Exact in the total, free in the profile

Something has been happening in this collection often enough to be worth stating on its own.

A wake survey measures a drag exactly and says nothing about the wake. The momentum theorem gives the force on a body as ρUΓ on every contour drawn round it and splits that force between pressure and momentum flux anywhere from three per cent to ninety-seven. A closed channel fixes the return transport under a wave exactly and leaves the depth at which a parcel turns round completely open. A body’s far field keeps three coefficients and loses the body.

Each of those was found separately, in a different field, and written up as a property of that field. They are one statement, it has a proof of two lines, and it comes with a number.

The statement #

A constraint is a linear functional. “The deficit integrates to this” is a number obtained by multiplying the profile by a weight and integrating; so is “the second moment is that”, and so is almost every exact statement in this subject, because almost every exact statement in this subject is a conservation law integrated over a region.

The quantity somebody actually wants is usually a linear functional too. Where does the wake reach half its peak? What is the velocity at this station? How steep is the profile at the wall?

Both live in the same space, so both have a representer — the function that, integrated against a profile, returns them. Write the constraint’s as $w$ and the wanted quantity’s as $j$. Split $j$ into the part lying in the span of the constraint representers and the part orthogonal to it, $j = j_\parallel + j_\perp$. Then for any two profiles $f$ and $g$ that both satisfy every constraint exactly,

$$J[f] - J[g] = \langle j_\perp,\, f - g\rangle,$$

because the parallel part sees only quantities the two profiles agree about. So by Cauchy–Schwarz

$$|J[f] - J[g]| \le \|j_\perp\|\,\|f - g\|,$$

and the residual freedom in the answer is one number: the length of the component of the wanted quantity that the constraints do not reach. It is zero exactly when the wanted quantity is a combination of the constraints themselves, and not otherwise.

That is the whole theorem, and the rest of this essay is about what it is worth.

The case with a body in it #

The far-wake momentum balance is the cleanest instance and it is exactly linear rather than nearly so. Where the deficit is small against the free stream, the drag per unit span is

$$D = \rho U \int (U - u)\,dy$$

with no shape assumption anywhere in it. So a wake survey measures a total, that total is a constraint of precisely the kind above, and the drag is the one quantity such a survey determines.

The four profiles above are a narrow Gaussian, a wide top hat, the two-lobed wake a bluff body with a splitter plate leaves, and a heavy-tailed profile. Every one of them is a wake somebody has measured. They are normalised to identical deficit integrals, and their drags agree to three parts in a hundred million million — which is the quadrature and not the physics, because the drag is the integral.

Their peak deficits differ by a factor of four and a half. Their half-widths differ by five. The kinetic energy left behind in the wake — which a reader might reasonably guess the drag was made of — differs by a factor of three.

There is nothing pathological here. A wake survey is a good measurement of drag and a widely used one; the point is only that it measures one number and returns one number, and everything anybody would like to infer from it about the body that made it has to come from somewhere else.

Where the freedom lives, quantified #

The theorem gives an inequality. An inequality is worth less than it looks: the true spread could sit far below the bound, in which case the number would be an upper limit nobody can reach, and it would flatter the constraint in exactly the direction that matters.

So the extremal perturbation is built explicitly. It is the wanted functional’s representer with its constraint component removed, which makes it orthogonal to every constraint by construction — the moments it moves are zero to ten decimal places, so it is admissible — and it moves the answer by exactly the predicted amount.

The bound is attained to nine figures. A random admissible perturbation of the same size reaches nine per cent of it, which is the other half of the picture: the freedom is real and reaching all of it takes a deliberate act. A profile drawn at random from the admissible set is not the worst case, and a worst case exists.

How much a constraint buys #

One constraint is one direction removed from an infinite-dimensional space, so the natural next question is what happens with more.

Four quantities are tracked as the first six moments of a profile are fixed exactly. The total is one of the constraints and is determined at every step — to three parts in 10¹⁷, which is the arithmetic. An average over the middle third of the interval is worth 81 per cent free after one constraint and 38 after six. A reading at a point, smeared over two per cent of the interval, falls from 96 to 85. A slope at the wall barely moves at all.

The ordering is the useful part. What a set of integral constraints reaches is whatever is smooth and spread out like the constraints themselves; what it does not reach is anything local. A moment is an average over the whole profile and a reading is a value at a place, and no number of the first is many of the second.

