewhere — a definition of the spectrum, a factor of $2\pi$, or a one-dimensional constant compared with a three-dimensional one.

The small-$x$ trap in that quadrature is worth recording, because it is the same one this collection has met from the other end. Near the origin $1-\cos x$ is the difference of two numbers that are both one, so evaluating it directly loses every significant figure exactly where the integrand is largest; the series is used below $x = 0.5$ instead. A quadrature that ignores this returns a plausible number and is wrong in the fourth figure, which is precisely the accuracy the comparison above is testing.

Where viscosity takes the law back #

The full balance carries the viscous term, and its size decides where the inertial range stops. With a K41 form for $S_2$ put into it, the two terms are equal at

$$r \approx 5.6\,\eta, \qquad \eta = (\nu^3/\varepsilon)^{1/4},$$

and the multiple is the same at every viscosity tried — $10^{-6}$, $10^{-7}$ and $10^{-8}$ give the same 5.643, to fifteen figures. That constancy is the check that matters, because nothing in the calculation was told what the Kolmogorov scale is: it was computed from $\nu$ and $\varepsilon$ and then the crossing was found by bisection.

That is a small result and it settles something. The inertial range is usually defined by saying that $r$ is much larger than $\eta$ and much smaller than the integral scale, with “much” left to taste. The balance above puts a number on the lower end — a few Kolmogorov lengths, not one and not a hundred — and it comes from the same equation the law does.

How it is measured, and the trick that made it possible #

Testing the law needs a third moment converged over a decade of separations in a flow with an inertial range, which is a demanding combination. Grid turbulence in a wind tunnel has the isotropy and not the Reynolds number; the atmospheric surface layer has the Reynolds number and not the homogeneity; a jet has both and a mean shear as well.

Two things made the measurements work. The first is patience — hours of data, so that the number of independent large eddies sampled runs into the millions rather than the thousands. The second is a change of variable: instead of plotting $S_3$ against $r$, plot each moment against $S_3$ itself, which is exact and therefore a perfect abscissa. Scaling ranges then extend far below where they have any business being, and exponent ratios can be measured in flows with almost no inertial range at all. The technique is called extended self-similarity, it was found empirically in 1993, and its justification remains the fact that it works.

Why an exact result exists here and nowhere else #

It is worth asking why this one statement escaped when everything else in the subject needed a model, because the answer is a general one.

The law is about a flux, and a flux in a steady state is fixed by what enters and leaves. The interior of the problem — how the energy actually gets from one scale to the next, which eddies do it, how long it takes — decides nothing. Every quantity in the derivation is either conserved or supplied at the boundary of the range, and the mechanism cancels.

This site has met the same shape of argument several times. The lift on a body is $\rho U\Gamma$ on every contour and by any route, while the split between pressure and momentum runs from three per cent to ninety-seven. The vorticity a wall makes is the pressure gradient along it, with no viscosity in the answer. In each case a conservation statement survives an interior nobody can compute, and the interior turns out to decide only how the answer is arranged.

The minus sign is the whole direction of the cascade #

Everything about the four-fifths law that is usually discussed is its magnitude, and the part that carries the physics is its sign.

A third moment is an odd moment, so it vanishes for any symmetric distribution. That the third moment of longitudinal velocity increments is not zero says the distribution is skewed: a fluid parcel is more likely to find the fluid ahead of it moving towards it than away. That it is negatively skewed says which way — the increments have a preference for compression along the separation direction, and compression along one direction is extension along the others.

That preference is the cascade. Nowhere else in the derivation does a direction appear. The Kármán–Howarth equation is exact and says nothing about which way energy travels; the dimensional argument fixes exponents and is blind to sign; the picture of eddies breaking into smaller eddies is a picture. The one statement in the whole subject that says energy goes from large scales to small rather than the reverse is the minus sign in front of the four fifths, and it is a consequence of the equations rather than an assumption about them.

Take the separation down to the dissipation range and the same statement reappears as a number people measure directly. The derivative skewness — the third moment of $\partial u/\partial x$ divided by the three-halves power of its second — comes out near $-0.5$ in every turbulent flow it has been measured in, and its negativity is exactly the statement that vortex stretching outweighs vortex compression on average. The enstrophy production term $\langle\omega_i\omega_j S_{ij}\rangle$ is proportional to minus that skewness, so a negative skewness is a positive rate of enstrophy production — a flow that is manufacturing velocity gradients faster than it destroys them, which is the local version of the same asymmetry the four-fifths law states globally.

The cleanest confirmation that the sign is doing the work is the case where it reverses.

