Nothing turns a sharp corner

Take any corner in any flow — the inside of a duct bend, the junction of a wing and a fuselage, the edge of a plate — and ask what the flow does within a small enough distance of it. The answer contains almost nothing about the rest of the problem.

Near a corner of interior angle $\alpha$ the boundary is two straight walls, and the solutions of Laplace’s equation with both of them as streamlines are

$$\psi = A_m\,r^{m\pi/\alpha}\sin\frac{m\pi\theta}{\alpha},$$

so the speed goes as $r^{m\pi/\alpha - 1}$. The exponent is set by the angle. Not by the size of the corner, not by the speed far away, not by the fluid, not by anything else in the problem.

Reading the exponent #

The number $\pi/\alpha - 1$ crosses zero at $\alpha = \pi$, which is a flat wall, and the sign of it on either side is the whole of the physics.

Below a straight line — a genuine corner, with fluid in less than half the plane — the exponent is positive and the speed goes to zero at the vertex. A right angle gives $q \propto r$; a 45-degree wedge gives $q \propto r^3$, which is very stagnant indeed. Corners fill up with slow fluid, which is why they collect dirt and why a duct bend’s inside corner is where a designer puts a fillet.

Above a straight line — a salient corner, with fluid in more than half the plane — the exponent is negative and the speed is unbounded. At 270 degrees it is $-1/3$; at 360, which is the flow round the end of a semi-infinite plate, it is exactly $-\tfrac12$.

Why the angle is the only thing in it #

The reason nothing else enters is worth stating, because it is what makes the result useful rather than merely tidy.

Close enough to the vertex, the two walls are straight and the region is a wedge. Laplace’s equation has no length scale in it, and a wedge has no length scale in it either, so the problem posed on the region within distance $r$ of the vertex is the same problem at every $r$ — self-similar, with the only freedom being an overall factor. A solution of a scale-free problem is a power of $r$, and which power is decided by fitting the angular part between the two walls.

That is also why the coefficient is not determined: the local problem is homogeneous, so any multiple of a solution is a solution, and fixing the multiple requires information from a scale the local problem does not contain.

The practical form of that observation is the one to carry away. Anything that depends only on the exponent is universal and anything that depends on the coefficient is not. Whether a corner is stagnant or singular, how fast a suction peak grows as an edge is sharpened, how a mesh must be refined — all exponent. How large the peak is on a particular wing at a particular incidence — coefficient, and therefore a global solve.

The condition that is not in the equations #

There is a step in the above that was taken quietly, and these rungs are about noticing exactly such steps.

Every negative $m$ satisfies the equations and the wall conditions too. $\psi = A_{-1}r^{-\pi/\alpha} \sin(-\pi\theta/\alpha)$ is harmonic, vanishes on both walls, and is a perfectly good local solution. So is $m = -2$, and $m = -3$.

They are thrown away by a condition that appears in neither the equation nor the boundary data: the kinetic energy near the corner must be finite. The energy in a disc of radius $R$ about the vertex is

$$E = \tfrac12\rho A^2 n^2\alpha\,\frac{R^{2n}}{2n},\qquad n = \frac{m\pi}{\alpha},$$

which converges as the disc shrinks for every $n > 0$ and diverges for every $n < 0$. The finite-energy requirement admits the whole positive family and kills the whole negative one, exactly.

And it is not enough #

Having thrown away the divergent-energy modes, the surviving one at a sharp edge still has an infinite velocity, with finite energy. Nothing in the ideal theory removes it.

What removes it is one of two things, and they are the same thing seen from different distances.

Viscosity. A real edge has a radius, and within a distance of order that radius the flow is governed by a local problem in which the speed is finite. The suction force it carries is the same for every radius, including zero, which is what makes the singular solution usable rather than merely wrong.

Or the Kutta condition. At a trailing edge, rather than resolving the local flow, an outer condition is imposed: the circulation is chosen so that the singular mode has zero amplitude. That is a global statement standing in for a local one, and it works because the amplitude of the singular mode is a linear function of the circulation and there is exactly one value that kills it.

What the Kutta condition means, at an angle #

The Kutta condition is usually stated as “the flow leaves the trailing edge smoothly”, and that phrase turns out to describe two different situations depending on the included angle of the edge.

Let the edge have included angle $\tau$, so the fluid occupies $\alpha = 2\pi - \tau$.

