The limit that is not the value

Write down the rate at which a fluid destroys kinetic energy and the viscosity is standing in front of it:

$$\varepsilon = \nu\left\langle \left(\frac{\partial u_i}{\partial x_j}\right)^2\right\rangle.$$

Halve the viscosity and — if nothing else changes — the dissipation halves. Take the viscosity to zero and the fluid stops dissipating altogether, which is exactly what the inviscid equations say: Euler’s equations conserve kinetic energy for ever.

Nothing else stays the same. As the viscosity falls the flow rearranges itself, the gradients grow, and the product does not move. This is the dissipation anomaly, and it is the most important fact about turbulence that can be stated in one line of arithmetic.

The arithmetic, which is exact #

Kolmogorov’s scales are defined from the two quantities the dissipation range has available — the viscosity and the dissipation rate itself:

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

Those are definitions, so the velocity gradient at that scale is $u_\eta/\eta = \sqrt{\varepsilon/\nu}$ and

$$\nu\left(\frac{u_\eta}{\eta}\right)^2 = \nu \cdot \frac{\varepsilon}{\nu} = \varepsilon$$

identically. There is nothing to check in that algebra, and the figure above checks it anyway, because the point of the figure is not the identity: it is the sizes of the two factors that cancel. Across the six decades drawn there, the viscosity falls by $10^6$ and the squared gradient rises by $10^6$, and $\eta/L$ falls as $\mathrm{Re}^{-3/4}$ exactly.

What is not arithmetic is the assumption underneath it: that $\varepsilon$ is fixed by the large scales, at about $u^3/L$, with no viscosity in it. That statement is the physical content, it is supported by every measurement that has been made of it, and it is not a theorem. It is sometimes called the zeroth law of turbulence — a name that concedes both that it is fundamental and that nobody has derived it.

A model problem where the same thing happens exactly #

The claim above is about turbulence and cannot be computed here. What can be computed, exactly, is the archetype of the behaviour — a singular limit in which a quantity refuses to follow the parameter that appears to produce it.

Take $\varepsilon u'' + u' = 0$ on the unit interval, with a condition at each end. It is the one-dimensional shadow of a boundary layer: the reduced equation with $\varepsilon = 0$ is first order, can satisfy only one of the two conditions, and misses the other by the whole range. Restoring $\varepsilon$ restores the second condition and pays for it with a layer of thickness $\varepsilon$, inside which the gradient is of order $1/\varepsilon$.

The stress — the product of the two — is the same number at every $\varepsilon$.

That is the whole structure of the dissipation anomaly in a problem with a closed-form solution. The limit $\varepsilon \to 0$ and the value at $\varepsilon = 0$ are different, because the solution develops a region whose thickness goes to zero and whose gradient goes to infinity in exactly compensating ways. Nothing about turbulence is needed to make it happen; what turbulence supplies is a three-dimensional version of the same trick, and no proof that it does.

Where the same statement has already appeared on this site #

As d’Alembert’s paradox. The exact inviscid theory gives a body no drag whatever, and the measured drag of that body does not fall towards zero as the Reynolds number rises: it flattens. The paradox is what the site’s inviscid figures are built around, and the usual resolution — that a boundary layer separates and the pressure never recovers — is the same statement as this essay’s, read from the force rather than from the energy. A drag coefficient that tends to a constant is a dissipation that tends to a constant, since the two are related by the speed and the size.

As the flat plate’s friction. A laminar plate’s drag coefficient falls as $\mathrm{Re}^{-1/2}$ and a turbulent one as $\mathrm{Re}^{-1/5}$: both fall, so a plate is a case where the anomaly is partial. The distinction matters and it is geometrical — a plate has no separation and no pressure drag, so its dissipation is confined to a layer whose thickness is set by viscosity, and the limit is not singular in the same way.

And as the cascade’s own book-keeping. The four-fifths law takes $\varepsilon$ as given and relates it to the third moment; the inertial range’s whole function is to carry that flux from the scales where it is supplied to the scales where it is destroyed. If $\varepsilon$ fell with the viscosity, the inertial range would carry less and less and the cascade would be a marginal effect rather than the organising one.

The same singularity, taken from the other end #

The limit of vanishing viscosity is not the only one in this subject that fails to commute with the answer. The limit of vanishing inertia does the same thing, and the pair make each other clearer.

