The paint that measures the wrong field

The rung below this one ends on a limit that is not a qualification: the optical techniques need a density gradient, and an incompressible flow has none. Half of this collection is invisible to all three.

Pressure-sensitive paint inverts every assumption in them. It does not look through the flow, it looks at the surface. It reports a scalar rather than a derivative. It works perfectly well at low speed. And what it measures is what a wind tunnel most wants measured — a pressure distribution, as a field rather than as a row of numbers from a row of holes — the quantity every force on a body is an integral of.

What the paint is doing #

The paint contains a luminescent molecule that absorbs light at one wavelength and re-emits at another. The re-emission competes with a second process: an excited molecule that meets an oxygen molecule gives its energy to the oxygen instead and emits nothing. That is quenching, and the more oxygen there is, the more of it happens.

Air is a fifth oxygen at every pressure, so the oxygen partial pressure is proportional to the total pressure. The paint glows less where the pressure is higher, and the relation is the Stern–Volmer form:

$$\frac{I_{\text{ref}}}{I} = A + B\,\frac{p}{p_{\text{ref}}},$$

a straight line in the pressure, with $A$ and $B$ calibration constants.

The technique’s cleverness is entirely in that ratio. A single image’s brightness depends on the illumination at that point, on how thick the paint happens to be there, on the angle the surface makes to the camera, and on the paint’s own age — none of which is known and all of which vary over a model. Taking the ratio of a test image to a reference image of the same model in the same place divides every one of them out.

Why the ratio removes what it removes #

The two-image ratio is the technique’s whole foundation and it is worth being precise about the class of error it defeats, because the precision explains why temperature survives it.

Write a single image’s brightness at a point as

$$I = \Phi\, \varepsilon\, C\, \frac{1}{A + B\,p/p_{\text{ref}}},$$

with $\Phi$ the illumination arriving there, $\varepsilon$ the collection efficiency of the optics at that angle, and $C$ everything about the paint’s own thickness and concentration. None of the first three is known and all of them vary strongly across a model.

Take the ratio of a reference image to a test one at the same point, with the model in the same place and the lighting unchanged, and $\Phi$, $\varepsilon$ and $C$ all cancel identically. Everything that is a property of the apparatus at that pixel goes, and what is left is the Stern–Volmer factor.

Temperature does not go, and the reason is now visible: it is not a property of the apparatus at a pixel. It is inside $A$ and $B$ — it changes the physics of the measurement rather than the brightness of the observation — and no ratio of observations can divide out a change in what is being observed.

That is a general rule about ratio methods and it is worth carrying. A ratio removes multiplicative factors that are common to both measurements and nothing else. An error that changes the calibration between the two survives it untouched, and the fact that a technique is a ratio is no protection at all against that class.

Registration is the practical qualification. The cancellation requires the same point of the model in both images, so a model that has deflected under load between the wind-off and wind-on exposures gives a ratio of two different places — and the resulting error is largest where the brightness gradient is largest, which is at the edges of features.

The field it did not mean to measure #

What the ratio does not remove is temperature, and the reason is that both coefficients depend on it.

Quenching is a collision process, so its rate depends on how fast the molecules are moving; and the radiative lifetime of the excited state depends on temperature too. Both $A$ and $B$ move, and a reduction that uses the reference temperature’s constants on a warmer surface reports a pressure that is wrong.

The sensitivity is 0.34 per cent of pressure per kelvin for the calibration drawn here, which is representative of a common paint. Ten degrees is 3.4 per cent. Twenty is 6.9.

Those numbers have to be read against what the technique is used to measure. A pressure coefficient of interest on a model might be a few tenths, and a few tenths of a dynamic pressure at a tunnel’s operating condition is a few per cent of the absolute pressure. So a single degree of temperature error is comparable with the pressure differences being measured, and ten degrees is larger than them.

Why the temperature is never uniform #

The awkward part is that a model’s surface temperature is not merely non-uniform; it is non-uniform in the same places the pressure is.

A wall told nothing about its temperature settles near the recovery temperature, which rises with the local speed. So the surface is warmer where the flow has been brought to rest — at the stagnation point, in a separation, under a shock — and cooler where it has accelerated — over a suction peak. Those are exactly the features a pressure measurement is being made to resolve.