In two dimensions the vortex-stretching term is identically zero, energy travels to large scales rather than small, and the corresponding exact law comes back with the opposite sign: in the inverse-cascade range the third moment is $+\tfrac{3}{2}\varepsilon r$, positive, with a different numerical coefficient because the dimension enters the isotropy algebra. The same equation, the same kind of derivation, the same absence of any adjustable constant — and a plus where the three-dimensional case has a minus.

That makes the third moment the only instrument in the subject that reports the direction of a flux without being told it. A measured spectrum with a $-5/3$ slope is consistent with a cascade running either way, which is exactly why the existence of the inverse cascade was argued about on theoretical grounds for a decade before it was measured. A measured third moment is not: its sign is the answer.

There is a companion law in the plane for the range above the forcing, where enstrophy rather than energy is what flows downscale, and it is cubic in the separation rather than linear — the exponent changing because the conserved quantity being fluxed has an extra derivative in it. Three exact laws, one in each of the three flux ranges the two dimensionalities offer between them, and each of them says the same kind of thing: given a range across which something conserved is passed at a steady rate, the odd moment that measures the passing is fixed, and everything about the mechanism that does the passing cancels out.

What the law does not say #

It does not predict $\varepsilon$. The dissipation rate appears in it as a given, and what fixes $\varepsilon$ is the large scales — the stirring, the shear, the grid — which is where every practical difficulty in turbulence lives. The law relates two unknowns; it does not remove one.

It does not survive to higher moments. K41 predicts $S_p(r) \propto (\varepsilon r)^{p/3}$ for every $p$, and measurement finds exponents that fall increasingly below $p/3$ as $p$ rises. That departure is intermittency: the dissipation is not spread evenly but concentrated in thin regions, so a high moment is dominated by rare intense events and scales differently. The four-fifths law is untouched by any of this, because $p = 3$ is exactly the moment the exact argument fixes — a coincidence with a reason behind it, since the flux is a third-order quantity.

It says nothing about vorticity. The mechanism usually drawn for the cascade is a vortex tube being stretched and intensified, and that picture is a good one, but the exact law neither uses it nor tests it. A different mechanism producing the same flux would satisfy the law identically, which is the same limitation the spectrum has in a stronger form.

And it is a statement about an ensemble. Every angle bracket in this essay is an average over realisations, and a single record gives a third moment with a scatter comparable to its own value — which is the difficulty the essay below this one is about. Verifying the four-fifths law experimentally took until the 1990s, not because anybody doubted it, but because converging a third moment over a decade of separations needs an enormous amount of data.

What the picture cannot show #

Nothing here is a measurement of turbulence. The figures draw the law, the balance between its two terms and the constant relating a spectrum to a structure function; the fields used to illustrate the second-order statistics are synthesised from a stated spectrum, and their third moments are zero in the ensemble. This collection cannot produce a turbulent field, and the arithmetic of why is elsewhere in this field.

The viscous crossing was computed with a K41 second-order function. Where intermittency modifies $S_2$, the crossing moves — by a few per cent at achievable Reynolds numbers, which is smaller than the uncertainty in $\varepsilon$ in most experiments and is a real limitation of the number 5.64.

And the isotropy assumption is the one that fails in practice. Every flow anybody wants to compute has a mean shear in it, which makes the statistics anisotropic at the large scales; the hope is that the small scales forget, and the extent to which they do is measurable, imperfect, and still being argued about. Where the flow is also stratified or rotating the hope fails outright, because buoyancy supplies a scale the argument assumed away and the range that would have been inertial is not.

Who found it, and when #

Kolmogorov’s three 1941 notes are four, five and six pages long between them. The four-fifths law is in the third, derived from the Kármán–Howarth equation of 1938; Kolmogorov himself regarded the dimensional arguments of the first two as the substantial contribution, and the exact law as a corollary. Landau’s objection — that the dissipation rate fluctuates, so an average over a large volume is not the same as the local value — arrived immediately and is the seed of everything later written about intermittency; Kolmogorov’s 1962 refinement is a response to it.

The surprising connection is with the boundary conditions an earlier group of these essays settled. The four-fifths law is a statement about the interior of a turbulent flow, and the reason it can be made at all is that the flux through a range of scales is set at the two ends of that range and not inside it — the same structure as an overlap layer, where the profile’s form is fixed by two limits and not by anything local. Turbulence yields exact results exactly where the answer is a boundary condition in disguise, which is a short way of saying why there is only one of them.

Where the ladder goes next #

The rung above is the assumption the law leans on hardest: that $\varepsilon$ stays finite as the viscosity goes to zero, which is not a theorem and is the strangest well-established fact in the subject. Beside it lies intermittency, where the exact law’s neighbours stop obeying the dimensional argument, and which this collection treats as an open subject rather than a settled rung.