How much more than the least

The flow with the least energy in it makes the case that what separates the real flow from the infinitely many pictures that look like it is a number, and that the number is an energy. Kelvin proved it in 1849: among all the fields that conserve mass and match the normal velocity on every boundary, the irrotational one carries strictly the least kinetic energy.

That essay stops where most accounts stop, at least. This one asks the next question, which turns out to have a completely exact answer and a thoroughly discouraging consequence.

The theorem in one line, and the line is an integral #

Take any field $\mathbf{u}$ that conserves mass and has the right normal velocity on the boundary. Write it as the potential flow plus whatever is left over,

$$\mathbf{u} = \nabla\phi + \mathbf{v}.$$

Both $\mathbf{u}$ and $\nabla\phi$ are solenoidal and both have the same normal component on the boundary, so the remainder $\mathbf{v}$ is solenoidal and has no normal component there. That is the whole hypothesis, and everything follows from it in one step:

$$\int \nabla\phi\cdot\mathbf{v}\,dA = \int \nabla\cdot(\phi\,\mathbf{v})\,dA = \oint \phi\,\mathbf{v}\cdot\mathbf{n}\,ds = 0.$$

The middle equality uses $\nabla\cdot\mathbf{v} = 0$; the last uses $\mathbf{v}\cdot\mathbf{n} = 0$ on the boundary. The cross term is not small. It is zero.

The domain here is an annulus between $r = 1$ and $r = 4$, carrying the flow past a cylinder cut off at four radii, and the perturbations are built as the curl of a stream function that vanishes on both circles — which makes them solenoidal to machine precision rather than to the accuracy of a projection, and makes their normal component vanish by construction rather than by assertion. Six of them, of angular orders one through eight, give inner products between three and eight parts in $10^{17}$ of the geometric mean of the two energies.

Which means the excess is the energy of the mistake #

With the cross term gone, expanding the square gives

$$E(\mathbf{u}) = E(\nabla\phi) + E(\mathbf{v}),$$

so the extra energy of any admissible field is exactly the energy of its own difference from the potential flow. That is a far stronger statement than “the potential flow is least”. It says by precisely how much anything else misses, and it says it in terms of the thing that is wrong with it.

Measured across amplitudes from 0.5 to 32 the two agree to two parts in $10^{13}$ at worst. It is an identity, and the residual is the arithmetic.

One numerical detail is worth a sentence, because it is the kind of thing that gets reported as physics. The angular integral here is over a periodic function, where uniform sampling is spectrally accurate and therefore exact to round-off however coarse it is; the radial one is not. With the midpoint rule in the radius, the cross term comes out at seven parts in a million, and seven parts in a million is a perfectly respectable-looking number to publish as the accuracy of a theorem.

It falls by exactly four for every doubling of the radial resolution — 256.1 over a sixteen-fold refinement — which is the signature of a second-order quadrature and not of anything about fluids. Four-point Gauss quadrature in each radial cell takes the same integral to round-off at every resolution, which is why the figure above quotes $10^{-17}$ rather than $10^{-6}$. The site’s standing warning is that a smooth picture proves nothing; the same is true of a smooth number.

And that is why energy cannot find a wrong flow #

Here is the discouraging part. The excess is the energy of the error, and energy is quadratic in velocity. So the relative energy error is exactly the square of the relative velocity error, with no constant in front:

$$\frac{E(\mathbf{u}) - E(\nabla\phi)}{E(\nabla\phi)} = \left(\frac{\|\mathbf{v}\|}{\|\nabla\phi\|}\right)^2.$$

Read off the figure: a field whose velocity is wrong by 3.9 per cent everywhere has an energy wrong by 0.15 per cent. One wrong by 15.4 per cent has an energy wrong by 2.4. Even a field wrong by 31 per cent — which is not a subtle error, which is a picture a reader would call visibly different — carries an energy only 9.5 per cent too high.

The measured ratio of the energy error to the square of the velocity error is 1.000000000000 across the whole sweep. This is an identity and not a coincidence, and its consequence is worth stating plainly: the quantity that identifies the true flow is the least sensitive available measure of how far from it anything else is. Any variational principle has this property — it is the definition of a minimum, and it is why the elliptic wing’s optimum is flat in exactly the same way — but it is easy to read the theorem the other way round and conclude that a flow with nearly the right energy is nearly the right flow. It is not. Halving the energy error only reduces the velocity error by thirty per cent.

The site has a companion warning about smooth streamlines. This is the same warning about a number.

