An instrument that takes a derivative

The first rung of this ladder asks what a photograph of a flow actually shows and answers it about smoke, dye and particles. The second shows that the exposure is part of the measurement. Both are about techniques that put something in the fluid and read where it went.

The optical techniques put nothing in. Light passes through the flow, is bent by it, and what a plate records is a property of the refractive index field — which for a gas is linear in the density, $n - 1 = K\rho$, so every optical technique is a densitometer.

And the three of them respond to three different derivatives of the same field.

Why each one takes the derivative it takes #

The three optical arrangements differ in one component each, and the derivative follows from it.

Interferometry splits the beam, sends half through the flow and half round it, and recombines them. The phase a ray accumulates is the integral of the refractive index along its path, so the fringe pattern is a map of $\int n\,ds$ — which for a two-dimensional field is the density itself, times a path length.

Schlieren puts a knife edge at the focus of the beam that has passed through the flow. A ray bent by a density gradient arrives at the focus displaced; the knife cuts some of it; and the amount cut, and therefore the darkness at that point of the image, is proportional to the deflection angle — which is the first derivative of the density, integrated along the ray.

Shadowgraph has no knife edge and no reference beam. It simply projects the bent rays onto a screen, and what shows is where rays have converged or diverged relative to their neighbours — a gradient in the deflection, which is the second derivative.

So the instrument’s optical complexity runs backwards to its mathematical order. The simplest apparatus, a light and a screen, produces the highest derivative; the most elaborate, with a split beam and a matched path, produces the field itself. That inversion is not a coincidence: a reference beam is exactly what is needed to recover a quantity rather than a rate of change of one.

What two of the three cannot see #

A uniform stream has no density gradient. Schlieren and shadowgraph therefore record exactly nothing in one, and it does not matter how fast it is going.

That is worth saying flatly because it is so easy to read past. A schlieren photograph of a wind tunnel running empty at Mach 3 is a photograph of nothing, and it is the technique working correctly. What appears in a schlieren image of a supersonic flow is not the supersonic flow; it is the shocks, the expansions and the layers — every one of which is a place where the density is changing — and the uniform regions between them are blank whatever their state.

A constant gradient sharpens the point further. A stratified layer at rest, with the density falling steadily upward, gives a schlieren image of a uniform grey: a number rather than a picture, from which the gradient can be read but nothing can be seen. The shadowgraph gives $8\times10^{-14}$, which is zero, because the second derivative of a straight line is zero.

Only interferometry reads either of them, and it reads both, because it responds to the density and not to a rate of change of it. That is why it needs a reference beam and why it is the hardest of the three to set up.

Which instrument for which job #

The scaling with feature thickness settles the practical question, and it is one line of calculus with a measured slope behind it.

Hold the density jump and shrink the thickness. The first derivative grows as $1/t$; the second grows as $1/t^2$. So the sharper the feature, the more the shadowgraph gains relative to the schlieren, and at the thicknesses on the left of that figure the ratio is a thousandfold.

Which gives the working rule. A shadowgraph is the instrument for finding a shock, because a shock is the sharpest feature a flow has and the shadowgraph is most sensitive to exactly that. A schlieren is the instrument for reading one, because its response is proportional to the gradient rather than to its rate of change, so a schlieren image is quantitative in a way a shadowgraph is not and its greys can be calibrated against a known deflection.

And interferometry does not move at all in that sweep, because the density jump has not changed — it is the instrument for measuring how much, and it is indifferent to how abruptly.

What “the thickness of a shock” means in a photograph #

There is a consequence for reading images that is worth stating because it is a genuine and common misreading.

A shock in a schlieren photograph is a line of finite width, and that width is not the shock’s. A shock is a few mean free paths thick — under a micrometre in air at ordinary conditions, which this collection computes elsewhere — and no optical system resolves it. What sets the width in the picture is the optical system’s own resolution, the shock’s curvature over the path the ray took, and any three-dimensionality that smeared the integral.

Every optical technique integrates along the ray. That is the assumption underneath all three and it is the one that cannot be checked from the image: a thin strong feature and a thick weak one, with the same integral, produce the same picture. A conical shock viewed side-on and a plane shock of a different strength are optically indistinguishable, and the reduction requires knowing which of the two is being looked at.

That is the optical version of what the shutter being part of the answer says about time. One instrument averages over an exposure; these average over a path; and in both cases what the picture shows is the flow convolved with something the experimenter chose and rarely states.

The instrument that answers the question this field keeps asking #

There is a reason the optical techniques belong in a field about misconceptions rather than in one about experiment, and it is worth making explicit.

Every other entry in this field concerns a picture that shows something true and is read as showing something else — a smooth photograph read as proof of smooth flow, streaklines captioned as streamlines, a long exposure read as an instant. The failure is always that the instrument’s transfer function is invisible.

