A layer with a kink in it

Most of what is known about which flows go unstable is empirical, environmental and specific to a geometry. There is one general statement, it is a theorem, and it is short.

An inviscid parallel shear flow cannot be unstable unless its velocity profile has a point of inflection.

Lord Rayleigh proved it in 1880 in about half a page. It has three properties that make it the right place to start this ladder: it is exact, it is about the shape of a profile rather than about a number, and it is necessary and not sufficient — a distinction that is dropped almost universally and that changes what may be concluded from it.

What the theorem says, and the one line it takes #

Take a steady parallel flow U(y) with no viscosity, perturb it by a small wave of wavenumber k and phase speed c, and linearise. What comes out is Rayleigh’s equation:

$$(U - c)\left(\phi'' - k^2\phi\right) - U''\phi = 0$$

with φ the amplitude of the disturbance streamfunction, vanishing at both edges of the domain.

Multiply by the complex conjugate of φ, integrate across the layer, and take the imaginary part. Most of the terms are real and drop out. What survives is

$$c_i \int \frac{U''}{|U - c|^2}\,|\phi|^2\,\mathrm{d}y = 0$$

so if the disturbance grows at all — if c_i is not zero — then the integral must vanish. The denominator and |φ|² are positive everywhere. Therefore U″ must change sign somewhere inside the layer.

That is the whole proof. It requires no approximation beyond linearisation, it holds for every wavenumber at once, and it says nothing whatever about whether an inflectional profile is unstable.

Necessary, and the difference that makes #

The theorem is an obstruction, not a mechanism. It rules out instability for a class of profiles and promises nothing for the rest.

The consequence in practice is that “this profile has an inflection point, therefore it is unstable” is a non-sequitur, and it is written down constantly. Tollmien tightened the statement in 1935: for a monotonic profile it is enough that the vorticity have an extremum at the inflection point and that a further inequality hold there. Fjørtoft strengthened it in 1950 to the condition that U″(U − U_s) < 0 somewhere, where U_s is the velocity at the inflection point — which rules out a further family of inflectional profiles that Rayleigh’s version admits.

The tanh layer above passes all three tests and is genuinely unstable, which is why it is the standard example. The figure locates its neutral wavenumber by putting the exact eigenfunction sech y back into the equation and scanning for the wavenumber that annihilates the residual; the answer is k = 1, and it is an answer rather than an input.

What the theorem is really about #

There is a physical reading of the criterion that makes it less of a conjuring trick, and it is worth having because it explains why the same condition keeps reappearing in problems that look unrelated.

U″ is the gradient of the vorticity. In a parallel shear flow the vorticity is −U′(y), so U″ = 0 is the statement that the vorticity has a maximum or a minimum somewhere inside the layer — a concentration of spin with less spin on both sides of it.

That is the structure that can roll up. A monotonic vorticity distribution has nothing to organise itself around: every parcel is being sheared past its neighbour in the same sense at every level, and a disturbance has no preferred place to grow. A vorticity extremum is a ridge, and a ridge of vorticity can concentrate into lumps in exactly the way a row of vortices concentrates.

Read that way, Rayleigh’s theorem stops being a condition on the second derivative of a velocity profile and becomes a condition on the shape of the vorticity distribution — which is the quantity that is actually conserved and actually transported, and the one this site’s own solver marches. It is also why the result survives so many changes of setting: put the same fluid on a rotating planet, add stratification, add a magnetic field, and the criterion changes its detailed form while keeping its character, because the underlying statement is about whether the transported quantity has a ridge in it.

The same name, and three criteria that are not this one #

The section above says that the criterion changes its form and keeps its character when the setting changes, and it is worth being specific, because one of the variants shares this one’s author and its name and is a different theorem about a different flow.

Rayleigh’s circulation criterion, from 1917, is about a rotating rather than a shearing flow. Fluid moving in circles is stable against axisymmetric disturbances if and only if the square of the angular momentum, $(rV_\theta)^2$, increases outwards everywhere. Where it decreases, a ring of fluid displaced outward arrives carrying more angular momentum than its new surroundings, is flung further out, and the flow overturns. That is what makes the inner cylinder of a Taylor–Couette apparatus the dangerous one to rotate and the outer one safe, and it is the criterion behind every statement about a swirling flow being centrifugally stable or not.

