The effect that is real, and where it stops

This collection has an essay saying that the Coandă effect does not explain lift, and it is right. What it leaves out is the half a reader is entitled to: what the effect actually is, and how to tell the two situations apart with a number rather than an assertion.

One mechanism, two flows #

A streamline that curves has a pressure gradient across it. That is the normal momentum balance,

$$\frac{\partial p}{\partial n} = \frac{\rho q^2}{R},$$

with $n$ pointing away from the centre of curvature, and nothing about it is in dispute. It applies to a wall jet on a cylinder and to a wing equally, and it is the same statement as fast meaning low pressure written across streamlines rather than along them.

So the mechanism the story names is real, and pointing out that it is real is not a defence of the story. The question is where the integral of that gradient comes from.

A body in a stream #

Take the ray straight up from the top of a cylinder in a uniform stream. By symmetry the flow there is exactly horizontal, so the ray is exactly normal to the streamlines and the balance is one term.

Integrating $\rho q^2/R$ outwards from the surface to infinity gives exactly $1.5\rho U^2$, which is the Bernoulli value — recovered here by quadrature and in closed form, by a route with no Bernoulli in it anywhere. So the curvature account and the Bernoulli account are the same account, which is worth establishing before asking where the difference lies.

The difference is in the cumulative curve. Twelve per cent of the deficit is generated within a twentieth of a radius of the surface, twenty-two within a tenth, and half of it beyond a third of a radius. A tenth of it comes from beyond one and a half radii.

There is no thin layer doing the work. The suction on the surface of a cylinder is made by the curvature of the whole outer field, over a region comparable with the body.

It is worth saying why the ray straight up from the top is the right place to do this, since a sceptical reader will wonder whether the answer depends on where the integral is taken.

The normal momentum balance is a statement along a line perpendicular to the streamlines, and on a general ray the streamlines are not perpendicular to it, so the balance acquires a second term and the accounting becomes messier. Directly above the top of a cylinder the symmetry makes the flow exactly horizontal, the ray exactly vertical and the two exactly perpendicular, so the balance is one term and the integral is unambiguous.

That is a convenience rather than a special case. The same integral taken along the orthogonal trajectories of the streamlines from any other point on the surface converges to the same kind of answer, over a similar distance, because the field it is integrating is the same field. The top of the cylinder is chosen because it makes the arithmetic exact, not because it is where the effect is.

A wall jet #

Now a jet of thickness $t$ running along a convex wall, with fluid at rest outside it. Inside the jet the flow is fast and curved; outside it there is nothing moving, so $\rho q^2/R$ is zero and the region contributes nothing.

All of the deficit is generated inside the jet — one hundred per cent, by construction, and half of it within half a thickness. For a jet two per cent of the radius thick, that is a layer two per cent of the radius thick doing the whole job.

That is the Coandă effect. It is why a jet will follow a surface round a corner and pull on it, why a Coandă surface on a blown flap works, and why the effect is used in fluidic devices.

Measured against the size of the surface each is attached to, the jet makes half of its suction within one per cent of the radius and the cylinder needs thirty-one per cent. That factor of thirty-one is the difference between the two explanations, and it is a measurement rather than a preference.

Why “the air follows the surface” is the wrong picture even where it is right #

There is a further point worth making, because the phrase carries an implication beyond the mechanism.

“The air follows the surface” suggests that the surface is causing the curvature and the curvature is then causing the pressure. In a wall jet that reading is nearly fair: the jet is a distinct stream with fluid at rest around it, the wall is what makes it turn, and the suction is the price of turning it.

Over a wing there is no such separation. The streamline at the surface curves because the whole field curves, and the whole field curves because the body is in it — every streamline out to many chords is displaced and bent. Attributing the curvature of one streamline to the surface it lies on, and the pressure to that curvature, mistakes a boundary condition for a cause.

The measurement makes that precise rather than rhetorical. If the surface were doing the work, the integral would be dominated by the region next to the surface. It is not: seventy-eight per cent of it comes from beyond a tenth of a radius, where the surface’s presence is felt as a displacement of the whole flow rather than as a wall to follow.

Which is the same conclusion this collection reaches by another route in where the reaction to a wing’s lift is: the force on a body is recoverable on any contour drawn round it, at any distance, and localising its cause to a thin region near the surface is a choice of accounting rather than a fact about the flow.

What the story gets right and what it borrows #

So the explanation “the air follows the curved upper surface, and that makes the low pressure” is not a wrong description of a wing. It is an accurate description of a different flow, applied to one where the same integral converges over a region the size of the body.

