The one thing that does not add up

Flows add up. Laplace’s equation is linear, so a solution plus a solution is a solution, and almost every classical result in this collection is built that way: a stream plus a doublet is a cylinder, a source plus a sink in a stream is a Rankine body, an aerofoil is a distribution of vortices, an image system is a body added to a flow.

The habit that produces is: the flows add, so the answers add. This rung is about how badly that fails and about exactly how much.

The identity #

Write $C_p = 1 - q^2/U^2$, which is the definition this collection uses. For two flows $A$ and $B$,

$$q^2 = |\mathbf{u}_A + \mathbf{u}_B|^2 = |\mathbf{u}_A|^2 + 2\,\mathbf{u}_A\cdot\mathbf{u}_B + |\mathbf{u}_B|^2,$$

so

$$C_p(A+B) = C_p(A) + C_p(B) - 1 - \frac{2\,\mathbf{u}_A\cdot\mathbf{u}_B}{U^2}.$$

Two terms of defect, and they are different in kind.

The constant $-1$ is bookkeeping. Each $C_p$ carries its own reference to the free stream, so adding two of them double-counts the reference. It is present between any two flows anywhere, including two that are not interacting at all.

The cross term is the interaction. It is $-2\mathbf{u}_A\cdot\mathbf{u}_B/U^2$, it can be of either sign, and it is unbounded — two flows that are both fast at the same place produce a large one.

The identity is exact and is checked here to $2\times10^{-16}$ at ten points, which is the last digit a double carries.

The check that had to be repaired #

The check on that identity was, in its first version, a test of nothing.

It required the departure between the measured defect and the predicted one to be tiny, which it is, and it also required the defect itself not to be tiny — because an identity verified only at points where both sides are near zero has not been verified. That second requirement was written against the defect, and the defect contains the constant $-1$.

So a set of points chosen far from both flows, where the two barely interact and the cross term is negligible, still reports a defect of one, sails past the guard, and demonstrates nothing. The check measures the cross term now.

A test whose non-triviality guard can be satisfied by a constant is not guarding anything, and that is the second time the same shape of mistake has been found in this corner of the site — the first was an orthogonality that killed a leak the check was supposed to detect.

The sharpest case #

Take a cylinder in a uniform stream. It has no lift: the pressure distribution is symmetric top and bottom, and the surface integral is zero to $6\times10^{-16}$.

Take a point vortex alone in an infinite fluid. It has no lift either — nothing acts on it and it does not move — and the same surface integral over the same circle gives $-5\times10^{-17}$.

Add them, and the lift is $\rho U\Gamma$: 3.00000 in the units of the figure, against a Kutta–Joukowski prediction of exactly 3.

No amount of care about the linearity of the equations would have predicted that, because the force was never linear in the field. It is a surface integral of a quantity quadratic in the velocity, and a quadratic form applied to a sum has a cross term whether or not anybody wants one.

What does add #

The left-hand column is worth having explicitly, and worth checking rather than remembering.

Everything linear in the velocity adds exactly: the velocity itself, the streamfunction, the potential, the circulation round a given loop, and the volume flux across a given curve. Those were measured rather than asserted for the figures here — the circulation of the sum comes to $-3.000000$ against $-8.6\times10^{-16}$ and $-3.000000$ for the parts, and the flux agrees to $10^{-16}$.

Everything quadratic does not: the pressure, the force, the moment, the kinetic energy, and the speed — which is not even quadratic, being a square root of one.

How big the cross term gets #

“It does not add” is a qualitative claim, and the size of the failure decides whether anybody has to care.

At the ten sample points the defect ranges from $-0.08$ to $-1.92$. Strip out the constant $-1$ and the interaction term alone runs from $+0.92$ to $-0.92$ — the same size as a whole pressure coefficient, and larger than the difference between a good aerofoil and a bad one at cruise.

Its magnitude is set by $2\,\mathbf{u}_A\cdot\mathbf{u}_B/U^2$, so it is largest where both flows are fast and aligned, and it is zero where they are perpendicular. On a lifting cylinder the two are aligned over the top surface and opposed underneath, which is why the same construction that produces the lift produces the largest cross terms exactly where the lift comes from. The interaction is not an incidental correction to the flow; it is the flow’s whole reason for being interesting.

