Not half a venturi

Of all the wrong explanations of lift, this one is the hardest to argue with, because almost every sentence in it is true.

The air above a wing is in a narrower channel than it was upstream. It does move faster. Its pressure does fall. Continuity is exact. Bernoulli’s theorem does hold along a streamline in a steady incompressible flow. Nothing in the chain is a mistake.

And it is not an explanation.

What a venturi has that a wing does not #

Start with the thing being compared to.

A venturi is a duct with walls. The walls are given: somebody made them out of brass, they do not move, and the cross-sectional area at every station is a number read straight off a drawing. Continuity then determines the velocity, Bernoulli determines the pressure, and the whole calculation runs forward from data.

A wing has one wall. The lower boundary of the “channel” above it is the wing’s upper surface, which is given. The upper boundary is a streamline, and where that streamline goes is not given by anything. It is part of the solution.

So the venturi argument, applied to a wing, has the shape: the streamlines arrange themselves like this, therefore the velocity is like that, therefore the pressure is like that, therefore there is lift. Every arrow is valid. The first premise is the thing being explained.

That is what makes it circular rather than wrong. It is a correct redescription of the answer, offered as a reason for the answer.

The tube is narrowed everywhere, including where there is no wing #

The clearest evidence that the constriction is not a duct effect is that it does not stop at any particular height.

Trace pairs of streamlines starting at increasing distances above the wing and measure how much each tube narrows as it passes over the section. Close to the surface the contraction is large. It falls off slowly, and it is still there at two and three chords, where the nearest solid object is a long way away and there is nothing whatever forming a channel.

The reason is that an incompressible pressure field is elliptic: the wing’s influence reaches everywhere at once, falling off as a power of distance rather than stopping at some boundary. The streamlines rearrange themselves throughout the whole flow, and the “narrowing” is a description of that rearrangement at whatever height it is measured.

A duct’s influence stops at its walls. A wing’s does not, and the constriction is a property of the flow rather than of the geometry.

The relation the argument uses is exactly true and says nothing #

It is worth checking the arithmetic the story rests on, because the check comes out perfectly and that is the point.

Measure the tube width and the local speed along a streamtube over the wing. They lie on the diagonal: the speed is exactly the inverse of the area ratio. Continuity is not approximately satisfied, it is an identity, and it holds along any streamtube in any incompressible flow whatever caused it.

There is a sharper way to put it. Continuity relates two unknowns — the tube’s width and the speed in it — and determines neither. To get a number out of it, one of the two has to come from somewhere else, and in a venturi it comes from the drawing. Over a wing there is no drawing, and the width has to be computed from the same solution that already contains the speed and the lift. The relation is being used as though it supplied information, and what it supplies is a consistency check that any correct answer passes automatically.

A relation that holds in every flow constrains nothing about a particular one. That is the same objection this site raises to the Coandă explanation — an identity that is true everywhere cannot explain anything in particular — and it is the same objection with a different identity in it.

A constriction with no lift #

Now the first of the two refutations, and it is the simplest possible one.

Take a symmetric section and put it at zero incidence. There is a constriction above it. Every step of the venturi argument applies to it in full: the tube narrows, the speed rises, the pressure falls, and there is a suction on the upper surface.

The lift is zero.

Because the identical thing is happening underneath, and the argument has no way of noticing. It is an argument about one surface, and lift is a difference between two. Anything a section does to the air above it, a symmetric section at zero incidence does equally below, and the venturi story — which is complete without ever mentioning the lower surface — cannot distinguish the case that lifts from the case that does not.

That is not a technicality. It says the argument is not merely circular but incomplete in its premises: it has no term in it that could be responsible for the asymmetry, which means it has no term in it that could be responsible for lift.

A lifting surface with no channel at all #

The second refutation goes the other way, and the site’s exact machinery supplies it without any special pleading.

