The number that does not depend on the tunnel

A pressure tap in a wind tunnel returns a number in pascals. That number depends on the speed the tunnel was running at, on the density of the air in the room that morning, and on the reference the gauge was zeroed against. Almost none of it is about the model.

The fix is one division, and it is the single most useful piece of bookkeeping in the subject:

$$C_p = \frac{p - p_\infty}{\tfrac{1}{2}\rho U^2}$$

The numerator is how much the pressure differs from the undisturbed stream. The denominator is the dynamic pressure — the pressure the stream would reach if it were brought to rest. What comes out is dimensionless, and for an incompressible flow it is a property of the shape and the angle of attack and nothing else.

Why exactly one, at the front #

The stagnation point is where the flow comes to rest, and Bernoulli’s equation along the streamline that arrives there says $p + \tfrac{1}{2}\rho q^2$ is the same at both ends. Far upstream that is $p_\infty + \tfrac{1}{2}\rho U^2$; at the stagnation point the speed is zero, so it is $p$ alone. Subtracting and dividing gives $C_p = 1$, exactly, with no properties of the fluid or the body left in it.

That is the strongest statement in the whole of low-speed aerodynamics. Whatever the shape, whatever the speed, whatever the fluid, the pressure coefficient at a stagnation point is one. Measured on the solved field it comes out at 1.000000.

It also gives the number a physical size. $C_p = 1$ is the most positive value achievable in an incompressible flow, because nothing can be slower than at rest. There is no corresponding bound below: the coefficient can go as negative as the flow can go fast, which is why the interesting half of every pressure distribution is the suction side.

Why exactly minus three, at the shoulder #

For a circular cylinder the surface speed is $2U\sin\theta$ — a result that falls out of the doublet-plus-stream solution in two lines. At the shoulder, $\theta = 90°$, that is $2U$: the flow is going twice as fast as the stream.

Putting $q = 2U$ into the coefficient gives $C_p = 1 - (2U/U)^2 = -3$. The measured value is −2.999999, and the residue is the finite-difference offset used to evaluate the field a hair outside the surface rather than exactly on it.

Minus three is a big number. It says the pressure at the shoulder is three times the dynamic pressure below ambient, which for a car at motorway speed is a suction of about two kilopascals and for an airliner’s wing is a great deal more. The whole reason aerodynamic loads are structural problems is on the negative side of this curve.

What the solver computed, and how it was checked #

Two computations, deliberately arranged not to share anything.

The first is the closed form, $C_p = 1 - 4\sin^2\theta$, evaluated at 361 angles. The second walks the same 361 points, asks the solved potential-flow field for the velocity a hair outside the surface, takes its magnitude, and forms $1 - q^2/U^2$. The largest disagreement anywhere round the body is $8 \times 10^{-7}$.

The claim that the coefficient does not depend on the tunnel is then tested directly, by running three tunnels.

The factor of 256 is $16^2$, which is the point: the pressures scale with the square of the speed and so does the thing they are divided by. Nothing about the shape entered that argument, so nothing about the shape can spoil it.

Where the pressures actually are, on a wing #

A cylinder is a convenient object with a symmetric distribution and no lift. On a lifting section the coefficient is where the whole argument about lift lives.

Sweeping the incidence shows how fast the suction peak grows. At zero degrees the minimum coefficient on this section is −0.741; at six degrees it is −2.345; at twelve degrees it is −7.444. The lift coefficient over the same range goes from 0.499 to 1.212 to 1.913 — so the lift roughly quadruples while the peak suction rises by a factor of ten.

That mismatch is the reason aerofoils stall. Lift is an integral over the whole surface and grows politely; the peak is a local quantity and grows violently, and it is the peak that the boundary layer has to survive. Everything in where the straight line stops is a consequence of these three numbers.

The instrument that lives on this definition #

The stagnation value being exactly one is not only a tidy fact. It is how aircraft know how fast they are going.

A Pitot tube is a tube pointing forward with a hole in the end. The air inside it is at rest, so the pressure there is the stagnation pressure, $p_\infty + \tfrac{1}{2}\rho U^2$. A static port elsewhere on the airframe reads $p_\infty$. Subtract them and the difference is the dynamic pressure, from which the speed follows.