That is a more precise complaint than saying the effect is irrelevant, and it is a more useful one, because it says what would have to be true for the story to work. It would need the flow outside a thin layer over the wing to be at rest, and it is not — it is a stream, moving at very nearly the free-stream speed, curving, and contributing most of the integral.

It also explains why the story is so persistent. It names a real mechanism, it is dimensionally sensible, and the flow it accurately describes — a jet following a spoon, a stream of air bending round a cylinder — is exactly the demonstration people are shown when they are taught it.

How to tell which flow is in front of one #

The cumulative curve is the discriminator and it needs no computation to apply, because what it is really measuring is whether the flow outside the layer is moving.

Is there fluid at rest outside the curved region? If so, the region is doing the whole job, the integral terminates at its edge, and the effect is the Coandă effect. A jet issuing into still air, a water jet on a spoon, a blown surface in a quiescent room: all of these.

Or is the outer fluid a stream? If so, it is curving too, it contributes to the same integral, and the suction is a property of the whole field. A wing in flight, a cylinder in a tunnel, a sail: all of these.

The distinction is not about the shape of the body and not about how strongly the flow is turning. It is about the boundary condition far away, which is exactly the thing that never appears in a demonstration — a spoon held under a tap and a wing in flight look alike, and the difference is what is happening a metre away from each.

That also explains the one genuine borderline case. A wing with a blown flap has both: a jet doing Coandă work over the flap, and an outer stream doing the ordinary work over the rest of the section. Which is why blown flaps are the place the term is used correctly in aeronautics, and nearly the only place.

Where the real effect stops #

The effect has a bound, and it is worth computing because “the jet follows the surface” sounds unlimited.

The wall pressure deficit rises as the square of the jet’s strength and the ambient pressure is finite, so there is a speed above which the wall pressure would have to be below vacuum. For a jet a millimetre thick on a fifty millimetre radius that speed is two kilometres a second, which is Mach six — far outside where an incompressible free vortex describes anything, and that is the honest form of the bound: the incompressible model reaches its own limit before it reaches the physical one.

The real limits arrive much sooner and are viscous. A wall jet separates from a convex surface when the pressure recovery it has to manage exceeds what a turbulent layer can take, which this collection has a number for; it is destabilised by the curvature; and it entrains ambient fluid, so its momentum flux is conserved and its speed falls as it goes. None of those is in the calculation here.

What the effect is actually good for #

Having separated the two, it is worth saying what the real effect does, because it is used deliberately and to considerable effect.

A blown flap puts a jet over the upper surface of a deflected flap so that the flow stays attached round a turn no unblown surface could manage. The jet’s own suction holds it to the surface and its momentum re-energises the boundary layer, and the lift coefficients that result are far beyond what the section could otherwise reach — a circulation that has been bought rather than satisfied.

A fluidic device uses attachment and detachment as a switch. A jet in a chamber with two walls attaches to one of them, stays there, and can be flipped by a small control flow — a bistable element with no moving parts, which is why such devices were built for environments where mechanisms fail.

And a Coandă ejector uses the entrainment that goes with attachment to pump: the attached jet drags surrounding fluid along with it, and the device moves far more air than it supplies. That is mixing used as a pump, and the attachment is what keeps the jet in contact with the surface long enough to do it.

In every one of them the jet is the essential ingredient and the surface is secondary. Which is the short way to tell whether the effect is being invoked correctly: if there is no jet, it is not the Coandă effect.

What the number says about the usual demonstration #

There is a demonstration that accompanies almost every telling of this story: a stream of air from a tube is blown past a ball or a cylinder, the stream bends round it, and the object is drawn towards the stream.

The demonstration is real, the mechanism is the one described here, and it is a jet — a stream of moving air with still air around it, which is the case where all of the deficit is made inside the moving layer. So the demonstration is a demonstration of the Coandă effect, correctly, and it does not demonstrate anything about a wing.

That is worth saying carefully because the demonstration is often the whole of the argument offered. A reader shown a jet bending round a cylinder and told “this is why wings work” has been shown a true thing and an inference that does not follow, and the inference fails at a specific point: the demonstration’s still surroundings, which a wing does not have.

The corresponding demonstration for a wing would be a whole wind tunnel with a section in it and a pressure survey out to several chords, and the reason it is never the demonstration offered is that it is not a demonstration — the interesting quantity is an integral over the whole field, and nobody can see an integral.