Set the Reynolds number to zero exactly and the Stokes equations are linear, reversible and beautifully behaved — and in two dimensions they have no solution at all for flow past a cylinder, because the neglected inertial term is never negligible far enough away. That is Stokes’ paradox, and this collection computes it as a drag that never settles. The structure is identical to the one here: a small parameter multiplying the highest derivative in one case and the largest region in the other, and a limit that is singular rather than regular in both.

A regular limit lets a small parameter be set to zero. A singular one does not, and fluid mechanics is a subject in which both of its two natural small parameters are singular. That is the reason the field is so much harder than its equations look, and it is why the counting of boundary conditions is where this collection starts its account of viscosity rather than where it finishes.

The dimension where it does not happen #

The strongest evidence about what causes the anomaly comes from the case where it is absent, and that case is not exotic: it is the same equations with one direction removed.

In two dimensions the vortex-stretching term vanishes identically, because a vortex line is perpendicular to the plane and the velocity has no component along it to stretch it with. Take that term away and the mathematics changes character completely. Solutions of the two-dimensional Navier–Stokes equations are smooth for all time from smooth data — a theorem, not a conjecture — and a smooth solution cannot dissipate in the limit, because Onsager’s criterion is a statement about roughness and there is none available. The energy dissipation of two-dimensional turbulence goes to zero with the viscosity, exactly as the naive reading of $\varepsilon = \nu\langle(\partial u)^2\rangle$ says it should.

So the anomaly is not a consequence of the equations being nonlinear, and it is not a consequence of their being hard. Both of those survive into two dimensions unchanged. It is a consequence of the one term that does not, and the whole of the three-dimensional cascade rests on the mechanism this collection draws as a stretched vortex tube: stretching a tube amplifies its vorticity and narrows it, which is a machine for manufacturing gradients out of a flow that had none.

What survives in the plane is a different anomaly with a different invariant. Enstrophy — the mean square vorticity — is a second inviscid invariant in two dimensions, and it is enstrophy rather than energy that finds its way to small scales and is destroyed there. So the plane has a direct enstrophy cascade and, in the other direction, an inverse energy cascade: the energy that cannot go down goes up, accumulating at the largest scale the domain has. That is where a two-dimensional flow’s energy actually ends up, and it is why the plane needs a large-scale friction to reach any steady state at all — a term that three-dimensional turbulence has no use for, because the small scales already dispose of everything.

Both halves of that contrast are worth carrying into a reading of any planetary or oceanic flow, which is where two-dimensionality is a decent approximation. A large-scale atmospheric flow does not dissipate its energy by cascading it; it hands energy upward to the largest available structures and loses it at the boundary, which is why an Ekman layer at the bottom of the atmosphere is a term in the energy budget of the whole circulation rather than a detail of the surface.

It also explains why the two-dimensional case is not the easy warm-up for the three-dimensional one that it looks like. It is a different problem with different invariants, a different direction of flux, and a regular limit where the real case has a singular one — and every intuition trained on it about where energy goes is exactly reversed. The plane is the control experiment for the anomaly, and the result of the control is that removing one term removes the phenomenon.

Why it is not a theorem #

The mathematical question behind the anomaly is whether solutions of the Euler equations can dissipate energy at all. A smooth solution cannot: energy is conserved exactly. Onsager conjectured in 1949 that a solution rough enough — with velocity increments scaling as $r^{h}$ with $h \le 1/3$ — could dissipate, and that the critical exponent is exactly the one Kolmogorov’s theory gives. Both halves of that conjecture have since been proved: solutions smoother than $1/3$ conserve energy, and dissipating solutions rougher than $1/3$ exist.

What remains unproved is the physically interesting statement — that the solutions of the Navier–Stokes equations actually approach such an Euler solution as the viscosity goes to zero. That is not a technicality. It is the question of whether the limit exists at all, and it is entangled with the regularity problem that has stood open since Leray.

There is a second reason the question resists computation, and it is not a matter of resources. To show that the dissipation tends to a non-zero limit one has to compute at a sequence of viscosities and watch a quantity that changes by nothing; the numerical error in $\varepsilon$ at the largest affordable Reynolds number is of the same order as the effect being looked for at the next one, so the sequence has to be long as well as fine. The measurements are better placed to settle it than the computations, and they have — over ninety years and across flows with no obvious resemblance to each other.

So the position is this. The anomaly is measured, it is consistent with a proved mathematical possibility, and it is assumed by every turbulence model in use. It is not derived from anything, and a subject whose central quantitative fact is an assumption is a subject that should say so in its figures.