The temperature error is therefore correlated with the pressure field rather than random over it, which is the worst possible arrangement. A random error averages down over an image; a correlated one does not, and it biases precisely the extrema a designer is reading off.

There is a second and larger contribution that has nothing to do with the flow. A tunnel run heats the air, and a model that has been sitting in a laboratory takes minutes to come to equilibrium — so the reference image, taken before the run, and the test image, taken during it, can differ by ten or twenty degrees for reasons that are about a schedule.

The remedy, which is another instrument #

Since the contamination is a second field, the standard answer is to measure the second field.

A temperature-sensitive paint is applied alongside — a second luminescent molecule, chosen to be insensitive to oxygen and strongly sensitive to temperature, emitting at a third wavelength. Two cameras with two filters then give two images, the temperature field is recovered from one, and the pressure reduction uses the local temperature rather than the reference one.

An instrument added to correct an instrument is a familiar shape and it is worth naming its relatives. Jones’ correction to a wake survey adds a static tapping to remove an assumption about pressure recovery. A total-temperature probe’s own recovery factor is a calibration constant absorbing three physical effects. In each case the second measurement removes an assumption rather than improving a number.

The alternative remedies are worth knowing because they are what is actually done when the second paint is unavailable.

Match the temperatures. Take the reference image at the same tunnel temperature as the test image, which means running the tunnel to a steady thermal state and taking a wind-off image at that state rather than before the run. This removes the schedule contribution and not the flow’s own.

Use a paint with a low temperature sensitivity, of which several exist and all of which are less sensitive to pressure as well — the two sensitivities are not independent, because both come from the same excited state’s lifetime.

Or measure the ratio at two emission wavelengths from the same paint, one temperature-sensitive and one not, which is the same idea as the second paint with one coating instead of two.

What is bought, which is the reason to bother #

The essay has been about an error and the technique is genuinely a large advance, so the balance is worth stating.

A row of pressure taps gives a row of numbers. Each is accurate to a fraction of a per cent, each requires a hole through the model and a tube to a transducer, and a model with two hundred of them is an expensive model that took months to make — sitting in a tunnel whose walls are in the answer as well. Between the taps there is nothing, and a feature that falls between two of them is invisible.

The paint gives a field. And a pressure field integrated over a closed body is a force, so a paint measurement can be reduced to a lift and a drag in the way a row of taps never quite can. Every pixel is a measurement, a feature narrower than a tap spacing is resolved, and the model is a shape with paint on it rather than a plumbing exercise. On a three-dimensional configuration — a wing-body junction, a store separation, a control surface — a field is not an improvement on a row of taps; it is the only way to see the thing at all.

So the trade is per-point accuracy for coverage, and it is the right trade for exactly the measurements a row of taps cannot make. The usual practice is to do both: a small number of taps as an in-situ calibration, and the paint to fill in between them — which turns the paint’s absolute accuracy problem into an interpolation problem, and interpolation is what it is good at.

What the technique is blind to #

Every instrument in this ladder has a blindness and it is worth naming this one’s, since the essay has otherwise been about a contamination rather than an absence.

The paint measures only where there is paint. It reports the surface pressure and says nothing about the flow above it — no velocity, no separation position except through its pressure signature, no vorticity. That is not a limitation for the measurement it was built for and it means the technique cannot substitute for the others rather than complement them.

It needs oxygen. In a nitrogen tunnel, in a cryogenic facility run on nitrogen, or in any inert working gas, the quenching mechanism has nothing to quench with and the paint reports nothing. That rules out an entire class of high-Reynolds-number facilities, which is a substantial part of where industrial testing is done.

And it needs light in and light out. A surface the camera cannot see is not measured, so an internal duct, a slot, the underside of a wing at the wrong angle, and any region shadowed by another part of the model are all blank — and the blankness is a geometric property of the setup rather than of the flow. Multiple cameras help and do not solve it.

The last of those is the one that constrains a test programme most. A model has to be painted, lit and viewed, and arranging all three for every surface of interest is a rigging problem that decides how many runs a campaign needs — which is a very different kind of limitation from a calibration error and is usually the one that costs the money.