For a cusp, $\tau = 0$ and $\alpha = 2\pi$ exactly. The singular mode has exponent $-\tfrac12$; the surviving mode, $m = 2$, has exponent exactly zero. So the speed at the edge is finite and non-zero, and equal on the two sides — which is the form of the condition that is usually stated.

For any wedge, $\tau > 0$ and $\alpha < 2\pi$. The $m = 2$ exponent is $2\pi/\alpha - 1 > 0$, so the speed goes to zero: a finite-angle trailing edge is a stagnation point.

Those are different physical statements, and both are called the Kutta condition.

An absolute distinction nothing can measure #

That last number deserves its own section, because it is the most useful thing on this page.

A ten-degree trailing edge is a stagnation point in the mathematics. The rate at which it approaches stagnation is $r^{0.0286}$ — which means falling to two-thirds of the coefficient takes six decades of distance, and falling to a half takes ten and a half.

A trailing edge is perhaps a hundredth of a chord thick, and a chord is a metre or two. Six decades below that is a micron, which is below the boundary-layer thickness, below the surface roughness, and below any scale on which a potential-flow solution means anything at all.

So the distinction between a cusped edge and a wedge is exact, absolute, and unobservable. Both edges produce the same measurable flow, and the difference between them lives entirely in a region no instrument reaches and no model applies to. That is a useful thing to know before spending effort on which form of the Kutta condition to impose.

What a duct designer does with it #

The exponents are not only about edges. The stagnant end of the range is where most of the engineering is, and it explains a set of practices that are otherwise a matter of taste.

Fillets. A ninety-degree internal corner has $q \propto r$, so the fluid in it is nearly at rest and the boundary layer there is thick, slow and prone to separating under any adverse gradient. Rounding the corner replaces the wedge solution with a locally curved wall and removes the stagnant region, which is why every wing-fuselage junction has a fairing and every duct bend has a radiused inside.

Corner separation. In a compressor or a diffuser, the corner between the blade and the endwall is where separation begins, and it begins there because the corner flow was slow before any adverse gradient arrived. The design response — corner suction, blade fillets, endwall contouring — is entirely about the number in the first panel of the figure above.

And sediment. A stagnant corner collects whatever the flow is carrying, which is why the inside corners of a channel silt up, why a pipe elbow’s inner radius fouls first, and why the corners of a settling tank are the only part of it that works as intended.

The same corner, on the other side of Mach one #

Everything above is elliptic, and the character of the answer changes completely when the equation becomes hyperbolic — while the reason the problem is soluble at all stays exactly the same.

A supersonic flow meeting a salient corner does not accelerate without bound. It turns through a centred expansion fan, and the fan is a solution of the compressible equations in which every quantity depends on the polar angle alone. That is the same self-similarity the incompressible case has, and it is there for the same reason: the corner supplies no length, so the solution cannot contain one. What differs is what the similarity solution turns out to be — a smooth, exact, continuous turn through a computable angle, rather than a power of the radius that runs away at the vertex.

The singularity is resolved by the equations rather than by importing something from outside them. No finite-energy condition is needed, no viscosity, no Kutta condition. Where an incompressible corner produces an unbounded velocity that the theory cannot dismiss, a supersonic one produces a Prandtl–Meyer fan whose every property follows from the Mach number ahead of it.

And the fan has a hard limit. Each increment of turning costs a definite increment of Mach number, and the total available turn is finite: for a gas with $\gamma = 1.4$ the flow can turn through at most

$$\nu_{\max} = \frac{\pi}{2}\left(\sqrt{\frac{\gamma+1}{\gamma-1}} - 1\right) = 130.45^{\circ}$$

from rest, and no further. Past that the density has fallen to nothing and there is a vacuum — the gas simply cannot follow the wall, and it separates from it with empty space in between. That is a threshold of the same kind as the ones this collection sorts as discriminants: exact, algebraic, with no tolerance anywhere in it, and produced by a solution ceasing to exist rather than by anything becoming inaccurate.

The concave corner behaves differently again and has its own limit. Supersonic flow into a compression corner turns through an oblique shock, and for a given upstream Mach number there is a maximum deflection the shock can achieve — about 23 degrees at Mach 2, rising to 45.6 degrees as the Mach number goes to infinity. Ask for more and no attached oblique shock exists: the shock detaches and stands off the corner as a bow wave, which is the same event a blunt body produces and is why a wedge sharp enough to be attached at one speed is detached at another.

