Nothing sucks

Every account of a wing says the same thing about its upper surface: it is under suction. The word is in the textbooks, on the figures, in the wind-tunnel reports, and it is doing real work — the suction peak is a genuine feature with a genuine location.

It also carries an implication that is false. Suction sounds like pulling, and a fluid cannot pull. A gas exerts a normal compressive stress and nothing else; there is no mechanism by which the air above a wing takes hold of it.

What is actually happening is a smaller push against a larger one, and it is worth doing the arithmetic, because the numbers are not close.

Where the minus sign comes from #

Every pressure on this site is normally reported as a coefficient,

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

which is a gauge pressure — a difference from the free stream — divided by the dynamic pressure. Over the upper surface of a lifting wing it is negative, reaching $-2.2$ on the shape drawn above.

That negative number is where the vocabulary comes from, and it is a bookkeeping convention. What it says is that the pressure there is below the ambient, which is true and useful. What it does not say is that the pressure is below zero, which would be a different claim entirely.

Convert it back with the numbers of an actual flight and the difference is stark. At $U = 60$ m/s and sea-level density, the dynamic pressure is 2.2 kPa, so a $C_p$ of $-2.2$ is a dip of 4.8 kPa in an ambient of 101.3 — five per cent.

The identity that settles it #

There is a stronger argument than the numbers, and it takes one line of vector calculus.

For any closed surface,

$$\oint \mathbf{n}\,\mathrm{d}S = 0$$

— the outward normals of a closed body sum to nothing, for every shape whatever. So a uniform pressure exerts no net force on anything closed, and therefore

adding a constant to a pressure field changes no force on any closed body.

Whether the pressure over the wing is called $-4.8$ kPa or $96.5$ kPa is a choice of datum, and the lift is the same either way. A quantity whose sign depends on where zero was put cannot be a mechanism.

What the solver computed, and how it was checked #

Three things, and the third has an interesting failure mode.

The identity, to machine precision. $\oint\mathbf{n}\,\mathrm{d}S$ over a 1200-point Joukowski outline comes to $3\times10^{-15}$, and the check first verifies that the outline actually closes — an open outline has no such identity, and the assertion refuses one whose ends are further apart than $10^{-9}$.

The two lifts. Integrating $-p\,\mathbf{n}$ over the surface with gauge pressures gives $9728.628291127$ and with absolute pressures $9728.628291128$. Twelve significant figures, on integrands that differ by 101,325 at every point.

The absolute pressure must be positive. The check refuses a state whose lowest absolute pressure is negative, because that is not a pressure a fluid can have — and the refusal fires on a real case, not a contrived one: the same aerofoil in water at ten metres a second comes out at $-8.6$ kPa, which means the model has run past the point where the liquid stays liquid.

Where the push does run out #

That refusal is the interesting part, because it locates the one place where the intuition of pulling has something behind it.

A fluid can be pushed less and less, but not past zero — and in a liquid, not even that far. Below the vapour pressure the liquid boils, and at 20 °C water’s vapour pressure is 2.3 kPa, so the margin is 99 kPa rather than 101.

So the difference between air and water here is not the fluid’s willingness to be pulled — neither is willing — but the size of the margin relative to the dynamic pressure. Water’s density is 815 times air’s, so at the same $C_p$ it takes 28 times less speed to use up the same ambient.

That is precisely the cavitation number of the applied field,

$$\sigma = \frac{p_\infty - p_v}{\tfrac{1}{2}\rho U^2}$$

and cavitation begins when $\sigma$ falls to $-C_{p,\text{min}}$. The whole of that field’s opening argument is this essay’s arithmetic run until the margin is exhausted.

How small the difference has to be #

The scale of the whole business is worth fixing with one number.

A wing carries its aircraft’s weight over its area, and the pressure difference required is the wing loading times $g$. For a light aircraft at 80 kg/m² that is 0.78 kPa; for an airliner at 600 kg/m² it is 5.9 kPa. Under six per cent of an atmosphere, for the heaviest wing in ordinary service.

So the picture to hold is not of a wing being pulled upwards by a partial vacuum. It is of a slab of air pressing on the wing from below at 101.3 kPa and from above at 95.4, with the difference — one part in seventeen — being the entire lift. A wing is a device for making a very small difference between two very large numbers, which is exactly why the difference is quoted as a coefficient and the absolute values are almost never mentioned.