The cross term is a theorem elsewhere #

The energy’s cross term is the same object that appears three rungs down this ladder wearing a completely different expression.

$$E(A+B) = E(A) + E(B) + \rho\!\int \mathbf{u}_A\cdot\mathbf{u}_B .$$

Here that middle term is the nuisance. In Kelvin’s minimum-energy theorem it is the whole argument: when $A$ is irrotational and $B$ is divergence-free with no normal component on any boundary, the integral becomes $\rho\oint\phi(\mathbf{u}_B\cdot\mathbf{n})\,dS$ and vanishes, which is what makes the irrotational field a minimum and what makes the solution unique.

So the cross term is not always there. It vanishes under a specific hypothesis about the second field’s behaviour on the boundary, and the general case — a vortex added to a stream, say — does not satisfy that hypothesis at all.

Knowing which is which is worth more than either result. The same integral is an obstacle and a theorem according to what the two fields do at the edges.

The other quantity that is not even quadratic #

Speed deserves separating from the list, because it fails differently and worse.

$q = |\mathbf{u}_A + \mathbf{u}_B|$ is not a quadratic function of the components; it is the square root of one, so it is not even expressible as a sum plus a cross term. Adding two speeds gives an answer that is right only when the two flows are parallel, and the error is the whole triangle inequality.

The practical version is that a velocity magnitude cannot be superposed at all, and this is the mistake that survives longest because it is made with vectors that have been quietly converted to numbers. A “5 m/s crossflow added to a 20 m/s stream” gives 20.6, not 25, and gives 15 if it is opposed — and gives something else again at every point of a body where the local direction differs.

Everything downstream of a speed inherits the failure: the dynamic pressure, the Reynolds number formed from a local velocity, the Mach number, and every correlation that takes a speed as its argument.

When pressures may be added after all #

The essay so far reads as a prohibition, and taken literally it would forbid thin-aerofoil theory, which adds pressures constantly and works. The reconciliation is worth having in full, because it supplies the criterion the prohibition is missing.

Split the velocity into the free stream and a disturbance, $\mathbf{u} = U\hat{x} + \mathbf{u}'$, and expand the pressure coefficient:

$$C_p = -\frac{2u'}{U} - \frac{|\mathbf{u}'|^2}{U^2}.$$

The first term is linear in the disturbance. So if two bodies each disturb the same free stream, and their disturbances are added, the leading term of the combined pressure coefficient is the sum of the leading terms of the separate ones. Pressures add to first order in the disturbance, and that is the whole licence under which a thickness distribution, a camber distribution and an incidence are computed separately and superposed.

What is dropped is the quadratic term, and within it the cross term $-2\,\mathbf{u}'_A\cdot\mathbf{u}'_B/U^2$ — the same object as before, now visibly of the same order as the terms the linearisation already threw away. Superposing pressures is exactly as legitimate as the linearisation itself and no more, which is a far more useful statement than either “pressures add” or “pressures do not add”.

That immediately gives the size of the error. For a twelve-per-cent section at four degrees the disturbance velocities are of order a fifth of the free stream, so the cross term is of order four per cent of a dynamic pressure against a $C_p$ range near one — a few per cent, which is why thin-aerofoil theory is useful and why anybody wanting better than that computes the pressure from the total velocity instead.

It also explains why the example this essay is built on is the worst case obtainable rather than a representative one. A cylinder is not a small disturbance: at its shoulder the disturbance velocity is $U$ itself, so the expansion parameter is one, the neglected term is the same size as the retained one, and the measured defect of 1.92 is what “the small parameter is not small” looks like when it is plotted. Choosing a bluff body was the right choice for making the point and it is not the situation most calculations are in.

So the rule from the previous section survives with a clause attached. Carry the field, not its consequences — unless the disturbance is small, in which case the consequences may be carried too, at a cost that is the same order as everything else already neglected. A calculation that superposes pressures is not thereby wrong; it has committed itself to a linearised theory, and it owes the reader the same statement of the small parameter that every other linearisation does.

The clause has one sharp edge worth naming. A disturbance can be small over most of a body and enormous somewhere on it — at a leading edge, at a sharp corner, at a suction peak — and the superposition then fails locally while looking sound in the mean. That is precisely where a section’s behaviour is decided, so a linearised pressure summed from parts is least trustworthy in the one place a designer is reading it.