Here the transfer function is a derivative and it is completely explicit. Nobody looking at a shadowgraph believes they are seeing the density; the picture is a set of bright and dark bands that resemble nothing in the flow, and its unreadability is a protection. The technique cannot be misinterpreted as a direct view because it plainly is not one.

That inverts the usual danger and produces a different one. An image that looks like the flow invites over-reading; an image that does not invites under-reading, and the under-reading of a shadowgraph is to treat it as qualitative — as a picture that shows where the shocks are and nothing more — when it is a quantitative measurement of a second derivative with a calibration behind it.

The general point is worth taking away from the whole field. The question to ask of any flow image is not does this look right but what operator was applied to the flow to make it, and for these three the answer is a number: zero, one and two.

A fourth technique, and why it is in a different family #

There is a fourth optical method that this essay’s arithmetic does not cover and that belongs beside the three, because it is the one a modern tunnel actually uses.

Background-oriented schlieren photographs a patterned background through the flow and computes the apparent displacement of the pattern. The displacement is proportional to the ray deflection, which is the first derivative — so it is a schlieren, and its response is the schlieren line of the sweep above.

What is different is everything else about it. It needs no knife edge, no matched optics and no precision alignment: a camera, a printed pattern and the same cross-correlation software particle velocimetry already uses. That has made it usable outdoors, at large scale, and on things a laboratory optical bench cannot be built around — the shock system of an aircraft in flight, photographed against desert terrain from another aircraft, is a background-oriented schlieren.

The physics is the rung below’s and the apparatus is a generation newer, which is the ordinary way a technique becomes widespread: not by measuring something new, but by measuring the same derivative with equipment somebody already owns.

The number an image is worth #

It is worth putting a magnitude on these responses, because the deflections involved are tiny and the fact that the techniques work at all is not obvious.

The Gladstone–Dale constant for air is $2.26\times10^{-4}$ cubic metres per kilogram, so air at sea-level density has a refractive index of $1.000277$ — a departure from vacuum in the fourth decimal place. A shock doubling that density changes the index by another $2.8\times10^{-4}$.

Over a ten-centimetre path, a ray crossing a shock of that strength is deflected by an angle of order $K L \,\partial\rho/\partial y$, which for a shock a fraction of a millimetre thick is a few tenths of a milliradian — a few arc minutes. That is what the knife edge has to cut, and it is why a schlieren system needs a long focal length and a small, well-defined source: the deflection is small and the only way to make it visible is to give it a long lever.

The corollary is that sensitivity is bought with size, which is why schlieren mirrors are large and why the largest ones are prized instruments with names. Nothing about the physics improves; the geometry is what is being paid for.

And it explains the one thing an experimenter notices first. A schlieren system is exquisitely sensitive to things nobody wanted to measure — the plume of a hand held near the beam, convection off a warm surface, the draught from a door. All of them are density gradients, all of them are far stronger than the flow being studied in the parts of the image away from it, and keeping them out of the beam is most of the work of setting one up.

The knife edge, which is a choice about what to see #

There is a detail of the schlieren arrangement that is worth a section because it is the one place an experimenter makes a decision that changes the picture, and it is rarely stated in a caption.

The knife edge has an orientation. A horizontal edge cuts rays deflected vertically and leaves rays deflected horizontally alone; a vertical edge does the reverse. So a schlieren system does not record the gradient of the density — it records one component of it, and which component is a decision somebody made when they set the bench up.

That has two consequences for reading an image.

A feature aligned with the knife edge is invisible. A shock lying parallel to a horizontal edge deflects rays vertically… no: it deflects rays across itself, so a shock parallel to the edge produces deflections perpendicular to the edge and shows strongly, while a shock perpendicular to the edge produces deflections along it and shows weakly or not at all. Rotating the knife edge ninety degrees changes which features are visible in the same flow.

And the sign is visible. A knife edge cut from one side darkens where the density rises and lightens where it falls, so a schlieren image has a direction in it — the greys distinguish a compression from an expansion, which a shadowgraph does not. That is the technique’s other advantage over the shadowgraph and it is a direct consequence of taking one derivative rather than two: a first derivative has a sign and a second derivative’s signature is symmetric.

The variants follow from the same idea. A colour schlieren replaces the knife edge with a filter of graded colour, so the deflection maps to a hue rather than to a grey and the dynamic range is much larger. A circular cutoff responds to deflection in every direction equally, at the price of losing the sign. Each is a different choice of what to throw away.

What all of them measure, which is not a flow property #

There is a final point that the three techniques share and that the essay’s framing may have obscured.

None of them measures the density. All three measure an integral along a ray of some function of the density, and the reduction to a density needs the path length — which for a two-dimensional model is the span and for anything else is a model of the geometry.

