The flow with the least energy in it

Here are two pictures of a flow round a cylinder. Both fields conserve mass everywhere. Neither has any fluid crossing the body’s surface or the outer circle. Both were drawn by the same routine, with the same streamline tracker, at the same resolution.

One of them is the flow, and the other is not a flow at all.

This is the site’s founding worry in its purest form. A wrong flow field is beautiful: streamlines are smooth whatever produced them, and the eye has no instrument for the difference between a solution of the equations and a function that satisfies none of them. What is needed is a number — one produced by an integral that shares no arithmetic with the drawing — and Kelvin supplied it in 1849.

The theorem #

Among all divergence-free velocity fields with the same normal component on every boundary, the irrotational one has strictly the least kinetic energy.

That is the whole statement, and the useful half of it is the strictly. It is not that the irrotational field is one of the cheap ones. It is that every other admissible field costs more, and the amount more is computable in advance.

Write the candidate as $\mathbf{u} = \nabla\phi + \mathbf{u}'$, with $\nabla\phi$ the irrotational field and $\mathbf{u}'$ whatever was added. For $\mathbf{u}$ to be admissible, $\mathbf{u}'$ must be divergence-free and must have no normal component anywhere on the boundary — it can churn the interior as violently as it likes and must not move anything in or out. Then

$$E = \tfrac12\rho\!\int |\nabla\phi|^2 + \rho\!\int \nabla\phi\cdot\mathbf{u}' + \tfrac12\rho\!\int |\mathbf{u}'|^2 .$$

Three terms, and the middle one is the theorem. It is

$$\rho\!\int \nabla\phi\cdot\mathbf{u}'\;=\;\rho\oint \phi\,(\mathbf{u}'\cdot\mathbf{n})\,dS\;=\;0,$$

because $\mathbf{u}'$ has no normal component on the boundary and there is nothing left inside — the divergence theorem applied to $\phi\mathbf{u}'$, and $\nabla\cdot\mathbf{u}' = 0$.

So the cross term vanishes, and the energy of any admissible field is the irrotational energy plus a number that is never negative.

What that buys, arithmetically #

The theorem is stated as a minimum principle and is used here as an instrument, because the same three integrals that state it also test whether a candidate is admissible at all.

Given a field, compute its energy, compute the energy of its departure from the irrotational solution, and compute the energy of the whole. If the first is not the sum of the other two, the departure is putting fluid through a boundary — which is to say the candidate is not one of the fields the theorem is about, however innocent the picture looks.

The absence of a linear term is what makes zero a minimum rather than merely a point on a curve, and it is the thing to watch. A field that is subtly inadmissible — one that leaks a little — produces an excess with a term proportional to the amplitude, and at small amplitudes that term dominates. So the curve tips, the minimum moves off zero, and the theorem appears to fail.

The trap in the test #

The check as first written passed a field that leaks.

The perturbation used for these figures is built from a streamfunction $\psi' = \varepsilon\, g(r)\sin m\theta$ with $g$ vanishing on both circles. Breaking it is easy: shift $g$ so that $g(R) \neq 0$, and fluid crosses the outer boundary. The cross term should then be non-zero and the check should refuse the field.

With $m = 2$ it does not. The base flow’s potential on the outer circle goes as $\cos\theta$, the leak goes as $\cos 2\theta$, and $\oint\cos\theta\cos 2\theta\,d\theta$ is exactly zero. The integral that is supposed to detect the leak is killed by orthogonality, and reports a perfectly admissible field.

This is the same shape as the degenerate pair that made the reciprocal theorem look sixty per cent wrong in the essay that priced it: a test on which both sides vanish for a reason unrelated to the claim is not a test. The check uses $m = 1$ now, whose leak the base potential can see, and it measures the flux across the boundary separately so that a blind cross term cannot be mistaken for a clean one.

The uniqueness proof is the same proof #

The theorem is usually met as a statement about minima, and the more useful reading is that it settles whether the problem has one answer at all.

Suppose two irrotational fields satisfy the same boundary data. Their difference is irrotational, divergence-free, and has no normal component anywhere on the boundary — so it is admissible in exactly the sense above, and the cross-term identity applied to it against itself gives its energy as zero. A field with zero kinetic energy over a region is zero throughout the region. The two solutions are the same solution.

