A force forgets the datum, a stress cannot

The rung below this one proves something clean. A fluid cannot pull; what is called suction on a wing’s upper surface is a lower push against a higher one; and the demonstration is that the force does not depend on where the pressure datum is — because the outward normals of a closed body sum to zero, so adding any constant to the pressure everywhere adds nothing at all.

That identity is exactly true and it is about a resultant. It is not about anything else, and the step that is routinely taken next — that the absolute pressure therefore does not matter — is a step too far.

Where the constant goes when it cancels #

It is worth watching the cancellation happen, because seeing what it consumes shows what it leaves.

The force on a body is $-\oint p\,\mathbf{n}\,dS$. Write $p = p_{\text{ref}} + p_{\text{gauge}}$ and the integral splits:

$$-\oint p_{\text{ref}}\,\mathbf{n}\,dS \;-\; \oint p_{\text{gauge}}\,\mathbf{n}\,dS.$$

The first term is $p_{\text{ref}}$ times $\oint \mathbf{n}\,dS$, and that integral is zero for any closed surface — a purely geometric fact with no fluid in it. So the datum leaves.

What made it leave is the closure of the surface. The cancellation is not a property of pressure, of Bernoulli, or of aerodynamics; it is a property of integrating a constant vector field’s flux over something with no boundary. Break the closure and it does not happen.

Breaking the closure #

A panel of skin is not a closed surface. It has an outside and an inside, and its equilibrium is local: the net load is the pressure on one face minus the pressure on the other.

$$q_{\text{net}} = p_{\text{inside}} - p_{\text{outside}}.$$

There is no datum in that expression to cancel. Both terms are absolute pressures and the answer is their difference — and if only one of them moves, the answer moves.

The numbers are worth reading rather than skimming. At sea level, a wing at sixty metres a second with seventy-five kilopascals inside has its upper skin pushed outward at 28.3 kilopascals — nearly three tonnes per square metre — and its lower skin near the stagnation point pushed inward.

And the resultant of those loads is the lift again, computed from the same numbers, unchanged. The figure and the identity are consistent and they are answering different questions.

The three flights that make the point #

The cleanest demonstration is three flights with identical aerodynamics.

Hold the dynamic pressure constant and climb. The true airspeed rises as the density falls, so every pressure coefficient is unchanged, every gauge pressure is unchanged, the lift is unchanged, and the airspeed indicator reads the same number because it measures dynamic pressure. From the wing’s point of view nothing whatever has happened.

The absolute pressures have all fallen, because the ambient has. The lowest pressure on the wing goes from 96.5 kilopascals at sea level to 49.2 at five kilometres to 21.6 at ten.

So:

• The lift is identical, computed from those absolute pressures rather than asserted — which is the identity, checked.
• The load on a vented panel is identical, at 4.86 kilopascals, because a vent puts ambient on the inside and both faces moved together.
• The load on a sealed panel goes from 28.3 to 53.7 kilopascals between sea level and twelve kilometres, because the inside did not move.

Three flights, one aerodynamic answer, and a structural load that nearly doubles.

What this is actually about, structurally #

Two real cases, and both are ordinary rather than exotic.

An integral fuel tank. A wing box holding fuel is sealed, vented to the atmosphere through a small line for tank breathing, and the vent’s job is precisely to keep the inside and the outside together so this does not happen. A blocked vent is a serious defect, and the reason is the difference between the two curves above. It is the same reason an aircraft’s static ports must be clear: a reference pressure that has stopped tracking the ambient is an instrument reading a difference from the wrong thing.

A pressurised fuselage. The cabin holds about 75 kilopascals while the outside falls to 22, and the resulting hoop stress is the design case for the whole structure. That load has nothing to do with aerodynamics — the aircraft would carry it sitting still on the ground with the cabin pumped up — and it is why a fuselage is a pressure vessel that happens to fly.

Neither of those loads appears in any lift calculation, and neither is affected by anything the lift calculation says. The two live in the same skin and are computed from the same pressures by different integrals.

Why the misreading is so easy to make #

Three reasons, and they are worth separating because two of them are good habits producing a bad result.

Coefficients are the right currency and they have no ambient. Working in $C_p$, $C_L$ and $C_m$ is correct, universal and scale-free — it is what makes a model in a tunnel say anything about an aeroplane — and every one of those quantities is a gauge pressure over a dynamic pressure. A person who has worked in coefficients for a decade has genuinely never needed an ambient, and the habit is not a mistake.

