What the airspeed indicator believes

An airspeed indicator has one measurement in it: the difference between the pressure at a forward- facing hole and the pressure at a flush one. That difference is measured accurately, by a mechanism that does not care how fast the aircraft is going.

Everything that goes wrong goes wrong afterwards, in the arithmetic.

What is actually measured #

The forward-facing hole brings the flow to rest and reads the stagnation pressure, $p_0$. The flush hole reads the static pressure, $p$. The instrument responds to $p_0 - p$.

Bernoulli’s equation for an incompressible flow says $p_0 = p + \frac{1}{2}\rho U^2$, so $p_0 - p$ is $\frac{1}{2}\rho U^2$, and the speed follows immediately.

That derivation assumes constant density. Take the assumption away and, as the compressible energy equation shows, the isentropic relation gives

$$\frac{p_0}{p} = \left(1 + \frac{\gamma-1}{2}M^2\right)^{\frac{\gamma}{\gamma-1}}$$

and expanding it,

$$p_0 - p = \tfrac{1}{2}\rho U^2\left(1 + \frac{M^2}{4} + \frac{M^4}{40} + \cdots\right)$$

So $\frac{1}{2}\rho U^2$ is the first term of a series, and every correction is positive. The compressible stagnation pressure is always higher than the incompressible formula predicts, so a speed inferred from that formula always reads high.

The size of it, checked #

The numbers are worth having, because the range where the error is negligible is narrower than the convention suggests.

At Mach 0.3, 2.3 per cent in the pressure is about 1.1 per cent in the inferred speed, which is comparable with the instrument’s own accuracy and is where the conventional threshold comes from. At Mach 0.85 nearly a fifth of the pressure reading is compressibility, and no instrument would be trusted with that.

assertPitotOverreads checks three claims about this curve at once: that the ratio exceeds one at every Mach number above zero; that it grows monotonically, including across the transition at Mach one where the underlying formula changes; and that the two-term series tracks the exact value to within 0.2 per cent below Mach 0.9. The rejection test hands it a set of rows whose error does not grow with Mach number, which is what a correction applied in the wrong place would produce.

Where the M²/4 rule comes from, and where it stops #

The rule of thumb “add about $M^2/4$” is the second term of the series, and the figure above draws it alongside the exact curve so the agreement can be inspected rather than asserted.

It is excellent up to about Mach 0.7 and degrades slowly after that. At Mach 0.85 the two-term series gives 1.1937 against an exact 1.1939 — a discrepancy of 0.02 per cent, which is far better than the rule deserves.

What kills it is not the third term but the shock. Above Mach one the series is not a truncation of the right expression at all; it is a truncation of an expression that has stopped applying.

Above Mach one the probe cannot see the free stream #

A pitot probe is a blunt body facing into the flow. Above Mach one a blunt body carries a detached bow shock, and the stagnation streamline — the one that actually reaches the hole — passes through the strongest, most nearly normal part of it.

So what arrives at the mouth has already crossed a shock and lost total pressure. The instrument is measuring the stagnation pressure of the flow behind the shock, not the flow in front.

Rayleigh’s pitot formula is the correction, and it is a statement about a shock rather than about a tube:

$$\frac{p_{02}}{p_1} = \left[\frac{(\gamma+1)^2 M_1^2}{4\gamma M_1^2 - 2(\gamma-1)}\right]^{\frac{\gamma}{\gamma-1}} \cdot \frac{1 - \gamma + 2\gamma M_1^2}{\gamma+1}$$

It composes a normal shock with an isentropic compression to rest behind it, and inverting it gives the free-stream Mach number from the measured pressure and the static pressure ahead.

The physical reading is worth having: a supersonic pitot tube is a shock-strength meter. It infers the flow it cannot see from the damage that flow did on the way in.

The whole range, on one axis #

Putting the subsonic and supersonic branches on one plot shows something the two halves separately do not: the curve is continuous through Mach one, and its slope changes character rather than its value.

That divergence of the dashed curve is worth pausing on, because it is a good illustration of what a truncated series can and cannot tell anybody. Below Mach one, dropping the $M^6$ term costs 0.02 per cent. Above it, the series is not converging slowly — it is expanding a function that is no longer the right function, and at Mach 2 it gives 2.40 against an exact 1.66.

Nothing in the series itself signals the change. Its terms remain small and well behaved, and it goes on producing plausible numbers indefinitely. A series has no way of knowing that its hypothesis has lapsed, and that is precisely the failure this essay is about, met a second time one level down.

