Equal on average, and nothing else

The price of a gradient is the collection’s account of what dissipation is and where it happens, and the limit that is not the value is about the anomaly in its average. This essay is about a quieter problem: there are two standard expressions for the local dissipation rate, they are described as equivalent everywhere, and they are equal in exactly one sense.

The two forms #

The rate at which viscosity converts kinetic energy to heat per unit volume can be written

$$\varepsilon_S = 2\nu\,S_{ij}S_{ij} \qquad\text{or}\qquad \varepsilon_\omega = \nu\,\omega^2,$$

the first from the strain-rate tensor and the second from the vorticity. For a homogeneous incompressible flow their averages are equal, because

$$2S_{ij}S_{ij} - \omega^2 = 2\frac{\partial^2 (u_iu_j)}{\partial x_i\,\partial x_j}$$

is a divergence, and a divergence integrates to zero over a periodic box or a domain with no flux through its boundary. Both forms are correct, both are used, and the choice between them is made on whichever is easier to measure or to compute.

That is where the textbooks stop, and it is where the trouble starts.

Where the identity comes from #

The derivation is three lines and worth having, because it shows exactly what is being moved around.

Start from the strain form and split the velocity gradient into its symmetric and antisymmetric parts, $\partial u_i/\partial x_j = S_{ij} + \Omega_{ij}$. Then

$$\frac{\partial u_i}{\partial x_j}\frac{\partial u_i}{\partial x_j} = S_{ij}S_{ij} + \Omega_{ij}\Omega_{ij} = S_{ij}S_{ij} + \tfrac12\omega^2,$$

since the cross terms vanish by symmetry. Separately, integrating by parts twice and using incompressibility gives

$$\frac{\partial u_i}{\partial x_j}\frac{\partial u_i}{\partial x_j} = \frac{\partial u_i}{\partial x_j}\frac{\partial u_j}{\partial x_i} + \frac{\partial}{\partial x_j}\!\left(u_i\frac{\partial u_j}{\partial x_i} - u_j \frac{\partial u_i}{\partial x_i}\right),$$

whose last term is a divergence. Combining the two produces $2S_{ij}S_{ij} - \omega^2 = 2\,\partial^2(u_iu_j)/\partial x_i\partial x_j$, which is the statement above.

Two things to notice. Incompressibility is used, so the identity has a hypothesis. And what is being discarded is a second derivative of a product of velocities — a quantity that is not small, does not have a sign, and integrates to zero only because of what happens at the boundary. Every disagreement in this essay is that quantity.

The fields #

The test needs a field that is exactly incompressible and exactly differentiable, because the identity uses incompressibility twice and a stencil’s truncation error would show up as a disagreement.

Each field is a finite sum of Fourier modes of a stream function with stated amplitudes and phases, so the velocity is the analytic curl of an analytic function and every derivative is written down rather than differenced. The divergence is at round-off — between $10^{-16}$ and $10^{-15}$ — which is what the identity’s hypothesis requires and is the first thing to check.

The fields are not solutions of anything. They are synthetic, with a power-law spectrum across wavenumbers one to eight, and that is deliberate: the identity is about any solenoidal field whatever, and using a solution would invite the reading that it is a property of the Navier–Stokes equations rather than of the algebra.

The averages are equal, and equal is not a figure of speech #

Between seven parts in $10^{16}$ and one in $10^{14}$. That is round-off in a sum of nine thousand terms, and it is the numerical signature of an identity rather than of an approximation.

Three resolutions, from forty-eight cells a side to a hundred and ninety-two, with the two means agreeing at all of them and neither drifting. A genuine imbalance would survive refinement and a quadrature error would fall; neither happens, because the grid is only being asked to add up a smooth periodic function whose integral it already gets right.

And point by point they are two different fields #

Between $-0.18$ and $+0.24$ across six fields. Not “different in detail” — uncorrelated. Knowing $\varepsilon_S$ at a point says essentially nothing about $\varepsilon_\omega$ there.

And it is not a small disagreement between two large similar things. The root-mean-square difference is 150 to 185 per cent of the mean: at a typical point the two forms differ by more than the average dissipation itself. The cancellation that makes their averages equal is a cancellation of large numbers, and it is exact only because the thing cancelling is a divergence.

Drawn out, the disagreement is not subtle. The strain form is bright where the flow is being sheared and stretched; the enstrophy form is bright where it is rotating. Those are, near enough, the two things a flow is made of, and a picture of one is not a smoothed version of the other.