What happens when the hypothesis is dropped #

The proof rests on one boundary term and it is worth watching it fail, because the failure is not subtle and the hypothesis is not decorative.

Take a field that satisfies everything except the last condition: solenoidal, finite, perfectly smooth — and with a normal component on the boundary. A uniform radial flow of strength 0.1 will do. It carries fluid in through one circle and out through the other, which is exactly what an admissible perturbation may not do, and its inner product with the potential flow is not $10^{-17}$: it is a number of ordinary size, comparable with the geometric mean of the two energies.

With that term present the energy does not split, the excess is no longer the energy of the difference, and nothing prevents the sum from being lower than the potential flow’s. Kelvin’s theorem has not been broken; it was never being applied. The comparison class is “fields matching the boundary data”, and a field that changes the boundary data is a comparison with a different problem.

This is the same distinction how many things a flow must be told is about. The boundary conditions are not a technicality attached to a solution; they are half of what the solution is, and a theorem about a comparison class is a theorem about which fields were allowed into the room.

The site’s gate for this module refuses exactly that case: it is handed the inadmissible field and required to notice, because a check that cannot tell an admissible perturbation from an inadmissible one is a check that proves nothing about the one it passes.

The perturbation is free to be anything #

It is worth looking at what an admissible perturbation actually is, because the theorem’s hypothesis is weaker than it sounds and the class it allows is enormous.

The only requirements are that it conserve mass and that it not cross either boundary. Inside, it may have any structure at all: vortices, jets, shear layers, anything. The theorem does not restrict its shape, only its flux, and it still concludes that adding it costs energy.

That is what makes the result useful as a proof and useless as a test. It covers everything, so it distinguishes nothing.

What a bound is actually worth #

The quadratic law cuts both ways, and the useful direction is the one variational methods live on.

Suppose the energy of a flow nobody can solve for is wanted, in a domain of awkward shape. Draw any field that conserves mass and matches the boundary — a hand sketch, a coarse interpolation, a superposition of the elementary flows that add up to something roughly right. Whatever is drawn, its energy is an upper bound on the true one, and the theorem says the bound is in error by the energy of the mistake in it.

Because that error is quadratic, the bound is far better than the sketch. A field wrong by ten per cent in velocity gives an energy one per cent high; wrong by three per cent gives one part in a thousand. Nobody can draw a velocity field to a part in a thousand, and almost anybody can draw one to three per cent, so the bound is cheap and tight — which is why the Rayleigh–Ritz method works at all, and why the same reasoning gives useful answers for the cheapest shape the walls allow.

Run it the other way and the same arithmetic is the disappointment above. An energy known to one per cent locates the flow to ten per cent. An energy known to a part in a million locates it to a part in a thousand. Half the significant figures are gone, and no amount of care in the energy measurement changes the exchange rate, because the exchange rate is the theorem.

That trade — a stationary quantity known well, its location known to the square root of that — is not peculiar to this theorem. It is what a minimum is.

The bound from the other side #

The complaint that half the significant figures are lost is a complaint about a one-sided estimate, and the repair is to get a bound from the other direction — which is available, from a companion theorem, and turns a bound into a bracket.

Kelvin’s principle is a statement about guessing a velocity field: any solenoidal field matching the prescribed flux gives an energy that is too high. Its companion is a statement about guessing a potential: on a boundary where the potential itself is prescribed rather than the flux, any single-valued function taking the right values there produces a gradient whose energy is too low. Same domain, same true flow, two crude guesses, and the answer is trapped between them.

That changes what the quadratic law costs. Each bound on its own is insensitive in the way this essay has been complaining about; the gap between them is not — it is a computed number, it shrinks as the guesses improve, and it is an error bar rather than an appeal to plausibility.

The pairing is not peculiar to fluids. It is how the classical bounds on a capacitance, a permeability and an effective conductivity are all obtained, and in each case it is these two theorems wearing different names.

Three orthogonal pieces, and one of them is invisible #

The annulus was chosen with a hole in it on purpose, because a hole is where the theorem’s careful statement starts to matter.

Add a circulation $\Gamma$ round the inner circle. Its velocity is $\Gamma/2\pi r$ in the azimuthal direction, so its normal component is zero on both boundaries: it changes nothing whatever about the boundary data. It is therefore admissible, by the theorem’s own definition — and it carries energy.