So a schlieren image’s grey level is a deflection angle, and a deflection angle is a density gradient times a length. Two flows with the same gradient over different spans give different images; a laboratory model and a full-scale object in the same flow field give different images of the same flow.

That is a much stronger statement than the two-dimensionality caveat above. It says the pictures are not comparable between facilities without knowing the path lengths, that a quantitative schlieren is a quantitative measurement of the product rather than of either factor, and that the technique’s sensitivity scales with model size — which is one of the few advantages of a large tunnel that has nothing to do with Reynolds number.

What the picture cannot show #

No flow was solved. Every field here is prescribed — a smoothed jump, a layer, a gradient — and scaled to stated values, because the essay is about the mapping from a field to an image rather than about any particular flow. A reader wanting the density field of a real shock will not find it here.

Two dimensions, and it is the load-bearing assumption. Every response is an integral along the ray and every figure treats that integral as the field times a path length, which is exact for a field with no variation along the beam and wrong otherwise. Three-dimensional optical measurement exists and needs many views and a tomographic reconstruction.

No optical system. There is no lens, no focal length, no knife-edge position, no film response and no light source. The responses computed are proportional to the derivatives; turning them into a grey level needs an apparatus, and the constant of proportionality is where a calibration lives.

And the Gladstone–Dale relation is linear and not exactly. $n - 1 = K\rho$ holds very well for air over a wide range and stops holding when the gas is hot enough to be changing chemically — which is exactly the condition a hypersonic experiment is in, and where gamma stops being a number as well, and it is why optical measurement in that regime is harder than the arithmetic here suggests.

The assertion behind these figures is the one that makes the essay’s headline a computation rather than a claim. Every response must vanish identically on a uniform field — as an exact zero, not a small number — the shadowgraph must vanish on a linear gradient while the schlieren does not, each computed response must match the closed-form derivative of the field it was given, and the two scalings must come out at exactly 1 and 2 rather than near them.

What the three cannot do at all #

A last limit, and it is the one that keeps these techniques out of most of aerodynamics rather than merely qualifying them.

All three need a density gradient, and an incompressible flow has none. The whole of the low-speed half of this collection — the boundary layers, the separations, the wakes, the vortices — happens at constant density, and every optical technique in this essay is blind to all of it. A schlieren image of a wing at thirty metres a second shows nothing whatever, and no improvement in the apparatus changes that.

That is a much harder limit than the ones above. The blindness to a uniform stream is a property of a derivative and can be worked around by looking at features; the blindness to an incompressible flow is a property of the fluid and cannot be worked around at all.

Which is why the optical techniques and the tracer techniques have divided the subject between them rather than competing. Smoke, dye and particles work where the density is constant and stop working where it is not — a particle in a supersonic flow has inertia the flow will not forgive and cannot follow a shock at all. The optical methods work where the density varies and see nothing where it does not.

There is one heating trick that bridges them and it is worth naming: a low-speed flow can be given an artificial density gradient by warming part of it, and a schlieren system will then show the thermal plume as a proxy for the flow. That is a tracer technique wearing optical clothes — something has been added to the fluid — and it inherits every question the first rung of this ladder asks about tracers.

Who built them, and when #

Toepler devised the schlieren method in 1864, for looking at flaws in glass — Schlieren is German for streaks and the word is about a defect in a transparent solid rather than about anything moving. Shadowgraph is older and simpler and has no single inventor; Hooke described the effect in the seventeenth century.

The aeronautical career began with Mach. Ernst Mach’s photographs of supersonic bullets in the 1880s are shadowgraphs, they show the bow shock and the Mach cone unmistakably, and they were the first direct evidence that a body moving faster than sound carries a discontinuity with it. The number named after him was named because of what a shadowgraph made visible.

That is a satisfying place for this ladder to arrive at. The rung below argues that a photograph of a flow is not a picture of the flow; here is a technique whose whole content is a derivative, whose images look nothing like the flow, and which nonetheless settled the central fact of an entire regime — because what it could see was exactly the thing that mattered, and what it was blind to was everything else.

Where the ladder goes next #

The rung above is the technique that inverts every assumption in this one. Pressure-sensitive paint does not look through the flow at all; it looks at the surface, it reports a scalar rather than a derivative, and it turns a pressure measurement from a row of taps into a field.

It also comes with an error of a kind none of the optical methods has. The paint’s luminescence is quenched by oxygen, which is what makes it a pressure gauge; it is also quenched by temperature, at a rate that makes a single degree worth several per cent of pressure — and a model’s surface temperature is not uniform. The instrument that turns a scalar into a field is contaminated by the other field the flow produces, and the standard remedy is a second paint to measure the contamination.

The one beside it is the tomographic problem this essay names and declines. Recovering a three-dimensional density field from optical images requires many views and an inversion, the inversion is ill-posed, and how ill-posed it is — how much a small error in the images moves the reconstructed field — is a computation with a number in it.