So the corner problem has three regimes and three characters. Subsonic: a power law with an exponent set by the angle, singular at a salient corner, needing an external condition to be usable. Supersonic expansion: an exact continuous turn with a maximum beyond which the gas leaves. Supersonic compression: an exact discontinuous turn with a maximum beyond which the shock leaves. One geometry, one question, and the equation’s type deciding not only the answer but what kind of thing the answer is.

Where else the same exponent appears #

The corner solution is local, which means it is portable, and it turns up wherever a boundary has a vertex.

At a leading edge, which is a 360-degree corner, giving the inverse square root that carries the suction force.

At the sharp edge of an orifice, where the same singularity is what makes a jet contract — the free-streamline solution is a device for replacing the unbounded velocity with a surface at constant speed, and the vena contracta is what the replacement costs.

At a crack tip in an elastic solid, where the stress goes as $r^{-1/2}$ for exactly the same reason — Laplace’s equation, a 360-degree corner, and the same eigenvalue problem. The energy release rate that fracture mechanics is built on is the finite integral of a singular field, and it is structurally the same quantity as the leading-edge suction.

And at a re-entrant corner in a viscous flow, where the exponent becomes complex and the solution is an infinite sequence of ever-smaller counter-rotating eddies. Moffatt found them, and the mechanism is this eigenvalue problem with the biharmonic operator in place of the Laplacian.

What the picture cannot show #

The size of the region. Every statement here is about the limit $r \to 0$, and how small $r$ has to be for the local solution to dominate depends on everything the local solution has thrown away. A figure drawn at one scale cannot say whether it is inside that region.

The amplitude. The exponent is universal and the coefficient $A$ is not: it is fixed by the flow far away, and getting it requires solving the whole problem. The local analysis says what shape the answer has near the corner and nothing about how big it is, which is why it is a complement to a global solve rather than a substitute.

And the flow does not actually go round a sharp edge. In a real fluid at any Reynolds number the layer separates at a salient corner, and the singular solution describes a flow that does not happen. It remains useful for the reason singular solutions usually are: the integral quantities it carries are right even where the pointwise ones are not.

That is the pattern these rungs keep finding, arriving here in its most abstract form. Laplace’s equation and the wall conditions admit a family; a requirement from outside them selects a subfamily; and even after the selection, one member of what remains is unphysical for a reason the theory has no means of expressing. Each layer of the argument imports something the previous one did not contain, and the imports are viscosity, energy and the past — the three things ideal flow was built by throwing away.

One practical note before the history, because it is the question a reader with a mesh in front of them will have. The exponents above are for the velocity potential’s local behaviour, so a computation that resolves a corner is resolving a field whose derivative is singular there. No amount of refinement fixes that — halving the cell size near a 360-degree corner improves the local answer by a factor of only √2 — and the standard responses are to grade the mesh geometrically towards the vertex, to enrich the basis with the known singular function, or to accept that the pointwise answer at the corner is wrong and take an integral of it instead. The third is what aerodynamics does, and the leading-edge suction is the integral it takes.

Who found it, and when #

The corner eigenfunctions are as old as potential theory — they are in Lamb, and they are the standard separation-of-variables solution in polar coordinates. The finite-energy condition as a selection rule was made precise in the twentieth century, in the theory of elliptic boundary-value problems on domains with corners, where it is the statement that the solution lies in a Sobolev space.

The surprising connection is that the same eigenvalue problem decides something structural about numerical methods. A finite-element solution of Laplace’s equation on a domain with a re-entrant corner converges more slowly than on a smooth one, and the rate is set by the exponent $\pi/\alpha$ — because the solution has a singular derivative there and a polynomial basis approximates it badly. Every mesh generator that refines towards corners is doing so because of the numbers on this page, and the optimal grading of that refinement is a function of the same exponent. The physics and the numerics have the same eigenvalue in them, and it is a property of the geometry rather than of either.

Where the ladder goes next #

Below this rung are the exact theory whose local behaviour this is, and the Kutta condition, which is the global device that removes what the local analysis cannot.

Beside it are the force that lives in the singularity and the viscous version of the same corner, whose answer looks nothing like this one.