The effect that explains nothing

Hold the back of a spoon against a running tap and the water wraps round it, pulling the spoon into the stream. Blow across the top of a curved sheet of paper and the sheet lifts. Aim a fan at a cylinder and the jet bends round it.

All three are real, all three are striking, and all three are routinely explained by naming them: the Coandă effect, after the Romanian engineer Henri Coandă, who patented an aircraft that used it in 1934.

Naming is not explaining, and this particular name has done real damage — it appears in popular accounts of flight as the reason air follows a wing’s upper surface, and therefore as the reason aircraft fly. What is actually going on is one line of Euler’s equation, it holds everywhere in every flow this collection has drawn, and an identity that holds everywhere cannot be the reason for anything in particular.

The one line #

Euler’s equation resolved along a streamline gives Bernoulli, which is where the fast-means-low-pressure statement comes from. Resolved across a streamline it gives something less famous and just as exact:

$$\frac{\partial p}{\partial n} = -\rho q^2 \kappa$$

with n the normal, q the speed and κ the curvature. In words: wherever a streamline is curved, the pressure rises going outwards from the centre of curvature, and it rises at a rate set by the speed and the tightness of the bend.

It has to. Fluid going round a bend is accelerating towards the centre of that bend, and something has to supply the force. In a flow with no viscosity the only candidate is pressure, so the pressure must be lower on the inside of every curve than on the outside.

That is the whole of the effect named after Coandă. There is nothing else in it.

Testing it rather than asserting it #

The site computes both sides of the identity by routes that know nothing about each other.

Route one, the pressure. Take Bernoulli’s p = p₀ − ½ρq², evaluate the speed at two points either side of the streamline, and difference them numerically. This never mentions curvature.

Route two, the geometry. Step along the streamline, measure how fast the direction of the velocity is turning, and multiply by ρq². This never mentions pressure.

They agree on the cylinder to 1.5 × 10⁻⁵ of the local scale — the residual of the finite differencing rather than of the physics. Running the same check on a Joukowski aerofoil, above and below, gives the same answer.

That last figure is the refutation. Under the wing the streamlines are curved the other way; the pressure gradient across them points the other way; the identity holds identically. If the balance above the wing is “the Coandă effect making the air follow the surface”, then so is the balance below the wing, and the wing would be pushed as hard downwards as upwards.

What actually decides the lift is the circulation, fixed by the Kutta condition at the trailing edge — a statement about the rear of the aerofoil, not about the curvature of its top surface, and one this identity has nothing to say about.

A bent jet, solved exactly #

The wall-jet case is worth solving properly rather than describing, because it can be, and because solving it turns the “effect” into a pressure with a value.

A sheet of fluid turning a corner has ambient pressure on its outer face and nothing acting on it but pressure. Its Bernoulli constant is therefore the same across the sheet, and a flow with constant Bernoulli constant and circular streamlines is a free vortex: V = C/r, exactly.

Everything follows in closed form. In particular the wall face must sit below ambient by

$$p_{\text{wall}} - p_\infty = \tfrac{1}{2}\rho C^2\left(\frac{1}{r_{\text{out}}^2} - \frac{1}{r_{\text{in}}^2}\right) < 0$$

Read the causation carefully, because it runs the opposite way to the popular account. The wall is not pulling the jet round. The jet is curved — because it was aimed along a curved surface and has to go somewhere — and a curved jet requires a low pressure on its inner side. A wall is a convenient place to put that low pressure, and the wall then feels a force because low pressure on one side of a solid surface with ambient on the other is a force.

What the textbook formula is #

The pressure across a bent jet is usually quoted as ρV²t/R, and it is worth being clear what that expression is: the first term of the free-vortex answer.

For a film it is exact for every purpose. For a jet as thick as the radius it bends round — which is what a spoon under a tap actually is — it is out by twenty per cent, and a figure showing such a jet with that formula beside it is quoting a limit as though it were a law. The exact answer is a free vortex and is no harder to write down.

The term that is always left out #

There is a second finding in the same computation, and it is the sort this field keeps producing: two routes to one force that do not agree, with the gap being the physics.

