Where the energy goes

A turbulent flow is being stirred at one end and heated at the other. Energy goes in at the scale of whatever is doing the stirring — the diameter of a pipe, the chord of a wing, the width of a jet — and comes out as heat at a scale small enough for viscosity to act on it.

Between those two scales there is a range in which neither the stirring nor the viscosity is directly involved. The energy is simply passing through, from larger motions to smaller ones.

Kolmogorov’s observation in 1941 was that in such a range there is almost nothing left for the statistics to depend on, and that “almost nothing” is enough to determine the answer.

The argument, in full, because it is three lines #

In the inertial range the available quantities are the wavenumber k, with dimensions 1/length, and the rate ε at which energy is being passed down the cascade, with dimensions of energy per unit mass per unit time — that is, length² per time³.

The quantity wanted is E(k), the energy per unit mass per unit wavenumber, with dimensions length³ per time².

There is exactly one way to combine ε and k to get those dimensions:

$$E(k) = C\,\varepsilon^{2/3} k^{-5/3}$$

and the exponents are forced. Only the constant C is left for an experiment to supply, and it comes out at about 1.5.

Three things about that argument deserve saying plainly, because it is often presented as a piece of magic and it is not.

The strength is in what was excluded. Viscosity was excluded because in the inertial range it is negligible; the stirring scale was excluded because the eddies there are supposed to have forgotten it. Both exclusions are hypotheses, and they are the content of the theory. The dimensional analysis afterwards is bookkeeping.

The exponent is not adjustable. Given the two hypotheses, −5/3 is not a fit to anything. Any measurement finding a different exponent is a measurement refuting one of the hypotheses.

It says nothing about the mechanism. No statement about how an eddy hands energy to a smaller one appears anywhere in the derivation, and the theory would be unchanged if the mechanism were entirely different.

Measuring the slope off the drawing #

The figure above is drawn from the model spectrum — the inertial power law with the standard production and dissipation roll-offs — and then the build fits a straight line to the drawn points over the middle of the inertial range, a decade in from each end so that neither roll-off is inside the window.

At Re = 10⁶ the fit returns −1.658 over 4.5 decades from 201 points, against −5/3 = −1.667.

This is not a formality. The first version of the low-wavenumber roll-off in this site’s spectrum function went as k^(−10/3) at large argument instead of tending to one, which meant it was still steepening the curve four decades above where it was supposed to have stopped. The drawn spectrum looked exactly right — a straight line on log axes is convincing at any slope — and the fitted exponent came out at −5.06. The check found it; nothing else could have.

The build now refuses a spectrum whose fitted slope differs from −5/3 by more than 0.03.

How much room the cascade has #

The second half of Kolmogorov’s argument fixes the scale at which the cascade ends.

At the small end the available quantities are ε and the viscosity ν, and there is one length that can be built from them:

$$\eta = \left(\frac{\nu^3}{\varepsilon}\right)^{1/4}$$

If the energy entering at the large scale L does so at a rate of order u³/L — which is the statement that the large eddies hand over their energy in about one turnover time — then

$$\frac{\eta}{L} = Re^{-3/4}$$

The site computes η by both routes and asserts that they agree, which is a check on the internal consistency of the estimate rather than on nature: quoting a dissipation scale computed one way beside a Reynolds number that means the other produces two numbers that look like a check and are one number twice.

The consequence is the figure above. An inertial range three-quarters of a decade wide per decade of Reynolds number. A laboratory jet at Re = 10⁴ has three decades between L and η, of which only about one survives after the roll-offs at each end are excluded, and that is why measuring the exponent well took until high-Reynolds-number atmospheric and tidal-channel data became available.

Why the cascade goes downward, and not up #

Nothing in the dimensional argument says which way the energy travels, and it is worth knowing that the direction is a separate fact with a separate cause — because in two dimensions it goes the other way.

The mechanism usually offered is vortex stretching. A vortex tube being stretched by the surrounding strain field conserves its circulation while its cross-section shrinks, so its vorticity rises and its characteristic length falls. Energy associated with the tube’s motion therefore appears at a smaller scale than it occupied before, and a field full of tubes being stretched by each other’s strain moves energy down the scales.