Everything in that chain is the same relation the coefficient is defined by, run backwards: rather than dividing a measured pressure by a known dynamic pressure to get a shape-dependent number, an airspeed indicator measures the dynamic pressure directly and infers the speed. The failure modes follow from the derivation. A blocked Pitot reads the stagnation pressure it was left with, and the indicated speed then rises with altitude as the static pressure falls, which is the classic and lethal instrument failure. A static port in the wrong place reads not $p_\infty$ but the local pressure of the flow over the fuselage, which is why the position error of a static port is something aircraft carry a correction table for.

Reading a distribution #

A pressure distribution plotted against chord is the working drawing of a wing section, and it is worth knowing how to read one.

Three features carry the argument. The stagnation point is where the coefficient reaches one, and it moves: at zero incidence it sits at the nose, and as the incidence rises it slides onto the underside, so the flow has to come round the leading edge to get onto the top. That journey round a tight radius is what produces the peak.

The suction peak is the minimum, and it sets two things — the critical Mach number, and how much pressure recovery the boundary layer will be asked to perform. Deep peaks are expensive in both currencies.

The recovery is the climb from the peak back to something near ambient at the trailing edge. The lift is essentially the area between the upper and lower curves, so a designer would like the peak deep and the recovery gentle, and those two wishes are in direct conflict: a deeper peak means a longer climb over the same chord.

The distribution integrated is the force #

Nothing about the coefficient is merely descriptive. Integrating it over the surface with the local normal gives the force, and doing that for the cylinder produces the site’s founding embarrassment.

The symmetry visible in that curve is the whole of d’Alembert’s paradox. The pressure recovers completely: whatever was spent accelerating the flow round the shoulder is repaid decelerating it at the back. Nothing in the coefficient is wrong; what is wrong is the assumption that the real flow manages the recovery, and it does not.

That gives the pressure coefficient a second job it is rarely credited with. Comparing a measured distribution against the inviscid one is the most sensitive detector of separation available: the inviscid curve recovers to near +1 at the rear stagnation point, and a separated flow flattens out at some constant negative value instead. The gap between the two curves, integrated, is the pressure drag.

What the coefficient buys #

Three things, and the third is the one that makes wind tunnels possible at all.

Comparability. Two measurements taken at different speeds can be plotted on the same axes. A century of aerofoil data is usable today because it was published as coefficients rather than as pascals.

A bound to reason with. $C_p = 1$ at a stagnation point is a fixed point in every distribution, so a plotted curve that exceeds it anywhere is wrong, and a measurement that fails to reach it at the nose says the tap is in the wrong place or the flow has separated.

Scaling. A model at a tenth of full size in a tunnel at three times the speed produces the same coefficients as the real thing provided the dimensionless numbers match — which is the whole principle of testing, and also the thing that cannot be arranged for two numbers at once.

The three pressures, kept apart #

Half the confusion around this quantity comes from the word pressure doing three jobs, and the three are worth separating because instruments measure different ones.

Static pressure is what the fluid exerts on a surface moving with it — the thermodynamic pressure, the one in the equation of state. A tapping flush with a wall reads it, and it is what $p$ means everywhere above.

Dynamic pressure, $\tfrac{1}{2}\rho U^2$, is not a pressure at all in the same sense: it is a kinetic energy per unit volume that happens to have the units of pressure. Nothing anywhere in the flow need be at that value. It is a scale, and its job is to be divided by.

Stagnation or total pressure is the sum of the two, and it is the value the static pressure would reach if the flow were brought to rest without loss. It is constant along a streamline in a steady inviscid flow — that is Bernoulli’s statement in a different dress — and it is destroyed by exactly the processes that make the equation inapplicable: friction in a boundary layer, work done by a fan, entropy generated across a shock.

Total pressure is therefore the best available diagnostic of loss. A wake survey is a survey of missing total pressure, and the amount missing is what the drag was spent on. It is also why the site’s own attempt to read drag out of a wake fails on this grid: the method needs the static pressure to have recovered before the survey station, and behind a bluff body at these Reynolds numbers it has not.

The ceiling of one has two conditions on it #

The claim that the coefficient cannot exceed one is the essay’s strongest, and it is worth stating what it rests on, because both supports fail in places a reader will meet.

It rests on Bernoulli’s constant being the free stream’s. The derivation traced a streamline from far upstream, where the constant is $p_\infty + \tfrac12\rho U^2$, to a point where the speed is zero. Take a stagnation point whose streamline did not come from the free stream and the argument is simply unavailable. The rear stagnation point inside a separated region is exactly that case: the fluid there has been recirculating, it never carried the free stream’s constant, and its coefficient is not one. Measured base pressures are strongly negative, which is not a violation of anything — it is a streamline with a different constant on it, and it is why the ideal theory’s recovery to +1 at the back is the thing a real flow refuses.