Three lines, and they settle well-posedness for the whole of ideal flow. Specify the normal velocity on every boundary, require irrotationality, and there is exactly one answer.

Which raises the obvious question, and the answer to it is the reason this rung exists.

The exception, which is where the subject actually lives #

The proof used the divergence theorem on $\phi\mathbf{u}'$, and that step needs $\phi$ to be a single-valued function. In a multiply connected region it is not.

Draw a loop round the cylinder. The potential can increase steadily along it and come back to a different value, and nothing about the flow is wrong: the velocity, which is the gradient, comes back to itself. The jump per circuit is the circulation. So in a region with a hole in it, the boundary data does not determine the flow — it determines it up to one number per hole, and the number is a circulation the equations have no opinion about.

That is what the panel method’s missing row is, and it is the opening the Kutta condition is invented to close. It is also the first instance of a pattern this collection returns to repeatedly: the equations and the boundary conditions are exact, and they are not enough.

What the minimum is worth, in joules #

The theorem is not only a uniqueness device. The minimum is a real number and it is the one that prices acceleration.

For a cylinder of radius $a$ moving at speed $U$ through fluid at rest, the energy of the whole exterior is

$$E = \tfrac12\rho\pi a^2 U^2,$$

which is exactly half the mass of fluid the cylinder displaces, times the square of its speed. That is the added mass written as an energy, and Kelvin’s theorem says it is the least energy any admissible motion of the fluid could have had. A body pushing through an ideal fluid moves the fluid in the cheapest way the boundary allows, and there is no arrangement of the same displacement that costs less.

The quadrature that produces the figures above is calibrated against that closed form and agrees with it to six parts in a million, which is the only reason any of the other numbers on this page are worth printing.

And the minimum depends on the shape, which is the practical content of the theorem. A body that is long in the direction it is going gives the fluid a gentler path to take, so less of it has to move, so the least energy is smaller — and the ratio of added mass to displaced mass, which is exactly one for a circle, falls steadily as the body is drawn out.

What happens if the missing number is left to the theorem #

The exception above says the boundary data leaves one number free per hole in the region. There is an obvious thing to try: the theorem is a minimum principle, so let it pick.

Do the arithmetic and the answer arrives immediately, because the same cross-term identity applies again. Write the field as the zero-circulation solution plus a pure circulatory flow. The circulatory part is divergence-free, it is tangential to both boundaries, and so the cross term between it and the rest is zero by exactly the calculation above. The energy of the family is therefore

$$E(\Gamma) = E_0 + c\,\Gamma^2,$$

a parabola again, with its minimum at $\Gamma = 0$.

The minimum-energy principle chooses the flow with no lift. Not approximately, and not as a matter of the shape or the boundary or the outer radius: circulation is a strictly positive addition to the energy, so the cheapest member of the family is always the one that carries none, and a variational argument allowed to select the free number will select a wing that does not fly.

That is worth sitting with, because it is the sharpest available statement of what this collection keeps calling the missing number. The equations are exact. The boundary conditions are exact. A perfectly respectable energy principle is available and it gives a definite answer. And the answer is wrong about every aeroplane, because the flow a wing is actually in is not the cheapest member of its family — it is whichever member the history put it in, and history is not a minimiser of anything. Kelvin’s other theorem is what carries that history, and the starting vortex is the receipt for the energy the cheap solution did not have to pay.

Bounding an answer without solving for it #

There is a second use of the theorem that is worth knowing because it was the only practical route to an added mass for a century.

Kelvin’s principle turns any admissible trial field into an upper bound. Draw any divergence-free field with the right normal velocity on the boundaries — sketched, guessed, assembled out of two simpler flows — compute its energy, and the true minimum is no larger. No solving is involved and the bound is rigorous.

Its complement runs the other way. Take a trial potential, which is irrotational by construction and therefore need not be admissible in Kelvin’s sense, and let it satisfy the boundary condition only approximately. The energy expression built from it, arranged so that the boundary shortfall is subtracted rather than ignored, is a lower bound. Neither trial has to be good; each merely has to be honest about which half of the problem it satisfies.