The force on the curved wall can be had from the pressure over it, or from the change in the jet’s momentum flux. They differ. The reason is that the end faces of the control volume are not at ambient pressure — the flow is curved there too, so the pressure across each end runs from the wall value up to ambient — and the suction on them is exactly t/2r_out of the momentum flux.

Counting it closes the balance to fourteen decimal places at every thickness. Leaving it out is invisible for a film and a quarter of the answer for a thick jet, which is precisely the case the popular demonstrations use.

What the effect actually names #

None of this means Coandă discovered nothing, and it is worth saying what the real content is.

The genuine phenomenon is attachment: a jet emerging beside a surface bends towards it and stays attached, even when the surface curves away from the jet’s original direction. That is not explained by the curvature balance, which says what pressure a curved jet needs and not why the jet chose to curve. The mechanism is entrainment — a turbulent jet drags surrounding fluid along with it, and beside a wall there is a limited supply, so the pressure between jet and wall falls until the jet is pushed against it.

Two things follow, and both are outside this site’s arithmetic.

Entrainment is a turbulent phenomenon. A laminar jet entrains far less and attaches far less readily, so the effect is a property of the flow regime rather than of the geometry — which the site’s turbulence field would have to compute and does not.

And attachment fails if the surface curves too sharply: past some curvature the jet detaches, and where that happens depends on the entrainment rate and the jet’s momentum. Nothing here computes the threshold.

So the honest division is: the suction on the wall is a curvature balance and is exact; the decision to attach in the first place is entrainment, is turbulent, and is not computed on this site.

The same jet, bent harder #

Thickness relative to the bend is the parameter that decides whether the quoted formula is a rounding error or a serious one, and it is worth seeing both ends.

A spoon under a tap is not that case. The jet is a few millimetres thick and bends round a radius of a centimetre or two, so t/R is of order a third: the velocity across the sheet varies by that much, the suction is a fifth larger than the quoted formula, and the end-face term that nobody counts is another sixth. The most-repeated demonstration of the effect is the one where the standard formula is least accurate, which is worth knowing before quoting a number from it.

This is a familiar pattern on this site. The airspeed indicator’s error is a per cent at low speed and a fifth at high; Newtonian impact theory is wrong by thirty at five degrees and exact at infinite Mach number. A formula’s accuracy is a function of where it is used, and a demonstration is usually chosen for being visible rather than for being in range.

Why the confusion is worth taking seriously #

It would be easy to treat this as a naming quibble. It is not, for three reasons.

It substitutes a label for a mechanism. “The air follows the surface because of the Coandă effect” has the grammar of an explanation and the content of a restatement. A reader who accepts it has acquired a word, and the word will not help with the next problem.

It hides where the lift is decided. A curvature argument applied to a wing’s upper surface is a statement about the pressure given the streamlines. What sets the streamlines is the circulation, which is fixed at the sharp trailing edge, which is why a wing has one. That is the site’s own account of lift and it is displaced entirely by the Coandă story.

It is the same shape as the other errors in this field. Equal transit time takes a true observation about a picture and turns it into a false mechanism. The suction misconception takes a real low pressure and gives it a causal role it cannot have. This one takes a real identity and treats a universal property of curved flow as a special effect belonging to walls.

In each case the repair is the same: write down the equation that holds everywhere, then ask what is special about this case. Here the answer is that nothing is, except the presence of a solid surface to feel the pressure the curvature required.

What the picture cannot show #

No entrainment is modelled anywhere. Every figure here is inviscid, so no jet in any of them would attach to anything on its own: they are drawn already curved, and what is computed is what pressure that curvature demands.

The free-vortex sheet has no free-surface stability in it. A real thin sheet on a curved wall breaks up, thins unevenly and can leave the surface in patches; none of that is in a two-dimensional steady solution.

Nothing here is turbulent. The genuine Coandă attachment is a turbulent-entrainment phenomenon and this site says so rather than approximating it.

Where the effect is used on purpose #

The name is a poor explanation and the phenomenon is a real engineering resource, and it is worth separating the two by naming what is actually built on it.

