The number that is not a number

Every textbook on this subject contains a table of transition Reynolds numbers, and every table gives values to two or three significant figures. Flat plate: 5·10⁵. Pipe: 2300. Sphere: 3·10⁵. Cylinder: 2·10⁵.

Every one of those numbers can be moved by more than a decade without changing the fluid, the shape or the equations. What changes is the experiment.

This essay is about what those numbers are actually reporting, because the distinction matters for anything built on them: part of a transition Reynolds number is a fact about fluid mechanics, and part of it is a fact about the room.

What is being quoted #

A transition Reynolds number is a statement of the form: at this value of Ux/ν, the flow ceased to be laminar. It has three moving parts and only one of them is about the fluid.

The velocity and length are the experimenter’s, and are usually unambiguous.

The viscosity is the fluid’s, and is known well.

“Ceased to be laminar” is a judgement, and it is where most of the spread lives. Transition is not an event at a point; it is a region, several times as long as the laminar layer’s own thickness scale, in which the flow is intermittently one thing and intermittently the other. Different experimenters have defined the transition point as: the first appearance of a turbulent spot; the station at which the intermittency reaches one half; the minimum of the skin-friction curve; and the point at which the skin friction first departs from Blasius’ law. These do not coincide, and the spread between them alone is a factor of two.

That much is definitional and could in principle be fixed by agreement. The rest cannot.

The spread, and what causes it #

Four things move the number, in rough order of how much.

Free-stream turbulence. A wind tunnel with 1% turbulence in its working section transitions a flat plate near Re_x = 10⁵. The same plate in a tunnel at 0.02% will hold laminar flow past 3·10⁶. This is the largest single effect and it is a property of the tunnel, not the model.

Surface roughness. A single roughness element of height comparable with the local displacement thickness will trip the layer where it stands. The critical roughness Reynolds number is around 600 based on the element’s height and the velocity at that height, and below it the element does essentially nothing — so roughness has a threshold rather than a gradient.

Sound and vibration. Acoustic disturbances enter the layer through the leading edge and through any discontinuity in the surface, and they are the reason transition data taken in the 1950s in tunnels driven by unmuffled fans are systematically early.

Pressure gradient. A favourable gradient delays transition and an adverse one hastens it, for reasons that are genuinely fluid mechanics and that the inflection criterion explains: an adverse gradient puts an inflection point in the profile and an inflectional profile is unstable to a far wider band of disturbances.

Only the last of those four is a property of the flow being studied. The other three are properties of where it is being studied.

The three decades between instability and transition #

Here is the part that most complicates the picture, and it is quantitative.

A flat-plate boundary layer is genuinely unstable — not marginally, not subcritically, but in the ordinary linear sense — above a Reynolds number of about 520 based on displacement thickness, which on the plate above is Re_x ≈ 9·10⁴. Tollmien predicted this in 1929 and Schlichting computed the neutral curve in 1933, and their result was disbelieved for fifteen years because experiments in ordinary tunnels showed nothing there.

Schubauer and Skramstad settled it in 1947 by building a quiet tunnel and a vibrating ribbon: the waves were there, at the predicted frequencies and the predicted growth rates, and they had been invisible because in an ordinary tunnel they are drowned by everything else.

So on a plate transitioning at Re_x = 5·10⁵, the instability began at 9·10⁴ and took most of a decade to do anything. On a plate transitioning at 3·10⁶ it began at the same place and took a decade and a half.

The instability sets the earliest possible transition and does not set the actual one. What sets the actual one is how large the disturbances were when they entered the layer, and how much amplification the layer supplied before nonlinearity took over.

Receptivity: the half nobody can compute #

Between “a disturbance exists outside the layer” and “a wave of that frequency is growing inside it” sits a step with a name and very little theory: receptivity.

The difficulty is a mismatch of scales. A sound wave in air at 100 Hz has a wavelength of about three metres. A Tollmien–Schlichting wave in the layer on the plate above, at the frequency that grows fastest, has a wavelength of a couple of centimetres. A disturbance cannot force a wave whose wavelength it does not contain, so a plane sound wave in a smooth uniform free stream drives almost nothing.