Use both and the answer is bracketed. That is how the added masses of ship hulls, airship envelopes and odd-shaped bodies were tabulated before anybody could solve a Neumann problem numerically — squeeze the two bounds together by improving the trial fields until the gap is smaller than the precision the answer is wanted to. Two variational principles that disagree by a known amount are a better instrument than one exact method that cannot be checked, which is the same preference the rest of this page argues for in a different form.

The two bounds also make a useful check on any numerical answer, which is how they are most often used now. A solver’s added mass should sit between them; if it does not, the error is in the solver rather than in the trial fields, since neither bound depends on solving anything. That is a rare thing in this subject — an independent test of a computation that costs less than the computation.

What the theorem does not say #

Three things, and the third is the one that matters most for the rest of this collection.

It says nothing about which flow is stable. A minimum of energy at fixed boundary data is not a minimum of anything the dynamics descends towards. Ideal flow has no mechanism for shedding energy at all — that requires viscosity — so the fluid does not seek this state, it merely occupies it if it started there.

It says nothing about a rotational flow being wrong. The comparison is between fields with the same boundary flux, and a rotational field is admissible in that comparison; it simply costs more. A real flow with vorticity in it is an exact solution of Euler’s equations and carries more energy than the potential flow with the same boundary data, which is a statement about energy and not about legitimacy.

And it is a theorem about a fixed instant. Nothing here evolves. The boundary data is given, the minimum is found, and the question of how the fluid came to be in that state — which is the question Kelvin’s other theorem answers — is not asked.

Why an integral and not a look #

The habit this rung is meant to install is not the theorem. It is the shape of the test.

Every figure on this site draws a field, and a field is a thing that can be wrong in ways no drawing shows. The instruments available are integrals: a divergence measured over a grid, a normal velocity measured on a surface, a force computed twice by routes that share no arithmetic, and — here — an energy computed three times, where two of the answers must add up to the third.

An assertion that has never rejected anything proves nothing, which is why the check that produces these numbers is fed a leaking field and required to refuse it, and why the refusal itself had to be repaired once it turned out to be blind.

What the picture cannot show #

The figures on this page are honest about what they draw and cannot be honest about what they leave out.

The outer boundary is a circle six radii out, and it is not infinity. Every energy quoted here is an energy inside that circle, with the tail beyond it added analytically where it matters. For the cylinder that tail is a known fraction and is small; for a field whose disturbance decays more slowly it would not be, and a comparison of two such fields inside a finite region can be dominated by where the region was cut.

The eddy is one shape out of infinitely many. Nothing here samples the space of admissible fields; it walks one line through it. The theorem covers the whole space and the measurement covers a line, and the line was chosen because it is easy to build rather than because it is representative.

And the energy is not drawn. It is printed. There is no way to shade a picture so that a reader can see an integral, and every attempt makes a figure whose bright regions are where the gradients are — which is a picture of the velocity, drawn a second time.

Who found it, and when #

Kelvin gave the minimum-energy theorem in 1849, three years before Helmholtz’s vortex laws and twenty before Kirchhoff’s free streamlines. It is one of a family of variational statements that arrived across physics in the same few decades — Dirichlet’s principle for the Laplace equation, Castigliano’s theorems in elasticity, Thomson and Tait’s whole programme — and they are all the same statement about a quadratic functional with a linear constraint.

The surprising connection is with electricity, and it is exact rather than analogical. The current distribution in a conductor is the one that dissipates least power for a given total current, which is Thomson’s principle and is the same algebra with resistance where the density is. The two problems share their mathematics so completely that ideal flows were computed on sheets of conducting paper for thirty years, by drawing the body’s outline in silver paint and measuring the potential with a probe. The technique is obsolete and the equivalence is not: it turns up again, in a stranger form, when an ideal flow is drawn by a fluid that is nothing but viscosity.

Where the ladder goes next #

Beside this rung sits the same cross term doing the opposite job: what adds when two flows are added, and what does not, where the term that vanishes here is the whole of the difference between a pressure and a sum of pressures.

Above it is the ambiguity this theorem cannot remove — the momentum of an unbounded ideal flow, which is finite, computable, and different depending on how the sum was taken.

Below it are the exact theory and how many things a flow must be told, which is where the boundary data this whole argument is conditional on comes from.