Circulation control. A thin jet blown tangentially over a rounded trailing edge stays attached round it and leaves at an angle set by the blowing rate — so the effective Kutta condition can be moved without moving any surface. That is a genuine application of the attachment, and note what it does: it changes the circulation, which is what actually sets the lift. The jet is a way of steering the circulation, not a way of replacing it.

The jet flap and blown flaps. The same idea on a hinged surface, used on carrier aircraft to double the usable lift coefficient at low speed. Again the mechanism is circulation: the blowing keeps the flow attached over a flap deflection that would otherwise separate, and an attached flap turns the flow further.

Fluidic devices. A jet in a chamber with two outlets can be flipped between them by a small control flow, and it stays where it is put because it attaches to whichever wall it is nearest. That is a bistable switch with no moving parts, which was a serious technology in the 1960s for environments where electronics could not survive.

In each case the useful content is attachment, which is entrainment and is turbulent. In none of them is the curvature balance doing anything except what it does everywhere else in fluid mechanics. The engineering is real; the explanation that gets attached to it is the problem.

The same identity holding up boundary-layer theory #

There is one place where this balance is not merely an identity to read pictures with but the assumption a whole theory rests on, and it uses the same ratio the thin-sheet formula does.

Boundary-layer theory proceeds by asserting that the pressure does not vary across the layer — that whatever the outer inviscid flow imposes at the edge is felt unchanged at the wall. Every boundary-layer calculation ever done depends on it, because it is what allows the pressure to be an input rather than an unknown.

That assertion is this essay’s equation, applied to streamlines that are curved because the wall is. The pressure change across a layer of thickness δ on a surface of radius R is of order ρU²δ/R, and against the dynamic pressure that is a change of order δ/R — the same small parameter as the bent jet’s t/R, doing the same job. Over most of a wing the layer is a per cent of the chord and the surface radius is of order the chord, so the pressure is constant across the layer to a per cent and the theory is safe.

Where the ratio is not small, the assumption fails, and it fails in the places one would expect. A strongly convex surface, where R falls towards δ. A separated shear layer curving sharply away from the body, which is why separation is where the boundary-layer approximation stops being a good bargain rather than merely difficult. And a concave wall, where the gradient points the wrong way for stability and lets a centrifugal instability grow.

So the effect that explains nothing is also the reason the most productive approximation in the subject is allowed. That is a fair measure of how ordinary the identity is.

Who found it, and when #

Henri Coandă observed the attachment in 1910 — reportedly when the exhaust of an experimental aircraft he was testing bent round its fuselage and set it alight — and patented applications through the 1930s. Theodore von Kármán named the effect after him.

The curvature balance is a great deal older: it is Euler’s equation from 1757, resolved in the direction Bernoulli’s version does not use. Which is a fair summary of the whole confusion — a seventeenth-century identity, a twentieth-century observation, and a name that was attached to the second and is now used to explain things that were settled by the first.

The identity as a reading tool #

Once the curvature balance is in hand it becomes a way of reading any flow picture, and that is probably its real value to a reader of this collection.

Look at any streamline plot on this site. Wherever the streamlines bend, the pressure is lower on the inside of the bend. That single rule reproduces, without any further computation:

• the low pressure over the top of an aerofoil, where the streamlines bend downwards over the front;
• the low pressure at the shoulder of a cylinder, where they bend most sharply;
• the high pressure at a stagnation point, where a straight streamline meets a wall and the neighbouring ones bend away from it;
• the pressure rise through a diffuser, where they straighten;
• and the suction under a curved jet, which is where this essay started.

It is a stronger reading tool than the Bernoulli one, because speed is hard to judge from a picture and curvature is not. Streamline spacing indicates speed only where the flow is two-dimensional and the plot is properly seeded; curvature is visible directly.

A reader who can see curvature can see the pressure field, and that is worth more than a name for one of its instances.

Where this ladder goes next #

Two more explanations in this field turn on the same move: taking something real and small and giving it the job of something real and large. The next asks whether the planet’s rotation decides which way a bath drains, and answers it with a ratio rather than an opinion.