What converts one into the other is any place where the flow varies rapidly along the surface — a leading edge, a step, a roughness element, a suction slot. There the boundary layer’s own properties change over a distance comparable with the instability wavelength, and the long external wave can transfer energy into the short internal one. This is why the leading-edge geometry and the surface finish over the first few per cent of chord matter out of all proportion to their size.

The consequence for the number under discussion is direct: the initial amplitude of the growing wave is set by geometry the Reynolds number does not describe, and the amplification that follows is what the theory computes. A transition Reynolds number is the product of a computed quantity and an uncomputed one, quoted as though it were a measurement of the first.

The e^N method, and its honest name #

The practical method used for transition prediction in aircraft design is not a criterion at all. It is a bookkeeping exercise called e^N.

Compute the boundary layer, solve the local stability problem at each station, and integrate the growth rate of the most amplified disturbance along the surface. The result is a number N: the natural logarithm of the total amplification a disturbance has received since it first became unstable. Transition is then declared where N reaches some value.

For a typical wind tunnel that value is about 9, meaning an amplification of e⁹ ≈ 8,100. For a quiet tunnel or flight it is 11 to 14. For a noisy tunnel it is 4 to 6.

The method works well and is used throughout the industry, and it is essential to be clear about what it is. It computes the amplification exactly — that part is a linear-stability calculation with no free parameters — and it then requires a calibration constant which encodes the disturbance level of the environment. The environment is exactly the part that is not fluid mechanics.

Calling N a transition criterion invites the belief that transition has been predicted from first principles. What has been predicted from first principles is the amplification; N is the measured amount of amplification the local noise floor needs before nonlinearity takes over, and it is fitted.

What the amplification actually looks like #

It is worth putting a number on “the amplification the layer supplied”, because the exponential makes the intuition unreliable in both directions.

At N = 9, a disturbance that entered the layer at one part in 10⁷ of the free-stream velocity — which is a very quiet tunnel indeed — leaves the amplification region at about one part in 1,200. That is still small. It is small enough that a hot wire would have to be looking for it, and it is large enough that the nonlinear terms are no longer negligible, because the relevant comparison is not with the free stream but with the local mean shear.

Raise the entry amplitude by a factor of a hundred — a tunnel at 1% turbulence rather than 0.02% — and the same N of 9 is reached after far less growth, which means far less distance, which means an earlier transition. That is the whole mechanism by which free-stream turbulence moves the number, and it is why the effect is so large: the amplification is exponential in distance, so a factor of a hundred in the starting amplitude is a fixed subtraction in N, and a fixed subtraction in N is a large change in x.

The arithmetic also explains the shape of the sensitivity. Doubling the free-stream turbulence does not double the laminar run; it shortens it by whatever distance supplies ln 2 of growth, which on a typical plate is a few per cent of chord. The response is neither linear nor catastrophic, and neither of the two intuitions the subject usually offers is right. It is the same exponential sensitivity that makes the cascade so wide and the grid so large: wherever a fluid problem turns on an exponent, small changes in the input arrive as large changes in the outcome and large changes arrive as absurd ones.

The route that skips all of it #

Everything above describes one path from laminar to turbulent: a disturbance enters, a Tollmien–Schlichting wave grows exponentially, and nonlinearity finishes the job. That path is what $e^N$ computes, and above a free-stream turbulence level of roughly one per cent it is not the path the flow takes.

What happens instead begins with a mechanism this collection has already met at the other end of the same argument. The linearised operator governing a shear layer is non-normal, so a disturbance can be amplified enormously without any eigenvalue having a positive real part — the same transient growth that makes pipe flow turbulent with no linear instability anywhere. In a boundary layer its preferred form is the lift-up of low-momentum fluid: weak streamwise vorticity in the free stream drags slow fluid up and fast fluid down, and what appears inside the layer is a set of long streamwise streaks of alternating fast and slow fluid, spaced a few layer thicknesses apart across the span.

Those streaks are not waves. They have almost no frequency, they grow algebraically with distance rather than exponentially, and they reach amplitudes of tens of per cent of the free stream — far larger than anything a Tollmien–Schlichting wave reaches before breaking down. Transition then occurs through a secondary instability of a streak, which appears as an intermittent burst rather than as a wave train.