It has nothing to do with points of inflection and it is constantly conflated with the theorem this essay is about, because both are necessary-and-not-sufficient conditions, both are one line, and both are called Rayleigh’s criterion. A flow can satisfy one and violate the other. The quantity with a ridge in it is different — vorticity in one case, angular momentum in the other — and it is that quantity’s shape, not the velocity’s, that each theorem is really about.

The compressible version keeps this essay’s structure and moves the point. Lees and Lin showed in 1946 that what matters in a compressible layer is not $U'' = 0$ but the vanishing of

$$\frac{d}{dy}\left(\rho\,\frac{dU}{dy}\right),$$

the generalised inflection point — because the transported quantity is now a density-weighted vorticity. That single change makes the criterion depend on the temperature profile as well as the velocity one, with a consequence a designer can use: cooling the wall moves the generalised inflection point inwards and can remove it altogether, so a cooled surface in supersonic flow is stabilising in a way it is not at low speed.

The qualification matters as much as the result, and it is the sort this essay exists to insist on. Wall cooling stabilises the first mode — the compressible descendant of the disturbance this criterion is about — and destabilises the second, an acoustic mode with no low-speed counterpart that becomes dominant above about Mach four. A hypersonic vehicle cooled on the reasoning of the paragraph above transitions earlier rather than later, and the criterion that predicted otherwise was being asked about a mode that had stopped being the relevant one.

And the stratified version replaces a necessary condition with a sufficient one, running the logic the other way. Miles and Howard proved in 1961 that a stratified shear flow is stable if the Richardson number exceeds a quarter everywhere — so buoyancy strong enough against the shear forbids the instability outright, and the number that stops the mixing is a guarantee rather than an obstruction. It is the only member of the family that promises stability rather than merely permitting instability, and the difference in logical direction is exactly the one this essay’s central caution is about.

The wall relation, which is where this stops being abstract #

The reason this criterion earns a place on a site about aerodynamics rather than about stability theory is a single line from the boundary-layer equations, evaluated at the wall.

At a solid surface both velocity components vanish, so every convective term in the momentum equation vanishes with them, and what is left is

$$\mu \left.\frac{\partial^2 u}{\partial y^2}\right|_{0} = \frac{\mathrm{d}p}{\mathrm{d}x}$$

The curvature of the velocity profile at the wall is proportional to the pressure gradient. Not approximately, not in some limit — exactly, in the boundary-layer approximation.

Far from the wall, at the edge of the layer, the profile has to flatten onto the free stream, which means its curvature there is negative. So:

In a favourable gradient, dp/dx < 0, the curvature at the wall is negative and it is negative at the edge, and there is no need for it to change sign anywhere between. There is no inflection point.

In an adverse gradient, dp/dx > 0, the curvature at the wall is positive and at the edge it is negative. It must change sign somewhere in between. There is an inflection point, necessarily.

The same sign that eventually separates the layer is the sign that puts the inflection point in it. Instability and separation are two consequences of one fact, and the consequence arrives long before the separation does.

Testing it on profiles the site already solves #

The Falkner–Skan family is the site’s parameterisation of boundary layers under a pressure gradient, and it is solved here by shooting rather than tabulated. That makes it the natural test bed: the profiles are answers, and the search for an inflection point is run on the answer.

The results are exactly what the wall relation demands. At β = 0.6 and β = 0.3 — accelerating flow — the search finds nothing inside the layer. At β = 0 — the Blasius plate, zero pressure gradient — the curvature at the wall is exactly zero, so the inflection point sits at the wall, which is the boundary case and the reason a flat plate is unstable only through the viscous mechanism. At β = −0.1 and at the separation value β = −0.1988, the search finds an inflection point standing clear of the wall, further out the more adverse the gradient.

One methodological note, because it caused a real error while these figures were being written. The solver returns the profile on a grid at spacing 0.002, and the search takes a second difference. Take that difference at the grid spacing and it measures the interpolation rather than the profile: it spikes at every node and vanishes between them, and the search dutifully reports dozens of inflection points that are artefacts of storage. The step has to be well above the grid spacing, and the figure’s is 0.05.

Why an inflectional profile is so much more unstable #

The quantitative difference between an inflectional and a non-inflectional profile is not marginal, and it is worth naming because it is what makes the criterion useful rather than merely true.

A Blasius boundary layer is unstable through the viscous mechanism — Tollmien–Schlichting waves — which is a subtle business in which viscosity, normally the stabilising influence, destabilises by introducing a phase shift near the wall. Its growth rates are small. A disturbance needs a decade or more of streamwise distance to grow by the factor of e⁹ that transition typically requires.