Along a single cut the two rise and fall in different places, cross one another repeatedly and have different numbers of maxima. Averaged along that line they are not equal either — only the average over the whole box is — so even a one-dimensional measurement of “the dissipation” is a measurement of which form was used.

The two maxima are 0.46 apart on a box of side $2\pi$, which is a seventh of the domain.

They are not even the same kind of distribution. The enstrophy form is the more intermittent: it spends more of its area near zero and reaches higher extremes, while the strain form is flatter. Two quantities with the same mean, no correlation and different distributions have nothing in common but a number.

What the difference field looks like #

It is worth asking what the discarded divergence actually is, since it is the whole content of the disagreement.

$2\,\partial^2(u_iu_j)/\partial x_i\partial x_j$ is, up to a factor, the source term in the pressure’s Poisson equation. So the difference between the two dissipation forms is the pressure source — the same quantity that decides where the pressure is high and low, and which this collection’s account of the maximum principle is built on.

That is a satisfying connection and a discouraging one. Satisfying, because it explains the structure of the two pictures: the strain form is large where the pressure source is negative, the enstrophy form where it is positive, and the pressure source changes sign exactly at the boundary between strained and rotating regions. Discouraging, because the pressure source is not small, is not confined, and has no sign — so the difference between the two forms has none of the properties that would let it be neglected.

It also means the two forms carry a physical distinction rather than an arithmetical one. Choosing the enstrophy form is choosing to attribute dissipation to rotation; choosing the strain form is choosing to attribute it to stretching. Both attributions add up to the same total and neither is the truth about any point.

What that costs #

The question “where is the dissipation?” is asked constantly — of a boundary layer, of a vortex, of a turbulent field — and it does not have a well-posed answer. Three places where that matters.

Conditional averages. Any statement of the form “the dissipation in regions of high vorticity is $N$ times its mean” is a conditional average of one of these fields on a property of the other, and the two forms give different answers by construction — the enstrophy form correlates with the vorticity perfectly, because it is the vorticity squared, while the strain form does not. That is a tautology dressed as a measurement, and it is common.

Intermittency. Much of the modern picture of turbulence rests on the distribution of the local dissipation: how often it is far above its mean, how that varies with the size of the region it is averaged over, and what the exponents of those moments are. Every one of those statistics depends on which form was used, and the two differ in exactly the tail the argument is about. The site’s own account of the resulting exponents is the exponents that stop being thirds, and the ambiguity here sits underneath all of it.

Subgrid models. A large-eddy simulation needs the dissipation at the smallest resolved scale, and models are built and calibrated against one form or the other. Two models fitted to the same data through different forms are fitted to different data.

And vortex-core arguments. “The dissipation is concentrated in the vortex tubes” is an enstrophy-form statement, and it is not true of the strain form. In fact for a straight vortex tube the strain and the rotation are equal in magnitude and the two forms agree — it is in the stretching and folding between the tubes that they part company, which is exactly the region the same arguments are usually about.

The same disease, three times over #

This is not an isolated awkwardness. It is a specific instance of something that happens whenever a quantity defined by an integral is written as a density, and this collection has three other cases of it worth naming together.

The pressure’s split into a “Bernoulli” part and the rest. The pressure satisfies a Poisson equation whose source is a velocity-gradient invariant, and any particular decomposition of the solution into contributions is a choice of how to distribute a harmonic function. The lowest pressure is on the body is about a consequence of that equation which is well posed, precisely because it is a statement about extrema rather than about local contributions.

Production and transport in the Reynolds-stress budget. The terms in that budget are individually convention-dependent for exactly this reason — a divergence can be moved from one to another — and which grouping is called “production” is a choice made differently in different textbooks. The ladder that never closes is about a different problem with the same equations.

And the local energy flux across a scale. The mean flux through the inertial range is well defined and its local version is not, which is a flux that runs both ways — where the instantaneous value is negative nine per cent of the time and the mean is not.

The pattern in all four: an exact statement about an integral, a convention about its density, and a literature that quotes the density.

A number for how much cancelling is going on #

One quantity is worth putting on the record, because it is the size of the thing being hidden.

The mean dissipation on these fields is about 3.2 in the units used. The root-mean-square of the difference between the two forms is about 5.4 — one and seven tenths times the mean. So at a typical point the divergence term is nearly twice the quantity everybody quotes, and it integrates to $10^{-15}$ of it.

That ratio is what “exact cancellation” costs to arrange, and it explains why the identity is so easy to state and so easy to over-read. A quantity that averages to nothing while being large everywhere is the same object as a flux that runs backwards nine per cent of the time and as an instantaneous transfer whose mean is the dissipation: the mean is a real statement and the field it is a mean of is not what the mean describes.