None of that is in the neutral curve. The stability calculation the $e^N$ method integrates says nothing about streaks, because streaks are not eigenmodes; the amplification is a property of the operator’s non-normality rather than of its spectrum. So in a noisy environment the method is not merely poorly calibrated — it is computing the growth of a mechanism that is no longer the one operating. Morkovin, who named receptivity, named this too: bypass transition, because the flow bypasses the route the theory describes.

The practical consequence is a whole class of machine. A turbine or compressor blade sits in a stream carrying five to twenty per cent turbulence, so its boundary layers are bypass-transitional from the leading edge and no calibration of $N$ can rescue the method there. Turbomachinery uses empirical correlations in free-stream turbulence and pressure gradient instead, which is an admission of exactly the kind this essay is about — and a more honest one than a first-principles calculation with a fitted constant hiding the same information.

What survives, and it is more than nothing #

The pessimistic reading of all this is that transition Reynolds numbers are useless. That is wrong, and the reason is similarity.

Two things do transfer.

The Reynolds number is still the right variable. A plate that transitions at Re_x = 5·10⁵ in a given tunnel will transition at Re_x = 5·10⁵ in that tunnel at half the speed and twice the length. The environment sets the value; the fluid mechanics says it is a function of that group and not of speed and length separately. That is a strong statement and it is the reason the number is worth quoting at all.

Comparisons within one environment are sound. A tunnel campaign that measures transition on six aerofoils under the same conditions produces a ranking that is meaningful even though every absolute value in it is tunnel-specific. Most of the practical use of transition data is comparative, and comparative use is exactly what survives.

What does not transfer is an absolute number taken from one environment and applied in another — which is precisely what a table in a textbook invites.

The failure this causes, and it is expensive #

The natural-laminar-flow aerofoil is where this bites hardest. A section designed to hold a favourable pressure gradient over the first half of its chord can keep the layer laminar to Re_x of a few million, and the drag saving is large — the figure above puts the laminar-to-turbulent ratio near eleven at Re = 5·10⁶.

The saving is also fragile in a way that is entirely about the environment. Insects on the leading edge, rain, ice crystals in cirrus, a repair with a proud rivet, or a step at a panel joint will each trip the layer, and the design reverts to a turbulent section with an unusually aft pressure recovery — which is worse than a conventional section, not merely no better.

That is not an argument against laminar flow. It is an argument for stating, in the specification, which of the two numbers a design assumes, and for knowing that the number came from a tunnel with a particular turbulence level in it.

Where the model stops #

Nothing in this essay computes a transition Reynolds number, and nothing on this site could.

What the site does compute is the laminar side: the Blasius layer’s thickness at every station, by shooting the similarity equation, and the effect of a pressure gradient on the profile, by solving Falkner–Skan. Every figure in this essay that carries a curve carries a solved one, and every transition point in it is a placement — a number put there by hand, marked on the figure as observed rather than derived.

The honest statement of the field’s position is this. Linear stability theory predicts, correctly and with no free parameters, which disturbances grow and how fast. Nothing predicts how large the disturbances were to begin with, because that is a question about the environment rather than about the equations. Receptivity theory — the study of how external disturbances enter a boundary layer — is the attempt to close that gap, and it is an active subject rather than a settled one.

Who found it, and when #

Rayleigh set out the inviscid criterion in 1880. Tollmien computed the viscous neutral curve for the flat plate in 1929 and Schlichting extended it in 1933, both against the prevailing view that viscosity could only stabilise.

Taylor argued in 1936 that free-stream turbulence, not the Tollmien–Schlichting mechanism, was what governed transition in the tunnels of the day — which was correct about those tunnels and wrong as a general claim, and the two were not disentangled for a decade.

Schubauer and Skramstad’s quiet-tunnel experiment in 1947 found the waves and vindicated the theory, and Smith and Gamberoni and, independently, van Ingen introduced the e^N correlation in 1956, explicitly as a correlation.

Morkovin coined receptivity in 1969 for the missing half of the problem, and named it as missing. It is still, in the sense that matters here, missing.

Where the ladder goes next #

The mechanism side of this argument is a layer with a kink in it, which is Rayleigh’s criterion: the one general statement about which velocity profiles can be unstable at all, and the reason an adverse gradient hastens transition as well as causing separation.

The other direction is the practical one. What transition costs, in drag, is the cost of going turbulent, where the two skin-friction laws are put side by side and one of them is a correlation.