The calibration that is done in place #

The remedy this essay has not yet named is the one most used, and it is a good illustration of what practice does with a technique whose absolute accuracy is imperfect.

An in-situ calibration uses a small number of conventional pressure taps on the same model, in the same run. The taps give absolute pressures at a handful of points; the paint gives an image; and the paint’s own calibration constants are then fitted so that the image agrees with the taps where both exist.

That changes what the technique is. It is no longer an absolute measurement with a laboratory calibration; it is an interpolation between a few accurate points, with the paint supplying the shape between them. And the temperature error largely disappears in the fit, because whatever surface temperature the model happened to be at is absorbed into the fitted constants — provided the temperature is roughly uniform over the region the fit covers.

The residual is exactly the temperature’s non-uniformity, which is the part the fit cannot absorb, and it is the part correlated with the pressure field. So the in-situ calibration removes the large, uniform, schedule-driven error and leaves the small, correlated, flow-driven one — which is a substantial improvement and is not a solution.

The habit is worth naming because it recurs across instruments. Fit an imperfect field measurement to a few accurate point measurements, and what survives the fit is whatever varies in the same way the quantity of interest does. A calibration removes what is common and cannot remove what is correlated.

What the picture cannot show #

No paint was modelled. The Stern–Volmer coefficients and their temperature derivatives are prescribed numbers, representative of a common formulation and not a measurement of one. A real paint’s calibration is a surface fitted over pressure and temperature, and the linear form here is the first term of it.

No photometry. There is no camera, no illumination field, no signal-to-noise ratio and no registration between the two images — and misregistration, where the model has moved between the reference and test exposures, is a large practical error source that this arithmetic has none of.

The reduction is the simple one. Modern practice uses a bi-luminophore paint, an in-situ calibration against a handful of taps, and a correction for the paint’s response time; each of those changes the error budget and none of them is here.

And the response is not instantaneous. The paint has a time constant set by oxygen diffusing into its binder, which is milliseconds for a thin porous coating and much longer for a conventional one — so an unsteady pressure measurement with paint is a filtered one, and the filter is the shutter of the rung two below in a different guise.

The assertion behind these figures makes the two claims the essay rests on. At the reference temperature the inferred pressure must be exact — not nearly, exactly, at four pressures — which is the statement that the reduction is a correct inverse. And away from it the error must be non-trivial and must match the sensitivity the function’s own derivative predicts, to two per cent, so that the number quoted in the caption is a computation rather than a quotation.

Who developed it, and when #

The oxygen quenching of luminescence is nineteenth-century chemistry and the Stern–Volmer relation is from 1919. Its use as an aerodynamic instrument is much later and its origin is unusual: the technique was developed in the Soviet Union in the 1980s, at the Central Aerohydrodynamic Institute, and reached the West at the end of that decade — after which it was taken up rapidly, because the problem it solved was one every large tunnel had.

What made it practical was not chemistry but photography. The measurement is a ratio of two images to a fraction of a per cent, which needs a detector with a large dynamic range, good linearity and low noise — and scientific CCD cameras arriving in the same decade is what turned a laboratory demonstration into an instrument.

That is worth noticing because it is the same story as the rung below’s. Schlieren became an aerodynamic instrument when photographic plates became fast enough for Mach’s bullets; paint became one when detectors became linear enough for a ratio. In both cases the physics was old and the sensor was what arrived.

Where the ladder goes next #

This anchor now has five techniques and one habit: ask what operator the instrument applied. Smoke marks the fluid and reports where it went; a shutter integrates over a window; schlieren differentiates the density once and shadowgraph twice; paint reports a scalar contaminated by a second field.

The rung above is the comparison itself, and it is the one this ladder has been assembling material for without making. Set the five side by side with their transfer functions written down — an exposure, a release window, a path integral, a derivative, a diffusion time — and ask which features of a flow each one can and cannot resolve. The bands overlap much less than the interchangeable use of the pictures suggests, and a table of what each is blind to would be more useful than any of the individual essays.

The one beside it is the technique none of these is: particle image velocimetry, which measures a velocity field directly and is the only one of the family that does. It has its own operator — a correlation window, a particle response time, a pulse separation — and the questions this ladder has asked of the others apply to it unchanged. That it measures the quantity everything else is a proxy for does not exempt it from being an instrument.