The demonstration that nothing sucks is a demonstration about a force. Its whole persuasive power is that the datum vanishes, and the conclusion drawn from it — that a low pressure is not a pull — is correct. What is easily carried along is the stronger and false statement that the absolute pressure is not a physical quantity at all, and the rung below’s own argument does not license it.

And gauge pressure is what instruments read. A pressure tap, a manometer, a transducer with one port open to the room: all of them subtract the ambient before anybody sees a number, so the ambient is absent from the data as well as from the theory. Recovering it means adding a barometer reading, which is a separate instrument and a separate habit.

The one place all three break down together is a question about the fluid’s state, and there are only a few of those: will it cavitate, will it boil, will it condense, will it liquefy, is it still a continuum. Every one of them needs an absolute pressure and every one of them is a question a force calculation never asks.

The same wing, three speeds, one datum #

The altitude sweep held the dynamic pressure and moved the ambient. The complementary experiment holds the ambient and moves the speed, and it separates the two contributions cleanly.

At sea level, going from sixty metres a second to eighty raises the dynamic pressure by 78 per cent, so every gauge pressure rises by that factor and the lift does too. The worst panel load on a sealed wing goes from 28.3 kilopascals to 30.0 — a rise of six per cent.

The load barely moved because most of it was never aerodynamic. Of the 28.3, about 26 kilopascals is the difference between the ambient and the tank, and only 4.9 is what the flow contributed. Speeding up by a third changed the small part.

That is the practical version of this essay’s whole point. On a sealed structure at low speed and high altitude, the aerodynamics is a correction to the pressure difference rather than the source of it — and a structural engineer who took the aerodynamicist’s coefficients and worked in gauge would have computed the correction and missed the load.

Where else the same over-reach happens #

The pattern — an identity about a resultant, read as an identity about everything — is worth naming because it recurs.

The moment. The pitching moment about any point is also datum-independent, for the same reason, which is why moment coefficients are quoted without an ambient. But the moment about a point is a resultant too, and the local twisting of a panel is not.

A pressure coefficient. $C_p$ is a gauge pressure over a dynamic pressure and it contains no absolute pressure at all — which is exactly right for computing forces and exactly wrong for asking whether the flow will cavitate, which the cavitation number is for and which needs the ambient — and which is the same margin a siphon’s crown is measured against.

And a wind-tunnel measurement. Pressure taps read gauge, referenced to the tunnel’s own static pressure, and every force derived from them is correct. Any question about the absolute state of the air — cavitation, boiling, condensation in the tunnel’s own expansion — needs a number the tap did not report.

The rule that covers all of them: the datum cancels in any integral over a closed surface and in no other calculation. Anything local, anything one-sided, anything about the state of the fluid rather than about the force it exerts, keeps it.

The identity’s other half, which the rung below also proves #

There is a second consequence of $\oint \mathbf{n}\,dS = 0$ that this ladder has not stated, and it is worth having beside the first because it is the same fact answering a question about buoyancy.

If a constant pressure exerts no net force, then a body in a fluid at rest with no pressure variation at all feels nothing — which is why a balloon in a sealed box on the ground floor of a building is not pushed anywhere. What produces buoyancy is not the pressure but its gradient: the hydrostatic variation with height, which makes the pressure on the bottom of a body larger than on the top by $\rho g h$, and the resultant is Archimedes’.

So the same identity that makes a datum irrelevant makes buoyancy possible, by leaving only the part of the pressure field that varies. A uniform field contributes nothing and a linear one contributes the displaced weight.

That is a useful thing to hold beside this essay’s argument because it says what survives a cancellation. When a constant goes, what is left is everything the constant was not — and in the panel case above, what is left is a difference between two sides rather than a variation over one.

And it identifies the third question the datum matters for. A force needs the gradient; a state needs the absolute value; a local equilibrium needs both sides. Only the first of those three cancels, and the rung below proves exactly that one.

Where the two calculations meet, and where they should #

There is a place in a design where the two integrals in this essay have to be done on the same pressures, and it is worth naming because it is where the handover this essay is about actually happens.

A wing’s structural loading case is a pressure distribution, not a lift. The spar caps are sized by a bending moment, which is an integral of the pressure weighted by distance along the span; the ribs by a local load; the skin panels by the difference across them. Every one of those is a different weighted integral of the same field, and only one of them — the total force — is the quantity the aerodynamicist reports.