Why the density in ½ρU² is a second problem #

There is a separate error in the same expression that is easy to miss because it hides behind the first one.

$\frac{1}{2}\rho U^2$ contains a density, and an airspeed indicator has no way of knowing it. The instrument is built with sea-level density baked into its scale, so what it displays is the speed that would produce the measured pressure difference at sea level.

An aircraft at eleven kilometres is flying in air of about a third of sea-level density, so at a true airspeed of 240 m/s it displays around 140. That is a discrepancy of seventy per cent, which dwarfs the compressibility correction this essay has spent its length on.

The two errors are unrelated and act in opposite directions: the density assumption makes the reading too low at altitude, and the compressibility term makes it too high at speed. Neither cancels the other, and both have to be applied in order, which is why the correction cascade below has four quantities in it rather than two.

Four airspeeds, and only one of them is a speed #

The correction cascade in aviation is a good worked example of the difference between a measurement, an inference and a quantity of interest.

Indicated airspeed is the instrument’s reading, computed from $p_0 - p$ using the incompressible formula and a fixed sea-level density. It is not the speed of anything.

Calibrated airspeed is indicated airspeed corrected for the errors of the particular installation — the static port never reads quite the free-stream static pressure, because the aircraft disturbs the field it is sitting in.

Equivalent airspeed is calibrated airspeed corrected for compressibility, which is exactly the series in this essay. It is the speed at sea level that would produce the same dynamic pressure.

True airspeed is equivalent airspeed corrected for the actual air density, which requires the altitude and the temperature. It is the speed the aircraft is moving through the air.

The instructive part is that the uncorrected reading is the one pilots fly by, and deliberately so: stall speed, manoeuvring limits and structural limits all depend on dynamic pressure rather than on true speed, so an instrument that reads dynamic pressure is reading the quantity the aeroplane cares about. The “error” is a feature at low speed and a liability at high, and both statements are about the same number.

The density change the formula is ignoring #

It is worth seeing where the extra term physically comes from, because “a series has more terms in it” is arithmetic rather than an explanation.

That is the whole mechanism. The incompressible derivation carries a fixed $\rho$ from the free stream to the stagnation point, and the real gas thickens on the way. A denser gas decelerating produces a larger pressure rise, so the measured difference exceeds $\frac{1}{2}\rho U^2$ — and the excess grows as $M^2$ because the density change does.

The other end, where the arithmetic is right and the signal is gone #

Everything above is about the fast end, where the instrument is faithful and the formula has lapsed. The slow end fails the other way round, and the boundary there is just as hard.

The measured difference goes as the square of the speed, so it collapses. At fifty metres a second in sea-level air it is 1,530 pascals, which is a comfortable signal for any transducer. At ten metres a second it is 61 pascals. At two metres a second it is 2.4 pascals — and the static pressure of the atmosphere falls by about 12 pascals for every metre of height, so at that speed the entire signal is the pressure change over twenty centimetres of altitude.

Which is the whole difficulty in one comparison. A pitot-static system measures a small difference between two large numbers, and at low speed the difference has shrunk below the things that disturb each of them separately: a metre of climb, a gust, a door opening in the room, a slight temperature gradient in the tubing. The instrument is not inaccurate; there is nothing left for it to be accurate about.

There is a second failure at the same end, and it is a genuine departure of the physics rather than of the signal-to-noise. The stagnation reading assumes an inviscid deceleration to rest at the mouth, and when the probe’s own Reynolds number falls low enough that assumption goes: viscous stresses at the mouth add to the pressure, and the tube overreads. The correction becomes appreciable below a probe Reynolds number of a few hundred, which for a one-millimetre tube in air is about a metre a second — squarely inside the range where somebody would want to measure a ventilation flow.

So the instrument has a working band with a wall at each end, and the two walls are made of different material:

• At the top, the measurement is exact and the equation is wrong, by an amount that is computable and correctable to any desired accuracy. That is the subject of this essay, and it is the benign failure.
• At the bottom, the equation is exactly right and the measurement is not there. No arithmetic recovers a signal smaller than the disturbances on top of it, and there is no correction to apply.

The lower limit is therefore the harder one, which inverts the usual expectation. Low speeds sound like the easy case for an instrument built on an incompressible formula, and they are the case where the instrument has to be abandoned — for a hot wire, a vane, an ultrasonic transit-time meter, or anything else that does not depend on a quantity vanishing as the square of what is being measured.

Why this belongs with the other misconceptions #

The claim being refuted here is not a mistake anybody makes carelessly. It is a definition applied outside its hypothesis, which is the characteristic failure this field collects.