That framing also explains the split between the surfaces. On the solved aerofoil here, integrating separately, 75 per cent of the lift comes from the upper surface and 25 from the lower. The upper surface contributes more not because it is pulling harder but because its departure from ambient is larger: the flow has to speed up more to get round the top than it slows down underneath, and Bernoulli’s relation turns that asymmetry into pressure.

Why fluid moves towards low pressure, which is a push as well #

The essay so far is about the force on a body. The same objection is usually raised about the fluid — air is said to be drawn towards a region of low pressure, into a nozzle, up a hose, out of a hole — and the answer is the same one, taken one level down.

A parcel of fluid accelerates according to

$$\rho\,\frac{D\mathbf{u}}{Dt} = -\nabla p,$$

and a gradient is a difference between the pushes on a parcel’s two opposite faces. The parcel has no way of knowing that there is a low pressure somewhere off in the distance; it knows what is touching it. If the face behind is pushed harder than the face in front, it accelerates forward. That is the whole mechanism, and there is nothing in it but contact.

Follow what happens when a hole is opened to a vacuum and the picture becomes concrete. The layer of gas at the hole is unbalanced — atmosphere on one side, nothing on the other — so it is pushed out by its neighbour. That leaves the neighbour unbalanced, so its neighbour pushes it. The disturbance travels outward from the hole one layer at a time, as a rarefaction wave, at the speed of sound, and gas far away carries on as it was until the wave arrives. Nothing reached out for anything. The vacuum did not act; the atmosphere acted, in sequence, on itself.

That finite speed is the tell, and it is what distinguishes a chain of pushes from a pull. A genuine attraction would act at a distance and would need no messenger; this needs one, it travels at a definite speed, and everything about the domain a signal can reach follows from that. The incompressible idealisation makes the messenger infinitely fast, which is exactly why an incompressible pressure field appears to act everywhere at once — and the appearance is a limit rather than a mechanism.

The same reading applies to every everyday case the word suction is used for. A vacuum cleaner does not pull the dust; it lowers the pressure inside, and the atmosphere pushes the dust in behind it. A jet does not pull the surrounding air into itself; it accelerates the fluid it touches, the pressure near it falls, and the ambient pushes more fluid in. Air does not flow towards a depression because the depression attracts it; each parcel on the way is pushed from behind by the parcel behind it.

So the general statement is stronger than the one about closed bodies, and simpler. A fluid transmits force only by contact, and every apparent pull in this subject is a chain of pushes with a messenger travelling at a finite speed. Where a fluid seems to reach out, what has happened is that the chain arrived faster than anybody was watching.

Why it matters, beyond pedantry #

Three consequences that follow from taking the pushing seriously.

It explains why a wing does not need the air above it to be attached to anything. If the upper surface were pulling, one would need a mechanism of adhesion, and the usual attempt to supply one — that the air “sticks” to the surface and is therefore pulled along with it — confuses the no-slip condition, which is about tangential velocity, with a normal force, which it is not.

It explains the failure mode. A structure loaded by suction fails when the pressure difference is large, and the largest available difference is one atmosphere, however fast the aircraft is going — for as long as the flow is incompressible. That is a useful bound: a panel on the upper surface can never see more than about 101 kPa of net outward load from the aerodynamics alone.

It explains cavitation as an ordinary consequence. If suction were a pull there would be no reason for a liquid to boil at ordinary temperature; it happens because the push has been reduced to the point where the liquid’s own vapour can hold the boundary.

The pull that does exist, and is not this #

There is one place in fluid mechanics where a genuine tension appears, and it is worth naming so that the general claim can stay clean.

A liquid can sustain a negative absolute pressure for a while. Water in a sealed tube with no nucleation sites has been held at tens of megapascals of tension in the laboratory, and the sap in a tall tree is under tension of a few megapascals as a matter of routine. What makes that possible is that a liquid has cohesion between its molecules — a real attraction, with an energy per unit area that shows up as surface tension.