There are not two forms, there is a line of them #

The essay has been arguing about a choice between two expressions, and the situation is worse than a choice: any multiple of the discarded divergence can be added to either one and the mean is unchanged. So what exists is not a pair but a one-parameter family of local densities, all sharing a single integral, of which the strain form and the enstrophy form are two points.

A third point on that line is already in wide use and has a name. The pseudo-dissipation is the viscosity times the sum of the squares of all nine velocity gradients, and the algebra above places it exactly:

$$\varepsilon' = \nu\frac{\partial u_i}{\partial x_j}\frac{\partial u_i}{\partial x_j} = \nu\!\left(S_{ij}S_{ij} + \tfrac12\omega^2\right) = \tfrac12\left(\varepsilon_S + \varepsilon_\omega\right).$$

It is the arithmetic mean of the other two — not a compromise between them, just the midpoint of the line. It is also the form most direct numerical simulations report, because it costs nine derivatives and no tensor assembly, while experiments overwhelmingly report a fourth thing again: an isotropic surrogate built from the single derivative a hot wire can reach.

The trap that follows is worth stating, because it looks like a validation and is not. Anyone checking the pseudo-dissipation against the strain form on the same field will find them strongly correlated — with two uncorrelated fields of equal variance, a midpoint correlates with each of them at about 0.71 by construction, whatever the physics. That agreement is arithmetic. It is the same kind of tautology as conditioning the enstrophy form on the vorticity, and it is available in any paper that compares a density against half of its own definition.

Which one is right #

Neither, and both, and the question is the wrong one.

There is no local quantity called “the dissipation” that the two forms are estimates of. Dissipation is defined by an energy budget, and an energy budget is a statement about a region: the rate at which kinetic energy leaves the resolved motion of a volume equals a surface term plus an interior term, and the split between them is not unique because a divergence can be moved from one to the other. What the two forms differ by is that divergence, and moving it is exactly the choice being made.

For a volume with no flux through its boundary the surface term vanishes and the two agree. For any smaller region — a point, a line, a cube inside a flow — it does not, and neither form has a claim to be the true one.

The correct statement is therefore the awkward one: the dissipation is an integral quantity, and its local density is a convention. The same is true of the pressure’s decomposition, of the Reynolds stress’s split between production and transport, and of most quantities in this subject that are written as a density and used as a total.

Why the equality is nevertheless worth having #

It would be wrong to leave this as a complaint. The identity is doing real work, and there are three places where it is exactly what is wanted.

Checking a simulation. Computing both forms over a periodic box and comparing their means is a sharp test of a code’s spatial discretisation: they agree identically for a solenoidal field, so any disagreement is the code’s divergence error made visible. It costs one extra pass over the field and it catches a class of bug that conservation of mass alone does not.

Estimating a dissipation from a measurement. A hot wire gives one velocity derivative, and both forms can be estimated from it under isotropy — but the two estimates use different isotropic relations, so comparing them tests the isotropy assumption rather than the instrument. Two estimates that should agree and do not are evidence about the flow.

And the anomaly argument. The claim that the dissipation stays finite as the viscosity goes to zero — the limit that is not the value — is a claim about the average, and the average is the thing the two forms share. That whole argument is untouched by anything in this essay, which is worth saying explicitly because the argument is the most important one in the subject.

What is not claimed #

The fields are synthetic and two-dimensional. They are exactly solenoidal and exactly differentiable, which is what the identity needs, and they are not turbulence. In three dimensions the identity carries an extra term involving the vortex stretching and the conclusion is the same; nothing here computes it.

“Uncorrelated” is measured on these fields. A correlation near zero on six synthetic fields with a $k^{-3}$ stream-function spectrum is evidence that the two forms are unrelated pointwise, and it is not a claim that the correlation is zero in real turbulence. Measured values there are small and positive.

The identity is exact only for homogeneous fields. In a boundary layer, where the flow is not homogeneous in the wall-normal direction, the divergence does not integrate to zero over a slab and the two forms give different profiles of dissipation against height — a well-known and much-ignored discrepancy near a wall.

The correlation is between two fields, not two measurements. Both forms are computed exactly from the same analytic field, so the zero correlation is a property of the mathematics rather than of noise. A pair of experimental estimates would correlate somewhat, because both would be contaminated by the same measurement errors — which is a reason to be careful about reading agreement between two estimates as evidence that either is right, and is the same hazard the instrument in the answer is about.

And nothing here says which form a measurement should use. Hot-wire anemometry reaches one velocity derivative and estimates both forms by assuming isotropy, which is a third assumption on top of these two.