Where this goes wrong in practice #

Four places, and the last one is the expensive one.

Adding pressure coefficients. A thickness distribution and a camber distribution are computed separately in thin-aerofoil theory and their loadings are added — which is legitimate, because the loading is linear in the first-order theory, and is not legitimate for the surface pressures at second order. Every panel code computes the pressure from the total velocity for this reason.

Adding interference corrections. A wall correction and a support correction and a blockage correction are each computed as a change of velocity and then applied to a pressure, and the corrections interact. Wind-tunnel corrections are linear in the velocities and not in what is done with them.

Adding gust responses. A wing meeting two gusts at once does not see the sum of the two loads unless the response is genuinely linearised, and the lift at the mean angle is not the mean lift whenever the relationship curves.

And adding energies to get a drag. Induced drag is a quadratic functional of the span loading, so the induced drag of two wings in formation is not the sum of their induced drags — which is precisely why flying in formation saves fuel, and the saving is the cross term.

What the picture cannot show #

The defect is a scalar field and only ten points of it are plotted. The choice of points matters — they were picked to include places where the two flows interact strongly and places where they do not, because a scatter plot of a field is only as honest as its sampling.

And the two component flows are not drawn. They could be, and the figure would be three panels of streamlines that look entirely reasonable and convey nothing about the arithmetic, which is the general difficulty this collection keeps running into: the failure being described is not a failure of the picture, it is a failure of an operation performed on numbers the picture does not show.

The cross term has no sign in general. In the cases drawn here it happens to be mostly negative, because the vortex reinforces the stream on one side and opposes it on the other and the sampling is not symmetric. A different pair of flows gives a different distribution, and no general statement about the sign is available.

Why the linear theory is still worth having #

It would be easy to leave this rung with the impression that superposition is a trap. It is not; it is a tool with a stated range.

Everything about the kinematics is linear. Where the fluid goes, what shape the body is, whether the wall condition is satisfied, how much flux crosses a surface, what the circulation is — all of it adds, and all of it is what a construction like an image system or a panel method is doing.

The dynamics is a separate step, taken once at the end. Build the velocity field by whatever superposition is convenient; then compute $q^2$ once, from the total; then get the pressure. The rule is not “do not superpose” but “superpose the field, never the consequences”, and every correctly written potential-flow code obeys it without comment.

The habit that follows from all of this is one line long and is worth stating as a rule, because it is what separates a correctly written potential-flow calculation from a plausible one.

Carry the field, not its consequences. Every construction in this collection — an image system, a panel method, a distribution of sources along an axis, a lifting line — assembles a velocity field out of parts, and every one of them is entitled to do so. What none of them is entitled to do is assemble a pressure, a force, a moment or an energy the same way. The pressure is computed once, at the end, from the total; the force is a surface integral of that pressure; and the intermediate quantities of the assembly appear nowhere in either.

That rule is invisible in a code that does it right, because a routine which takes a field and returns a pressure has no opportunity to make the mistake. It becomes visible only in a calculation done by hand, or in a spreadsheet, or in the moment somebody adds two published pressure distributions to estimate a third — which is where it is made.

Who found it, and when #

There is no discovery to date here — the algebra is immediate and has never been in doubt. What has a date is the recognition that it matters, and that came from aerofoil theory in the 1920s, when thin-aerofoil results computed by superposing thickness and camber effects began to be compared against measurements accurate enough to see the second-order terms.

The surprising connection is with how often the same structure is the point rather than the problem. A cross term between two fields is what an interaction is, in every part of physics that has the word: the interference term in optics is $2\mathbf{E}_1\cdot\mathbf{E}_2$ and is the whole of Young’s experiment; the exchange integral in quantum chemistry is a cross term and is the whole of the chemical bond; the covariance of two random variables is a cross term and is the whole of a portfolio’s risk. In each case a quadratic quantity is being computed from a sum of two linear ones, and in each case the term that appears is not an error in the addition but the thing being studied. Here it is both — an error, when a pressure is wanted; and a theorem, when the boundary conditions are right.

Where the ladder goes next #

Below this rung are flows adding up, which is what linearity does buy, and the trade between speed and pressure, which is the quadratic step that loses it.

Beside it is the cross term as a theorem, where the same integral vanishes and settles uniqueness for the whole of ideal flow.