A circle centred on the origin, taken through the Joukowski map, becomes a straight segment. The same flow.js routine that draws every cambered section on this site produces, with the camber and thickness parameters both set to zero, a flat plate — with the Kutta condition applied, an exact velocity field, and a lift coefficient of $2\pi\sin\alpha$.

A flat plate has no thickness. It has no upper surface and no lower surface, only two sides of one line. It has no curvature to squeeze anything against, and no channel above it that is narrower than the channel below it — the gaps between any pair of streamlines above and below are what the solution made them, and the plate itself contributes no geometry to either.

And it lifts. At five degrees the exact solution gives 0.548, which is a perfectly respectable lift coefficient and is very nearly what a cambered section makes at the same angle.

The venturi argument has nothing to point at. There is no constriction because there is nothing to constrict with, and the lift is there anyway.

The version with a real wall, and what it shows #

There is a way to make the story literally true, and doing it is instructive because of what has to be given up.

Put an actual wall where the streamline was. Take the streamline that passes a fixed distance above the section, replace it with a rigid surface, and the flow inside is now a genuine duct: the walls are given, the areas are data, and continuity and Bernoulli run forward exactly as in a venturi.

The flow does not change — a streamline can always be replaced by a wall in an inviscid flow without altering anything, which is the trick the image system and the half-body are both built on. So the numbers are identical and the argument is now valid.

And it has stopped being about a wing. What has been constructed is a wing in a tunnel of a particular shape, and the constriction is now genuinely a cause — of the flow in that tunnel. Change the wall’s position and the flow changes, which is exactly the blockage correction a real wind tunnel needs. Take the wall to infinity and the constriction persists, as the reach figure shows, which is the signature that it was never the wall doing it.

The story becomes true at the moment it stops being about the aeroplane. That is a precise statement of what is wrong with it and it is more useful than saying it is circular.

What is actually going on #

The site’s own account is circulation, and it is worth stating what makes it an explanation where the venturi story is not.

The lift is $\rho U \Gamma$, and Γ is fixed by the Kutta condition — one statement about the flow at the trailing edge, which is enough to select one member of the one-parameter family of ideal flows round the section. That chain has a premise which is not the conclusion: the Kutta condition is a statement about what a viscous fluid does at a sharp corner, and it is not derivable from the potential flow it selects.

The streamtube narrowing is then a consequence. Once the circulation is fixed, the whole velocity field is fixed, the streamlines are where they are, and the tube above is narrower than the tube below. Everything the venturi story observes is correct and is downstream of the answer.

The test for whether an account explains is whether it has an input that is not the output, and that is a better criterion than any amount of arguing about intuition.

Press it for a location and it does make a prediction #

The story is safe from measurement only while it stays qualitative. Ask it one more question — where on the wing is the pressure lowest? — and it has to answer, because a channel’s narrowest point is the one thing a duct argument is unambiguous about.

The answer it gives is the point of maximum thickness, since that is where the section bulges furthest into the air above it and the gap is smallest. For an ordinary section that is a quarter to a third of the way back from the leading edge, and it is a fixed property of the shape: it does not move when the wing is tilted, because the shape has not changed.

The measured suction peak is not there. On a lifting section it sits close to the leading edge, in the first few per cent of the chord — and it moves forward and deepens as the incidence rises, while the point of maximum thickness stays exactly where the draughtsman put it. By ten degrees the minimum pressure is within a per cent or two of the nose, at a station where the section is barely thicker than the plate this essay has already shown lifts without any thickness at all.

The mechanism is the one the venturi story has no term for. What produces that peak is the flow whipping round the curvature of the nose to satisfy the trailing-edge condition, so its depth is governed by the incidence and by the leading-edge radius — and by neither the thickness nor its location. A thinner nose gives a sharper, deeper peak at the same lift, which is the opposite of the duct reading, where less bulge should mean less constriction and a gentler suction.