Integrating all three together as one field gives a total of 25.480592004605490. Adding the three energies separately gives 25.480592004605487. Every pairing of them has an inner product at round-off: $-3.8\times10^{-18}$ between the stream and the circulation, $-1.4\times10^{-17}$ between the stream and the remainder, $-7.1\times10^{-21}$ between the circulation and the remainder. The split into three pieces is not a decomposition somebody chose; it is a property of the fields, and it is the same orthogonality that Helmholtz’s decomposition rests on.

The circulation’s own piece is $\Gamma^2\ln(R/a)/4\pi$, which quadrature recovers to sixteen figures.

The number the boundary does not contain #

So in a doubly connected domain the boundary data does not determine the flow. One number per hole is left free, it is the circulation, and it is invisible in every boundary measurement.

That is not a failure of the theorem. Kelvin’s statement is that the minimiser is unique among fields of a given circulation, and the annulus is where the qualification earns its keep. Drop it and the minimiser is the circulation-free flow, and every lifting flow in this collection is a non-minimiser — what actually holds a wing up is precisely a flow that carries more energy than the boundary conditions require.

What the free number costs is a logarithm.

At unit circulation the energy is 0.055 out to two radii, 0.183 out to ten, 0.366 out to a hundred, 2.932 out to $10^{16}$. It does not converge. It grows by exactly $\ln 10/4\pi = 0.1832338997$ per decade, for ever, which is the energy a vortex cannot have seen from the outside: a quantity whose integral diverges logarithmically is not merely large, it does not exist.

Plotted this way the distinction is visible. A quantity that converges bends over; one that grows as a power curves upward; one that grows as a logarithm is a straight line with the same slope over the sixteenth decade as over the first. Over sixteen decades the energy grows by a factor of 26.6, and nothing about the arithmetic suggests it is going to stop.

What the theorem is good for, then #

Three things, and none of them is checking a picture.

It proves uniqueness. Two irrotational flows with the same boundary data and the same circulations differ by a field with zero boundary flux and zero energy, and a field of zero energy is zero. That is the shortest uniqueness proof in the subject and it is a corollary of the orthogonality above.

It bounds a quantity nobody can compute. Any admissible field that can be written down — however crude, however hand-drawn — gives an upper bound on the true energy, and the bound is as good as the field is close, squared. Because the error is quadratic, a crude field gives a surprisingly tight bound, which is the same fact as the useless test read the other way round. That is what variational methods live on.

And it says what a circulation costs. The energy of a lifting flow above the non-lifting one is a computable, positive number — and in an unbounded domain it is not a number at all. Every finite statement about the energy of a two-dimensional lifting flow is a statement about where the counting stopped, which is the same disease as the momentum with no value and has the same cure: quote an impulse, or quote a domain.

Why the theorem is about rotation, said once more #

There is a way of reading all of this that makes the arithmetic feel inevitable, and it is worth having.

The remainder $\mathbf{v}$ is solenoidal with no flux through the boundary, so in two dimensions it has a stream function that is constant on each boundary component. Every such field is, in the sense the site uses the word, rotational: if it were irrotational as well, with the same zero flux, it would be a harmonic function with constant boundary values and therefore constant, and its velocity would be zero. So the class of admissible perturbations is the class of rotational corrections, and Kelvin’s theorem says that vorticity costs energy.

That is a claim about what spin is worth rather than about optimisation, and it is why the theorem holds in the annulus with its awkward extra constant: the circulation is the one rotational-looking thing that is not rotational — its vorticity is zero everywhere in the fluid, and what it has instead is a multivalued potential. The hole is where the distinction between “curl-free” and “the gradient of something” stops being a distinction without a difference, and the free constant is the size of the gap between them.

What is not claimed #

The domain is two-dimensional and annular. The orthogonality argument is dimension-free and the figures are not: the numbers quoted are for one geometry, and the three-dimensional statement about the free constants is about the number of independent circuits rather than the number of holes.

The perturbations are a family, not the class. Six angular orders with one radial shape are enough to demonstrate an identity that holds for every admissible field, and they are not a survey of what admissible fields can look like. Nothing here bounds how large $E(\mathbf{v})$ can be.

The circulation figures are for a single hole. A domain with $n$ holes leaves $n$ free circulations and their energies do not simply add: the cross terms between two circulations round two different holes are not zero, because neither vanishes on the other’s boundary in the way that made the split above exact. That case is the constant a hole leaves behind, and it is a different computation.

And the energy of the true flow is itself cut off. The 25.03 above is the energy in the annulus, not the energy of the flow past a cylinder, which in two dimensions has the same difficulty as the circulation does. Every energy in this essay is a statement about a domain.