The reason this matters is that vortex stretching requires three dimensions. In a two-dimensional flow the vorticity is a scalar carried by each parcel and cannot be intensified by stretching at all; what is conserved instead is enstrophy, the mean square vorticity, and the resulting cascade runs downward in enstrophy and upward in energy. Two-dimensional turbulence forms larger structures rather than smaller ones, which is why planetary atmospheres organise into a few big vortices rather than dissolving into fine grain.

That contrast is the strongest available evidence that the cascade is a real mechanism rather than a restatement of the dimensional analysis: the dimensional analysis is identical in two dimensions and the answer is different, so something other than dimensions is doing the work.

The dissipation anomaly, which is the surprising part #

The most surprising consequence of the argument is easy to state and hard to believe on first hearing.

ε, the rate at which energy is dissipated, contains no viscosity. It is u³/L: a property of the large eddies alone.

So lowering the viscosity of a turbulent flow does not lower its dissipation. The energy still goes in at the same rate and still comes out as heat at the same rate. What changes is η — the scale at which the conversion happens moves further down — and the inertial range gets wider.

This is called the dissipation anomaly, and it is why the drag coefficient of a bluff body is roughly constant over decades of Reynolds number instead of falling as viscosity becomes relatively less important. The viscosity is what does the dissipating, and it is not what sets how much.

It also gives the estimate that decides what a wind tunnel can and cannot reproduce. Two flows at the same Reynolds number have inertial ranges of the same width and small scales in the same proportion, which is why the Reynolds number is the right variable for matching a model to full scale — and why matching it is so often impossible once compressibility is in the problem too.

The same statement in reverse is the reason the grid nobody can build is as large as it is. A calculation that resolves the dissipation must resolve η, and η retreats as the viscosity falls, so making the problem more inviscid makes it computationally harder rather than easier.

What is actually being claimed about the small scales #

The two hypotheses deserve stating in Kolmogorov’s own terms, because they are stronger than the dimensional analysis suggests and they are the part that can be wrong.

Local isotropy. At scales much smaller than L, the statistics of the velocity differences are isotropic — the same in every direction — however anisotropic the large scales are. A boundary layer is emphatically not isotropic at large scale; the claim is that the small eddies have forgotten which way is up.

Local homogeneity and universality. Those small-scale statistics depend only on ε and ν, and therefore have the same form in a jet, a pipe, a boundary layer and the atmosphere.

Both are hypotheses about forgetting. The cascade is supposed to pass energy down through so many steps that the information about how it started is lost, in the way that a long chain of collisions loses the memory of an initial condition.

The second-order statistics support this well. The −5/3 spectrum is observed, over the ranges where there is enough Reynolds number to see it, in flows that have nothing in common at large scale.

The hypothesis every measurement of it rests on #

There is a third assumption in play whenever the $-5/3$ law is confirmed, and it belongs to the experiment rather than to the theory. It is worth separating out, because it is doing as much work as either of Kolmogorov’s.

A spectrum is a function of wavenumber — of space. Almost every instrument that has ever measured one is a probe at a fixed point recording a time series: a hot wire, an anemometer on a mast, a current meter in a channel. The two are connected by an assertion rather than by a measurement. Taylor’s, from 1938: if the turbulence is carried past the probe faster than it evolves, then what the probe records in time is a spatial cut through a pattern that has not changed while it went by, and

$$k = \frac{2\pi f}{U}.$$

The turbulence is treated as frozen, swept rigidly past at the mean speed. Every published inertial range is a frequency spectrum with that substitution applied to its abscissa.

The condition for it is that the fluctuation be small compared with the sweeping speed, $u'/U \ll 1$, and its failure has a specific character rather than being a general fuzziness. What actually carries a small eddy past the probe is not the mean velocity but the instantaneous velocity of the large eddy it is sitting inside, which is sometimes faster and sometimes slower. So each part of the record is stretched or compressed by a random factor, and the effect of that on the spectrum is a smearing that grows with wavenumber — worst precisely at the small scales the hypothesis is being used to examine.

This decides where a good measurement can be made. The requirement is a large Reynolds number and a small turbulence intensity at once, and those pull against each other in every laboratory apparatus: a jet or a wake reaches a high Reynolds number by having vigorous fluctuations, which is exactly what breaks the hypothesis. A tidal channel and an atmospheric surface layer have both — a strong steady mean flow with fluctuations at a tenth of it, and a Reynolds number no laboratory can approach — which is a second reason the convincing observations came from there, alongside the width of the inertial range.