That has consequences the story cannot reach either. The leading-edge peak is what stalls a wing: the layer separates where it must climb the steepest pressure rise, and the steepest rise is immediately behind the sharpest peak. So the location the venturi argument gets wrong is precisely the location that decides the wing’s maximum lift, and a designer working from the story would set out to reduce the thickness bulge when the quantity to manage is at the nose.

And there is one configuration where the story locates the suction correctly. On a symmetric section at zero incidence, with no circulation to whip the flow round the nose, the minimum pressure really does sit near the point of maximum thickness — above and below, equally. It is the case from two sections ago: the one arrangement in which the venturi picture is quantitatively right is the one that makes no lift.

Why the story survives #

Three reasons, and they are worth knowing because they are the reasons most durable wrong explanations survive.

It uses correct physics. Continuity and Bernoulli are both exactly right, and a reader checking the steps finds nothing wrong. The error is structural rather than factual, and structural errors are much harder to see.

It has a picture. Streamlines over a wing really do bunch up, every visualisation shows it, and a picture that matches the story is very convincing. This site’s own caution about that applies: a smooth picture proves nothing, and a picture consistent with an explanation is not evidence for it.

And it makes a correct prediction about one case. In a wind tunnel with the model close to the walls, the tunnel is a duct, the blockage is a constriction with real walls, and the flow really does speed up for the reason the story gives — which is why tunnel corrections exist. A story that is right about the apparatus and wrong about the aeroplane is a difficult one to dislodge.

What to say instead #

A refutation is more useful with a replacement, and the replacement has to be sayable in the same number of sentences or it will not be used.

The short version. A wing turns the air. Turning air requires a force, and the force on the air is downward, so the force on the wing is upward. The pressure difference is how that force is delivered, and the amount of turning is fixed by the sharp trailing edge, which the flow cannot go round.

That has an input which is not the output — the sharp trailing edge — and it is the ordinary account this site derives at length. It also connects immediately to two things a reader can check: a wing with a rounded trailing edge lifts far less, and a wing at zero incidence with no camber lifts nothing.

What not to do is replace one slogan with another. Lift is circulation is true and is not an explanation either unless the listener knows what fixes the circulation. The explanatory content is entirely in the Kutta condition, which is the one piece of the story that is not deducible from the rest and is therefore the piece worth the sentence.

What the model does not contain #

No viscosity. Every field here is an exact potential flow. That is not a limitation for this argument — the Kutta condition stands in for the viscous effect that matters — but the flat plate’s infinite leading-edge velocity is a model artefact and a real plate at five degrees would have separated at the nose.

Two dimensions. Everything is a section.

The streamtubes are traced by integration and the widths are read off at stations, so the narrowing percentages carry the integrator’s error. They are not close calls: a fifth is a fifth however it is measured.

And the flat plate is a limit rather than an object. A physical plate has an edge radius, a thickness and a surface, and its lift is close to but not equal to the exact zero-thickness result. What the calculation shows is that the limit lifts, which is what the refutation needs.

Who said it, and when #

The venturi analogy appears to have entered aeronautical teaching early and from wind-tunnel practice, where the analogy is genuinely apt for the apparatus. It is in flight-training material throughout the twentieth century and remains in a large fraction of it today, frequently alongside the equal-transit-time story, which is a different error and is often presented as though the two were one.

They are worth separating. Equal transit time makes a false prediction — it says the flow over the top arrives at a particular time, and it does not — so it can be refuted by measurement. The venturi story makes no false prediction at all: everything it says is true, and the objection is that its premise is its conclusion. The two require completely different kinds of argument, and a refutation of one is not a refutation of the other.

A story that cannot be refuted by measurement is not thereby a good one, and that is the general lesson worth carrying away. Circular explanations are consistent with every observation, which is precisely what is wrong with them.

Where the ladder goes next #

The next misconception is about the other end of the lift curve. A wing is said to stall at a speed, that speed is printed in the handbook and marked on the airspeed indicator, and the wing does not know what speed it is going.