An angle, and what it costs to be nearly parallel #

The two-line proof has a geometric reading that is worth spelling out, because it turns a question about flows into a question with an answer.

The constraints span a subspace. The wanted quantity is a vector. Everything about how much the first determines the second is the angle between them: at zero degrees the constraints give the answer outright, at ninety they give nothing, and in between they give the cosine.

That is why the ordering in the sweep above is what it is. The representer of a moment is a smooth function spread over the whole interval; the representer of a reading at a point is a narrow spike. Their inner product is small because the spike is narrow, and it stays small however many smooth representers are added — six moments reach 15 per cent of a reading’s length, and the seventh will not do much better.

It also explains the one case that is not gradual. The drag is not nearly parallel to the deficit integral, it is the deficit integral, so the angle is exactly zero and the freedom is exactly nothing. There is no continuum between “determined” and “not determined” for a single constraint and a single quantity: either the wanted functional lies in the span or it does not, and if it does not, what is left over is a length that has to be computed rather than guessed.

The practical version is a question to ask before any measurement is designed. Not is this quantity conserved — many are — but is the thing being asked for a combination of the things that are conserved. When it is, the measurement is exact and needs no model. When it is not, the model is carrying the whole of the inference, and the amount it is carrying is the number this essay computes.

The half that is not about the constraint at all #

Now the correction, and it is the reason this essay exists rather than a fifth worked example.

The bound is per unit of $\|f - g\|$ — per unit of how different the two profiles are — and which norm that is has been quietly doing half the work.

Measured over all profiles of unit mean-square size, the freedom in a wall slope grows without limit as the basis is refined: 66 at ten modes, 615 at forty, 5,047 at a hundred and sixty, and still climbing. That is not a numerical artefact. A derivative is not a bounded functional on the space of square-integrable functions, and no number of integral constraints makes it one — a profile can wiggle arbitrarily fast at arbitrarily small amplitude and change its wall slope by anything at all while moving every moment by nothing.

Measured over profiles of bounded curvature, the same quantity under the same constraints converges, and settles at 0.201.

So the freedom is not a property of the constraint. It is a property of the constraint and the admissible set together, and a calculation that quotes an insensitivity without saying which family it swept has quoted half of an answer.

This is exactly what makes the momentum integral work. Four assumed boundary-layer profiles give a drag within a few per cent not because the momentum integral is powerful — it is one linear constraint — but because the profiles anybody would offer are smooth, monotone and satisfy the same wall conditions. Widen the family and the agreement goes; the constraint has not changed.

What the same distinction does to an inverse problem #

There is a second thing worth separating, and it goes the other way.

Six constraints written as powers of $x$ and six written as orthogonal polynomials span the same subspace. They must therefore leave the same freedom, and they do — the two answers agree to fifteen figures.

What is not the same is the inverse problem: recovering a profile’s coefficients from its measured moments. There the Gram matrix is inverted, the monomials’ is the Hilbert matrix, and six of them have a condition number near ten million. Moments measured to twelve digits recover the coefficients to twelve in one spelling and to five in the other.

So the two halves separate cleanly, and the separation is the finding. A constraint’s reach into an answer is well conditioned and basis-independent. The reconstruction of what it constrains is neither. Somebody who has confused the two will report that the choice of moments matters to how much the data determines, which is false, or that it does not matter to the fitting, which is worse.

What a probe would have returned #

It is worth turning the wake result round. A single hot wire on the wake centreline returns one of four numbers spanning a factor of four and a half, and the drag is identical for all of them. So the reading carries information the drag does not — and the drag carries information no single reading can supply.

Neither measurement is better. They are measurements of different linear functionals of the same profile, they are nearly orthogonal, and an experiment that takes one and infers the other is inferring across the gap this essay is about.

The four cases, and why they are one case #

The four results this essay opened with sort themselves once the question is asked properly.

The drag from a wake survey has no freedom left in it, because the wanted quantity is the constraint. That is the extreme case and it is the one people generalise from.

The split of the force on a contour into pressure and momentum flux is a second functional of the same field, not in the span of the first, and it runs from three per cent to ninety-seven as the contour is changed. The total is exact on every contour; the accounting is a property of the contour.