An inflectional profile is unstable through the inviscid mechanism, which needs no such subtlety and is far more vigorous. The growth rates are larger by one to two orders of magnitude, and the band of unstable wavenumbers is wider.

The practical consequence: a boundary layer that has entered an adverse gradient does not merely become more likely to transition, it changes which mechanism is operating, and the new one is fast. On an aerofoil this is why transition so often sits within a per cent or two of the pressure minimum — the layer is stable-ish while the flow accelerates and becomes violently unstable the moment it does not.

The one case where the criterion is exactly right #

It is worth putting one case beside the shear layers where linear stability theory gets the whole answer, because it stops the criterion from reading as a piece of theory that never quite predicts anything.

A layer heated from below is stable below a definite Rayleigh number and unstable above it, the threshold is an eigenvalue with a closed form, and the experiment agrees. Nothing about the environment moves it by a decade. There is no receptivity problem, no subcritical route and no thirty-to-one spread.

The difference is structural rather than lucky. Convection’s instability is supercritical: at threshold the growing mode saturates at a small amplitude that increases smoothly from zero, so the first thing that happens after the threshold is a small steady flow that linear theory has correctly identified. Shear instabilities are not like that, and pipe flow is the extreme case where the linear threshold does not exist at all.

So the honest summary of the criterion’s standing is: it says exactly what it says. Where the subsequent behaviour is smooth, the threshold it gives is the observed one. Where the subsequent behaviour is violent, the threshold is the beginning of a story rather than its end.

Where the model stops #

Rayleigh’s equation has viscosity set to zero, and the flows it is being applied to do not.

That is not the contradiction it looks like. The criterion is used as a statement about the inviscid limit of a viscous problem, and the justification is that the inviscid instability, where it exists, is so much stronger than the viscous one that it dominates. Where it does not exist, the viscous mechanism is all there is, and Rayleigh’s theorem is silent about it — correctly, since it assumed viscosity away.

Two further limits are worth stating plainly.

Parallel flow. The theorem assumes U depends on y alone. A boundary layer grows, so it is not parallel, and the whole apparatus is a local approximation justified by the layer growing slowly compared with the instability wavelength. It is a good approximation and it is an approximation.

Two-dimensional disturbances. Squire’s theorem in 1933 established that for parallel flow the most unstable disturbance is two-dimensional, so nothing is lost by considering only those — for the linear problem. The transition process that follows is emphatically three-dimensional, and the theorem says nothing about it.

What it is safe to conclude #

Collecting the qualifications into a single statement, because the criterion is quoted far more often than it is stated correctly.

Safe. A parallel inviscid flow whose profile has no inflection point is stable. An attached boundary layer in a favourable pressure gradient therefore cannot go unstable by the inviscid mechanism, and whatever transition it suffers must come through the slower viscous route or through a disturbance large enough to skip the linear stage entirely.

Safe. An adverse pressure gradient guarantees an inflection point, by the wall relation, with no computation required beyond the sign of dp/dx.

Not safe. That an inflectional profile will go unstable. Fjørtoft’s condition rules out a whole family of them, and even where the mode exists its growth rate may be too small to matter over the distance available.

Not safe. That the wavelength predicted here is the structure that will appear. The linear problem selects a wavelength; what a finite-amplitude disturbance settles into is a different question and one this site does not compute.

The pattern is the fleet’s usual one and is worth naming as such: a theorem is exactly as strong as its own statement, and the commonest way to misuse one is to run it backwards.

Who found it, and when #

Helmholtz posed the discontinuous-shear problem in 1868 and Kelvin analysed it in 1871, giving the result that a vortex sheet is unstable at every wavelength.

Rayleigh’s inflection theorem is from 1880, and his 1887 paper added the constant-vorticity examples that are still the standard teaching cases. Tollmien supplied the partial converse in 1935 and Fjørtoft the stronger necessary condition in 1950.

The relation between the wall curvature and the pressure gradient is Prandtl’s, and comes with the boundary-layer equations themselves in 1904 — which is to say that the connection between instability and separation has been available for as long as boundary layers have.

Where the ladder goes next #

The other rung of this anchor takes the extreme case: a shear layer of zero thickness, which is unstable at every wavelength and most unstable at the shortest. Every wavelength at once is about what a model is saying when its answer has no smallest scale in it.

The application on this site’s own ground is the gradient that does both, which takes the wall relation above and follows it into the separation ladder — where the same sign, further along, stops the layer entirely.