So the exponent is confirmed subject to three hypotheses and not two, and the third is the one that would leave no trace if it were wrong. A distorted spectrum is still a smooth curve on logarithmic axes, and its slope is still measurable to three figures. The corrections to it were first worked out by Lumley in the 1960s, and the modern alternative is to abandon the substitution entirely and measure two points at once, which is what particle image velocimetry does and what a single probe never could.

Where the model stops: the higher moments do not obey it #

The 1941 theory makes predictions beyond the spectrum. The nth-order structure function — the mean of the nth power of the velocity difference across a separation r — should go as (εr)^(n/3) in the inertial range.

For n = 2 this is the spectrum in another guise and it works. For n = 3 there is an exact result, Kolmogorov’s four-fifths law, derived from the Navier–Stokes equations without any dimensional argument: ⟨Δu³⟩ = −(4/5)εr. It is the only exact non-trivial result in the whole subject and it holds.

For n larger than 3 the measured exponents fall progressively below n/3, and the discrepancy grows with n. The cause is intermittency: the dissipation is not spread evenly through the flow but concentrated in thin sheets and filaments, so ε varies enormously from place to place and the average of a high power is dominated by the rare intense regions.

Kolmogorov himself proposed a refinement in 1962 to accommodate it, and the subject of intermittency corrections has been active ever since without a derivation from first principles emerging.

The honest position is therefore: the cascade picture is right about the mean energy flux and the second-order statistics, and incomplete about the higher-order structure of the small scales. That is a very good position for a hypothesis to be in after eighty years, and it is not the same as being established.

What this site can and cannot do with it #

Nothing here computes a turbulent velocity field, so nothing here measures a spectrum from data — for the reason the closure ladder and the grid estimate both give, from opposite directions. The spectrum in the first figure is the model spectrum — the inertial power law with the standard roll-offs — drawn from the formula and then measured, which checks the drawing rather than the physics.

That distinction is worth keeping sharp because the figure looks exactly like a measured spectrum would look. What justifies drawing it at all is that the shape is the content of the theory: a reader can see how narrow the inertial range is at accessible Reynolds numbers, which is the fact that most changes how the −5/3 law should be read.

The arithmetic in the range and cost figures is a different matter, and it is as solid as the compressible-flow arithmetic elsewhere on this site. Re^(3/4) and the width of the inertial range are consequences of the definitions, and they are as solid as anything on this site.

Who found it, and when #

There is one further consequence worth drawing out before the history, because it settles a question this site keeps meeting. If the small scales are universal, then the only thing a wind tunnel has to match to reproduce them is the Reynolds number — and if they are not, matching it is not enough. The evidence at second order says they are, which is a large part of why similarity works as well as it does for drag and rather less well for anything that depends on the tails.

Richardson wrote the verse in 1922 — big whirls have lesser whirls that feed on their velocity, and so on to viscosity — as a description of atmospheric diffusion, and it is a statement of the cascade before there was a theory of it.

Kolmogorov’s three 1941 papers contain the hypotheses, the spectrum and the four-fifths law. Obukhov obtained the spectrum independently the same year, and Onsager in 1945 and Heisenberg and von Weizsäcker in 1948 arrived at it again by other routes — four independent derivations in seven years, which is itself evidence of how little the result depends on the details.

Grant, Stewart and Moilliet’s measurements in a tidal channel in 1962 are usually taken as the first convincing observation of the inertial range, because a tidal channel supplies a Reynolds number no laboratory could.

It is also worth saying which of the two hypotheses the intermittency corrections damage. Local isotropy survives them: the small scales do appear to forget direction. What fails is the assumption that ε may be treated as a constant rather than as a fluctuating field, and the failure grows with the order of the moment because a high power weights the rare intense events. The theory is wrong about the tails of a distribution and right about its middle, which is a common and unglamorous position for a physical theory to be in.

Kolmogorov’s 1962 refinement is the response to Landau’s objection, made at a seminar in 1944, that the fluctuations in ε itself had been ignored. Landau was right and the refinement is still not settled.

Where the ladder goes next #

The other rung of this anchor takes the scale separation seriously as a cost. The grid nobody can build turns Re^(3/4) into Re^(9/4) grid points and Re³ point-updates, and states plainly what that means for the figures on this site and for computational fluid dynamics generally.