So the honest deliverable is the field rather than the coefficient, and modern practice is exactly that: a pressure distribution, at absolute values, at a stated condition, handed over as data rather than as three numbers. That is a comparatively recent arrangement and it exists because the numbers that were being handed over were not sufficient.

The essay’s argument is then a statement about what is lost in a summary. A lift coefficient is a perfectly good summary of a pressure field for the one question it summarises, and it is silent about every other integral of the same field — including the one that decides whether the skin holds.

Which is a general caution about coefficients rather than a complaint about this one. Every non-dimensional group discards something, the thing it discards is chosen for a purpose, and using it for a different purpose is where the discarding shows.

What the picture cannot show #

No structure is analysed. The figures compute a pressure difference across a skin. Turning that into a stress needs the panel’s dimensions, its support conditions and its material, none of which is here, and the numbers above are loads rather than stresses.

The section is exact and inviscid. Every pressure is from a Joukowski solution with the Kutta condition applied, so there is no boundary layer, no separation and no wake — and near the trailing edge a real section’s pressures depart from these.

The atmosphere is the standard one. The altitude sweep uses the standard atmosphere’s pressure and takes the dynamic pressure as held; a real climb at constant indicated airspeed does exactly that, which is why the case is the ordinary one rather than a contrivance.

And the cabin pressure is a constant. A real cabin is scheduled — it falls with altitude to a cabin altitude of a couple of kilometres and then holds — so the sealed curve above is a caricature of the schedule rather than any aircraft’s.

The assertion behind these figures is the one that could reject, and it makes three demands at once: the lift must be identical at every altitude, computed from the absolute pressures; the vented panel’s load must be identical too, to a part in a billion; and the sealed panel’s must change by at least a fifth, because a demonstration in which nothing moved would demonstrate nothing.

Who said it, and when #

The identity $\oint \mathbf{n}\,dS = 0$ is elementary and old, and the practice of computing aerodynamic forces in gauge pressure — or, better, in coefficients — is as old as the pressure coefficient itself.

What is recent is the separation of the disciplines. In the era when the same person sized the structure and computed the lift, the distinction this essay is about did not need stating, because both calculations were in the same notebook and both sets of pressures were in view. Aerodynamics reporting coefficients to a structures group is a twentieth-century arrangement, and it is the arrangement in which the ambient can quietly go missing: a coefficient carries no ambient by construction, and a recipient who needs one has to know to ask.

That is the same handover the recovery ladder describes on the thermal side, where the fluid supplies two numbers and no temperature, and it has the same failure mode — an error crossing the boundary with no way to notice it on the far side.

A number for the whole aeroplane #

The panel figures are per square metre and the quantity that decides a design is a total, so it is worth carrying the arithmetic one step further.

A narrow-body airliner’s fuselage is roughly four metres across and thirty long, with a skin area of about 380 square metres. At cruise the differential is about 55 kilopascals, so the total load the pressurisation puts into that skin is of order twenty meganewtons spread over it — two thousand tonnes of force, held by aluminium a millimetre or two thick, in an aircraft whose entire weight is seventy tonnes.

The pressurisation load is therefore an order of magnitude larger than the weight, and it is completely absent from every aerodynamic calculation the aircraft’s shape was arrived at through.

The wing case is smaller and the same in kind. A wing with 120 square metres of skin and a worst differential of 28 kilopascals carries several meganewtons of net panel load, against a lift at cruise of about 700 kilonewtons — so the load that presses the skin outward is several times the load that holds the aircraft up, and only the second of the two is what anybody means by “the aerodynamic load”.

That comparison is the reason this rung exists. The identity in the rung below is correct and it disposes of the largest number in the problem, which is fine as long as nobody needed it — and the people who needed it are one office away.

Where the ladder goes next #

The rung above is the case where a fluid can be pulled, and it is the one exception this ladder has been circling.

Water that has been carefully degassed sustains tension — a genuinely negative absolute pressure — because there is nothing in it for a cavity to nucleate on, and the tensile strength of pure water is tens of megapascals in careful experiments. That is not a low push; it is a pull, and every sentence in the rung below about a fluid being unable to pull has an exception with a measured value.

Where it happens matters more than that it does. Trees lift water a hundred metres up a trunk at pressures well below zero, in vessels narrow enough and clean enough that nucleation never starts; the cohesion-tension theory is the standard account of it and it is a fluid mechanics of negative pressure. Computing the profile of that column, and the margin it holds against the cavitation the rung below prices, is a rung with a real calculation in it and an object nobody would have predicted.