That asymmetry is worth carrying beyond this argument. A vivid demonstration is available for the mechanisms that are local, and not for the ones that are not. Which is a reason for the local explanations to be more popular than the correct ones, quite independently of whether they are right.

Where this sits among the collection’s other refutations #

It is worth placing this one, because it is a different kind of refutation from the others in its field and the difference is the interesting part.

The equal-transit story is refuted by arithmetic: it makes a definite prediction and the prediction is wrong by a factor of thirty. The theory that forbade flight is refuted the same way. The cushion that is not there is refuted by computing the flow and finding no cushion.

This one cannot be refuted that way, because the mechanism it names is correct. What is wrong is the domain it is applied to, and the only way to show that is to compute where the effect’s own integral comes from in each case — which is why this essay is a measurement of a cumulative distribution rather than a computation of a wrong number.

That makes it the harder kind of misconception to dislodge, and the more interesting one. A story that is simply wrong can be shown to be wrong; a story that is a correct description of a neighbouring situation has to be shown to be misapplied, and doing that requires saying precisely what distinguishes the two situations. Here that is one number: how much of the pressure deficit is made within a layer.

It is also why the verdict on this claim is “misapplied” rather than “false”. Nothing in the sentence “the air follows the curved surface and that makes the low pressure” is untrue of a wall jet. Every word of it is true, in the wrong place, which is a different failure from an error and needs a different answer.

What the curvature account is good for #

Having spent the essay separating the two flows, it is worth saying that the curvature account is genuinely useful where it applies, because the separation is not a dismissal.

The normal momentum balance is the tool for any question about pressure across streamlines, and this subject has several. Why is the pressure lower at the inside of a bend than the outside — that is the balance, integrated across the bend. Why does a free vortex have a pressure minimum at its centre — the balance again, with the curvature the vortex’s own. Why does a curved duct develop a secondary flow — because the balance holds in the core and cannot hold in the slow boundary layer, so the layer is driven inwards.

It is also the right way to read a streamline picture. A region where the streamlines are tightly curved has a strong pressure gradient across them, and one where they are straight has none — which is why the pressure in a parallel shear flow is uniform, and is why the total pressure across such a flow varies without the static pressure moving at all.

So the account is not a rival to the Bernoulli one and is not inferior to it. It is the same physics written across streamlines rather than along them, both are exact, and the two together give a more complete reading of a flow picture than either alone. What it is not is an explanation of lift, and that is a statement about which integral converges where rather than about the account itself.

What the cumulative curve is, as an instrument #

The measurement used here is worth abstracting, because it applies to any argument about where a force comes from.

A force or a pressure that is expressed as an integral has a cumulative distribution: how much of it has been accumulated by a given distance, angle or scale. That distribution is not a rhetorical device; it is a computable object, and it settles questions of attribution that are otherwise a matter of preference.

“The suction is caused by the surface curvature” and “the suction is caused by the whole outer field” are not competing mechanisms — they are competing claims about where the integral comes from, and the integral has an answer. Half from beyond a third of a radius is that answer, and it does not depend on which explanation somebody prefers.

The same instrument settles other attribution disputes in this subject. Where does a wing’s lift come from along the chord — the cumulative load answers it. How much of a boundary layer’s drag is made in the first ten per cent of a plate — the cumulative skin friction answers it. How much of a spectrum’s dissipation is below a given scale — the cumulative dissipation answers it.

When an explanation locates a cause, ask for the cumulative distribution. If the explanation is right, most of the integral is where it says.

What is not claimed #

The wall jet is modelled as a free vortex. A real one has a viscous profile with a maximum inside it, entrains ambient fluid, grows, and slows. What that changes is the shape of the cumulative curve inside the jet and not the fact that it reaches one at the jet’s edge, which is the whole of the argument.

The cylinder is potential flow. A real cylinder separates, and the rear half of the surface pressure is nothing like the ideal one — which does not affect the calculation, since the ray used is at the top, upstream of separation, where the potential solution is good.

Nothing here is an account of why a jet attaches in the first place. Attachment is an entrainment argument: the jet entrains fluid from the gap between itself and the wall, the pressure there falls, and the jet is pushed against the surface. That is a genuinely separate mechanism from the curvature balance, and it is the part that makes the effect useful.

And the factor of thirty-one is for the geometry chosen. A thinner jet gives a larger factor and a thicker one a smaller; a slender body gives a different cumulative curve from a cylinder. What is general is that a jet’s integral terminates at its own edge and a body’s does not.