The reversal depth under a wave is a nonlinear functional — where a sum of two profiles crosses zero — and the constraint reaches it not at all. Three return currents with exactly the right transport put it at three different depths, and the sign of the drift at the bed changes between them, which is what decides where sediment goes.

The shape of a body from its far field is the limit: the constraint is every value of the field outside a contour, which is as much data as a flow can carry, and two bodies with nothing in common produce the same far field.

What this is not #

It is not a claim that exact results are weak. The opposite: the reason a wake survey is used at all is that the one thing it determines, it determines exactly, with no model, no calibration and no assumption about the body. Kutta and Joukowski’s formula is worth having precisely because it asks nothing about the shape.

It is not dimensional analysis. Counting the dimensionless groups says how many numbers an answer can depend on and this says how much of a profile a constraint reaches; the two are unrelated, and a problem can be reduced to one group and still leave everything about its profile free.

It is not about measurement error. Every number here is exact. The freedom is what remains after a constraint has been satisfied to the last bit, and refining the measurement does nothing to it — which is the difference between this and the ordinary business of not having enough data.

And it is not a statement about any particular constraint being badly chosen. The moments used here are as good as constraints get. What the sweep shows is that the class of quantity being asked for decides the answer: smooth averages are reached, local values are not, and derivatives are not reached at all without a smoothness assumption that has to come from physics rather than from the constraint.

Which quantity to ask for, then #

The practical content is a habit rather than a formula, and it is short.

Ask whether the wanted quantity is a combination of the constraints. If it is, the constraints determine it exactly and no model is needed — that is the wake survey, and it is why the momentum theorem is the most useful tool in this subject.

If it is not, say what family the answer was swept over. “The drag is insensitive to the profile” is not a statement until the profiles are named, because over the whole of $L^2$ almost nothing is insensitive to anything.

And treat a point value or a derivative as a different kind of request. Those are the quantities integral constraints cannot reach, and getting at them needs either a smoothness assumption stated out loud or a measurement of the same kind — a local one.

The last of those is why a boundary-layer method that predicts drag within three per cent predicts separation thirty per cent late, and why a photograph of a flow can be beautiful and prove nothing: both are asking a local question of machinery that only ever knew a total.

Two ways a constraint can be strengthened #

If the freedom is too large for the answer that is wanted, there are exactly two things that can be done about it, and they are worth separating because only one of them is usually available.

Add constraints. Each one removes a direction, and the sweep above prices what each is worth for a given quantity. Six moments take an average over the middle third from 81 per cent free to 38, and a reading at a point from 96 to 85. Constraints are cheap when they come from conservation laws and expensive when they have to be measured, and the returns diminish in a way the projection makes explicit: a new constraint helps in proportion to how much of the wanted quantity’s representer it happens to reach.

Shrink the admissible set. This is what smoothness does, and it is by far the more powerful of the two here — the wall slope’s freedom goes from unbounded to 0.201 without a single constraint being added. The price is that the shrinking has to be justified from physics rather than from the data, because it is an assumption about what the profile can be rather than a measurement of what it is.

Most working practice is the second, applied silently. Every profile family, every closure, every correlation and every basis expansion is a statement about the admissible set, and the accuracy that follows is attributed to the method rather than to the assumption. The measurement on this page is a way of asking how much of a result’s precision is coming from each — and the honest answer, in the boundary-layer case at least, is that nearly all of it comes from the family.

What is not claimed #

The theorem is about linear functionals. The reversal depth is not one, and neither is a separation point or a stall angle; those are handled here by sampling an admissible family rather than by projection, which gives a spread rather than a supremum. The linear statement bounds them only when they can be linearised, and near a fold they cannot.

The norms used are choices. Mean square and bounded curvature are two of infinitely many, and picking a different smoothness measure would give a different number for the same wall slope. The finding is not that 0.201 is the answer; it is that no answer exists until the family is stated, and that the two extremes differ by orders.

Nothing here bounds the error of any method. A method that gets the drag right to two per cent is not being validated by this argument, and one that gets it wrong is not being excused: what is being computed is how much room a constraint leaves, not how much of that room a particular approximation actually occupies.

And the wake profiles are constructions. They are chosen to be different, they all carry the same deficit integral because they were built to, and no real body was solved to produce them. The statement they demonstrate is exact; the demonstration is deliberately assembled, which is what a demonstration of a non-uniqueness has to be.