None of that applies to the case in this essay. A gas has no cohesion worth the name; its molecules are free, and the only stress it can transmit is the momentum flux of molecules arriving on a surface, which is a push. And a liquid under tension is metastable: the moment a nucleus appears, it boils, which is what cavitation is. Real water round a real propeller has plenty of nuclei, so it cavitates at its vapour pressure rather than at any tension at all.

So the honest general statement is: a gas cannot pull, a clean liquid can pull briefly and metastably, and no fluid in any engineering flow this site describes pulls on anything.

What the picture cannot show #

The atmosphere is not drawn. Every figure here is a slice through a fluid that extends upwards for kilometres, and the reason the ambient is 101 kPa is the weight of all of it. Nothing in these pictures shows where the number comes from.

Nothing here is compressible. At 274 m/s — the speed at which the incompressible calculation says the absolute pressure reaches zero — the Mach number is 0.8, and the whole calculation has been wrong for a long time. The right statement is that air never approaches zero absolute pressure on a wing, because compressibility changes the answer well before it could.

And the wing is inviscid. A real upper surface has a boundary layer, and the pressure through a boundary layer is essentially constant across it, so the surface pressures drawn here are close to right until the flow separates. After separation the whole distribution changes and the peak disappears — which is what a stall does.

The same reasoning, applied to a familiar object #

A vacuum cleaner is worth thirty seconds because it is the everyday example everybody reaches for and it makes the point better than the wing does.

The machine does not pull air up the hose. It lowers the pressure inside, and the atmosphere pushes air in — which is why a vacuum cleaner cannot lift anything with more than an atmosphere of force per unit area behind it, and why the same machine at the top of a mountain is weaker. The limit is set by the ambient rather than by the motor, and a stronger pump past the point of a perfect vacuum buys nothing at all.

The same reasoning caps every siphon, every suction cup and every straw at roughly ten metres of water. None of those is pulling either, and each has a hard limit that a pulling mechanism would not have.

A wing’s limit is of the same kind but is never reached, because the compressibility of air changes the problem long before the pressure runs out — which is the difference between an aerodynamic argument and a hydraulic one, and the reason cavitation is a marine engineer’s problem and not an aeronautical one.

Where the model stops #

The identity $\oint\mathbf{n}\,\mathrm{d}S = 0$ holds for closed bodies, so everything above is silent about anything that is not closed: a wing with an open cavity, a body venting to the atmosphere, a sail with two sides connected round its edge. In each of those the datum can matter, because the “body” being integrated over is not a closed surface.

The argument is also about force, not about stress. A local structural stress absolutely does depend on the absolute pressure — a panel with a vacuum behind it and one atmosphere in front is loaded, and a panel with 96 kPa on one side and 101 on the other is loaded much less. The datum cancels in the total force on a closed body and does not cancel anywhere else.

And nothing here says the word “suction” should be abandoned. It is a useful name for a region of below-ambient pressure and every aerodynamicist uses it. What it should not do is carry the implication of a pull, and the difference matters the moment somebody tries to explain why a wing works.

Who found it, and when #

The physics is old and uncontroversial. That a fluid at rest exerts only a normal compressive stress is in Pascal, and that a uniform pressure exerts no force on a closed body is a corollary of the divergence theorem, which arrives with Gauss and Ostrogradsky in the 1820s and 1830s.

What is modern is the vocabulary. “Suction” as a term for below-ambient surface pressure is standard by the time of the earliest wind-tunnel reports, and by then the misreading is already in the popular accounts. It survives because the word is genuinely useful and because nothing about the mathematics objects to it — only the explanation built on top of it is wrong, and that explanation is not in the technical literature at all. It is in the retellings.

Cavitation, the one place where the floor is reached, was identified in the 1890s from the propellers of fast ships — Barnaby and Parsons on HMS Daring and Turbinia — where the blades were eroding for reasons nobody could explain until Reynolds and Parsons connected the damage to the boiling of water at ordinary temperature.

Where the ladder goes next #

This field now has equal transit time, Bernoulli misapplied, the momentum sum done badly, Newton’s sine-squared law and the word “suction” all tested. What they have in common is worth naming: each is a correct piece of physics carried somewhere it does not apply, and none of them is refuted by an experiment that the original argument could not have predicted. They are refuted by arithmetic on a solved flow, which is the one thing every retelling of them omits.