And it rests on the flow being steady. Bernoulli’s unsteady form carries a term in the rate of change of the potential, and that term has no sign. A body that is accelerating can have a coefficient at its own stagnation point above one, and one that is decelerating below it, with no fluid moving faster than the stream anywhere. The bound is a property of steady flow rather than of pressure.

Both failures are diagnostic rather than embarrassing. A tap that ought to read one and reads less says either that the flow has separated upstream of it or that the measurement is not steady, and those are the two things anybody would most want to be told. What the bound cannot survive is being quoted as a law of nature: it is a consequence of two hypotheses, and naming them is what makes it useful when it breaks.

What the picture cannot show #

A coefficient field looks like a pressure field and is not one. Two flows with identical $C_p$ distributions can have wildly different absolute pressures, and the difference matters for anything structural: a wing panel does not care what fraction of the dynamic pressure is pushing on it, it cares how many kilonewtons.

The figures here also hide the sign convention trap that catches people reading published distributions. Aerodynamicists conventionally plot $C_p$ with the axis inverted, negative upwards, so that the suction side of a wing appears on top. A curve read off such a plot without noticing has the pressure distribution upside down.

Where the model stops #

The definition given here is the incompressible one, and it fails in a specific, well-signposted way as the Mach number rises. Once the flow is fast enough that density changes, $\tfrac{1}{2}\rho U^2$ is no longer the pressure rise to stagnation, so $C_p = 1$ stops being exact at a stagnation point. The compressible definition keeps the same denominator and accepts that the stagnation value drifts above one.

Worse, the minimum coefficient acquires a critical value. When the peak suction is deep enough that the local flow reaches the speed of sound, the section is at its critical Mach number, and everything downstream of that point is a different subject. The relationship between the −2.345 above and the speed at which that section can no longer be used is direct, and it belongs to the compressible field rather than to this one.

There is a third limit, quieter than the other two. The coefficient is defined against a free-stream pressure and speed, and in a flow with no free stream — inside a duct, in a rotating machine, near a ground plane where the reference is ambiguous — there is a choice to be made about what to divide by, and different fields have made it differently.

An aside on what “one” is worth #

Because the coefficient is dimensionless, it is easy to lose track of how large the forces it describes actually are, and a single conversion is worth carrying.

At sea level the density of air is about 1.225 kg/m³, so the dynamic pressure at a speed of $U$ metres per second is $0.61 U^2$ pascals. At 30 m/s — a fast cyclist, a light aircraft on approach — that is 550 Pa. At 250 m/s, an airliner in the cruise, it is 38 kPa, which is more than a third of an atmosphere.

So a $C_p$ of −3 at the shoulder of a cylinder in a 30 m/s wind is a suction of 1.7 kPa, or about 170 kilograms per square metre trying to pull the surface outward. That is the number behind roofs leaving houses in storms: the coefficient over a roof ridge is not far from −3, and the wind does not have to be extraordinary for the total to exceed what the fixings were designed for.

It also explains the shape of the design problem for a wing. The lift coefficient of a section in the cruise might be 0.4, and the peak suction coefficient perhaps −1.2. Both numbers are modest. Multiplied by 38 kPa, the first is a useful force and the second is a structural load, and the whole drag budget is arithmetic on quantities of that size.

Who found it, and when #

The grouping is implicit in Bernoulli’s 1738 relation and became a convention with the rise of systematic wind-tunnel testing in the first two decades of the twentieth century, when Prandtl’s group at Göttingen and the Royal Aircraft Factory in Britain both needed to publish results somebody else could use.

The habit of thinking dimensionlessly is older and better founded: it is Rayleigh’s, and the general argument for why it works — the Buckingham π theorem — was formalised in 1914, at almost exactly the moment the aerodynamicists were reaching for it out of necessity.

Where the ladder goes next #

Below this rung, fast means low pressure establishes the trade the coefficient measures. Beside it, d’Alembert’s paradox is the same distribution integrated to a force, and the reason it comes to zero is the fore-and-aft symmetry visible in the first figure here.

Above it are the two places the coefficient becomes the subject rather than a unit. One is compressibility, where a critical value of $C_p$ marks the end of the linear world. The other is separation, where the recovery from a peak of −7.4 back to ambient is more than a boundary layer can manage, and the flow leaves.