“The dynamic pressure is $\frac{1}{2}\rho U^2$” is a correct statement about an incompressible flow and a definition in that context. Carried above Mach 0.3 it becomes a claim about a quantity it no longer describes, and it fails quietly: the number is plausible, monotone in speed, and wrong by an amount that grows in exactly the range where being wrong matters most.

The shock the probe makes, drawn #

The supersonic case is worth seeing as a flow rather than as a curve, because the mechanism is not subtle and the curve hides it entirely.

Two consequences follow that are easy to state and were not obvious to the people who first met them.

A supersonic pitot reading cannot be corrected without knowing the static pressure ahead of the shock, which is why supersonic aircraft carry static ports well away from the nose and why calibrating them is difficult.

And the reading is not wrong in the sense of being noisy or biased. It is a precise measurement of a different quantity, and the difference is a computable function of the very Mach number one is trying to find — which makes the inversion possible and makes an uncorrected reading systematically misleading rather than merely imprecise.

What the picture cannot show #

The curves here are properties of an ideal probe in an ideal flow, and a real installation has several errors this essay does not touch.

The static port is the difficult one. It has to read the free-stream static pressure while mounted on a body that is disturbing the pressure field everywhere around it — so its position is chosen empirically, by finding somewhere on the fuselage where the local static happens to match the free stream over the useful range. That match is never perfect, changes with incidence, and is the largest term in the calibration of most aircraft.

Nothing in the figures shows it, because it is a property of an airframe rather than of a fluid.

The figures are also silent about what happens when the holes block. Ice, insects, and covers left in place have all produced accidents, and the failure signatures are unintuitive: a blocked static port makes the airspeed indication vary with altitude, and a blocked pitot with a drained line makes it read zero while a blocked pitot with a sealed line makes it behave like an altimeter. Those are consequences of the arithmetic in this essay applied to a pressure that has stopped tracking anything, and the arithmetic is what makes them predictable.

What the solver computes, and how it is checked #

pitot returns, at a given Mach number, the exact ratio of the measured pressure difference to $\frac{1}{2}\rho U^2$, the two-term series, and — above Mach one — the Rayleigh value and the fraction of total pressure lost crossing the probe’s own shock.

The subsonic branch composes the isentropic relations; the supersonic branch composes a normal shock with an isentropic compression behind it. Those are different calculations, and the check that they join up properly is the monotonicity requirement in assertPitotOverreads, which runs across a set of rows spanning Mach one. A discontinuity at the join — which is what an error in either branch would produce — fails it.

Where the model stops #

One-dimensional and ideal. A real probe has finite size, viscous effects at the mouth, and a sensitivity to being misaligned with the flow, all of which are calibration terms rather than physics.

The shock is taken as normal. The bow shock ahead of a probe is curved, and the stagnation streamline crosses it where it is normal, so this is a good assumption — but only for a probe pointing into the flow. At incidence it is not.

Perfect gas. At hypersonic speeds the gas behind the probe’s shock dissociates and Rayleigh’s formula, which assumes constant $\gamma$, overestimates the recovered pressure.

Steady flow. Everything here assumes the pressures have settled. In gusty air or during rapid manoeuvres the pneumatic lag of the tubing is a real term, and it is a property of the plumbing.

Who found it, and when #

Henri Pitot presented the tube to the Académie in 1732, using it to measure the flow of the Seine and to settle an argument about whether river water moved faster at the surface or at depth. He got the right answer to that question with an instrument nobody could yet analyse — Bernoulli’s Hydrodynamica was six years away and Euler’s momentum equation twenty.

Rayleigh’s supersonic formula dates from 1910, and it is a good example of a result arriving long before anything needed it: nothing flew supersonically for another thirty-seven years, and the formula was worked out because the shock relations made it answerable.

The airspeed correction cascade was assembled by aviation through the 1930s and 1940s, as aircraft started reaching Mach numbers where the second term of the series mattered. It has the shape of most engineering standards: a sequence of named quantities, each one an earlier quantity plus a correction, preserved because the sequence itself is what the instruments and the regulations are written against.

Where this goes next #

This closes the compressible phase, and it closes it on an instrument rather than on a flow, which is the right place for a site whose organising idea is that a picture proves nothing on its own. The pitot tube is honest at every speed. What is unreliable is the equation somebody applies to it.

The nearest other rungs are the flow that a photograph does not show, which makes the same argument about instruments that measure fields rather than points, and the four places Bernoulli’s constant does not survive, of which the shock is now the fifth and the one this phase added.