There is a third quantity that behaves like the energy does in three dimensions and unlike it in two, and it is worth naming because it is the one case where something has been proved. A passive scalar stirred by a turbulent flow — a dye, a temperature that does not push back — dissipates its variance at a rate that also refuses to follow its diffusivity to zero, and in Kraichnan’s model of a flow with no time correlation that scalar anomaly is a theorem rather than an assumption. It is the only member of this family whose central fact has been derived, and the flow it is derived for is one nobody claims is turbulence.

What it means for anything that has to be computed #

Three practical consequences follow, and they are the reason this belongs in an engineering collection rather than only in a mathematical one.

A drag prediction cannot be made by refining a viscous calculation until the viscosity stops mattering. It never stops mattering: it decides the scale of the dissipating structures, and those have to be resolved or modelled at every Reynolds number. This is why a calculation of a wing at flight conditions is not a calculation of a wing in a tunnel with the viscosity turned down.

A model that makes eddy viscosity large is not making a mistake about the physics, and the reason is worth stating in the same breath as the caution that an eddy viscosity is not a viscosity. The dissipation has to come out at $u^3/L$ regardless of the molecular viscosity, so a model that reproduces the right dissipation with a viscosity a thousand times too large will get the large scales right — which is the entire justification for large-eddy simulation, and the reason it works better than it has any right to.

The two drags a wing pays behave differently, and the anomaly says which. Induced drag is a large-scale quantity and has no viscosity in it at all; friction drag is a boundary-layer quantity and falls slowly with Reynolds number. The split between them therefore moves with speed, and the reason a wing’s total drag coefficient is nearly flat across a flight envelope is a cancellation between two effects with different origins rather than a single insensitive number.

And a laboratory measurement at one Reynolds number is worth more than it looks. If dissipation were proportional to viscosity, a drag coefficient measured in a tunnel would need correcting to flight by a large factor. Because it is not, the coefficient is nearly the same number, and the whole practice of testing scale models rests on that flatness — with the notorious exception of the transition Reynolds number, which is not flat at all, and which is the largest single reason a model in a tunnel cannot be matched to the aeroplane. A ball is the extreme case: its drag coefficient is flat over three decades and then falls by a factor of four across a factor of two in speed, which is a change in where the dissipation happens rather than in how much of it there is.

What the picture cannot show #

The hero figure is an identity, not a measurement. It draws what follows from the definitions of $\eta$ and $u_\eta$ once $\varepsilon$ is assumed independent of $\nu$; the assumption is the content and it is imported. A figure showing the anomaly itself would need turbulent fields at several viscosities, which this collection cannot produce.

The model problem is one-dimensional and linear. Its stress is $\varepsilon$-independent for a reason that is easy to see and has nothing to do with vortex stretching. It is an archetype, not a proof by analogy.

And the drag data are borrowed. The sphere’s drag curve is a correlation of measurements, and it is labelled as one wherever it is drawn on this site. Nothing here computes a bluff-body drag coefficient at $\mathrm{Re} = 10^5$, because that requires resolving a turbulent separated wake.

Who found it, and when #

Taylor’s 1935 papers on grid turbulence contain the first clear statement that the dissipation is fixed by the large scales, in the form $\varepsilon = A u^3/L$ with $A$ a constant of order one, and the measurements of that constant have been repeated for ninety years across a dozen kinds of flow; its value drifts with the flow’s own geometry and does not drift with viscosity. Onsager’s conjecture is from 1949, published as an aside in a paper about statistical hydrodynamics, and it was not taken up for forty years.

The surprising connection is with the site’s own opening argument. The exact theory of an inviscid fluid predicts no drag, and this collection introduces that as the useful failure in fluid mechanics — the thing whose wrongness taught the subject what to look at. The dissipation anomaly is the same failure stated as a limit rather than as a paradox: the inviscid answer is not the limit of the viscous ones, and the difference between the two is where all of the interesting physics has been hiding since 1752.

Where the ladder goes next #

Beside this rung sits the question of what the dissipating structures actually are — thin sheets, tubes, or something with no simple description — which is the subject of intermittency and which this collection treats as open. Above it lies the practical version: a model whose eddy viscosity is chosen to deliver the right $\varepsilon$ and which therefore cannot be wrong about the large scales, which is where large-eddy simulation begins.