What "of order one" is worth

Every dimensionless group in this subject is introduced the same way. Two terms in an equation are compared, their ratio is formed, the ratio is given somebody’s name, and the reader is told that the behaviour changes when the group is of order one. It is a good story and it is told about the Reynolds number, the Mach number, the Knudsen number, the Bond number, the Weber number, the Rossby number and every other number that has ever been named after anybody.

The story is exactly one sentence too long. Forming a ratio of two terms and saying it is one where the terms are equal is a tautology — it is what “ratio” means. Saying that the behaviour changes there is a claim about the solution of a differential equation, and nothing in the ratio establishes it.

The two numbers, and why they are different questions #

Take any group $\Pi$ formed as $A/B$ from two terms in an equation. Two entirely separate questions can be asked about it.

Where are $A$ and $B$ equal? At $\Pi = 1$. This is arithmetic and there is nothing to compute.

Where does the answer stop being what the $B$-only theory says? This depends on the whole solution. If some observable $Q$ has an expansion $Q = Q_0(1 + c\Pi + \dots)$, then the $B$-only theory is wrong by a fraction $\varepsilon$ when $c\Pi \approx \varepsilon$, which puts the threshold at $\Pi \approx \varepsilon/c$. The tolerance is in the answer and it was never in the group.

So for any group of this shape the two numbers are a factor of $c/\varepsilon$ apart, and asking for one per cent rather than ten moves the threshold by a decade. That is not a defect in the group. It is a statement that the group answers a question nobody asked.

The extreme case, which is a channel #

The Knudsen number is the mean free path over the size of the thing, and its story is the cleanest of all: at $\mathrm{Kn} = 1$ a molecule crosses the whole channel between collisions, so the fluid picture of a continuum with a velocity at every point cannot possibly survive. Every textbook draws the same ladder — continuum below 0.01, slip to 0.1, transition to 10, free molecular above.

Poiseuille flow with Maxwell’s slip condition is exact in closed form, and it says the flow rate is the no-slip one times $1 + 6\mathrm{Kn}$ for a channel. So the error in the no-slip answer is $6\mathrm{Kn}/(1 + 6\mathrm{Kn})$, which reaches one per cent at

$$\mathrm{Kn} = \frac{1}{594}.$$

Not at one. At one part in five hundred and ninety-four, and the continuum rung of the standard ladder runs to a hundredth, by which point the error is six per cent. Where a fluid stops being one takes that apart properly, and the number quoted there is this one.

The six is worth a sentence of its own, because it is not a property of rarefaction. Maxwell’s condition puts the wall velocity at $\sigma_v\lambda$ times the velocity gradient there, with $\sigma_v = (2-\sigma)/\sigma$; carrying that through the Poiseuille integration gives $1 + 6\sigma_v \mathrm{Kn}$ between flat plates and $1 + 4\sigma_v\mathrm{Kn}$ down a tube. The same gas in the same state has two different onsets depending on the shape of the hole it is in, a factor of three-halves apart, and a wall that accommodates imperfectly moves it again: at $\sigma = 0.8$ the onset falls to two-thirds of what a perfectly accommodating wall gives.

And the same expansion offers two more candidate thresholds, neither of them one either. The first-order term reaches the size of the term it corrects — the slip doubles the flow — at $\mathrm{Kn} = 1/6$. The second-order term reaches the first at $\mathrm{Kn} = 1/2$, which is where an expansion in the Knudsen number stops converging usefully whatever its coefficients are. So one group has four numbers attached to it, at 1/594, 1/6, 1/2 and 1, and the folklore quotes the only one that is not the answer to any question about the flow.

Four kinds, and the sort is by ratio #

The interesting thing is not that the two numbers differ. It is that sorting the fourteen groups by how much they differ sorts them by what kind of threshold they are, and there turn out to be exactly four kinds.

A term ratio #

Eight of the fourteen. The observable is smooth in the group, the onset is the tolerance divided by a slope, and the two numbers are as far apart as the tolerance is small. The Knudsen number above, the Reynolds number for creeping flow, the reduced frequency for quasi-steady aerodynamics, the Bond number for a drop’s shape, the Rossby number for geostrophic balance.

For these, “of order one” is not so much wrong as empty. It names the balance and says nothing about the accuracy, and the accuracy is what anybody wanted.

An eigenvalue #

Two of the fourteen, and they behave in exactly the opposite way. A layer of fluid heated from below has a Rayleigh number, and the Rayleigh number is a ratio of terms like any other: buoyancy against the two diffusions that fight it. Nothing whatever happens at $\mathrm{Ra} = 1$. Nothing happens at 10, or 100, or 1,700.

At 1707.762 the layer starts to convect.

That number is not a comparison of two terms and cannot be made into one. It is the smallest eigenvalue of a sixth-order linear operator, minimised over the horizontal wavenumbers the layer is allowed to choose, and what decides it is the boundary condition rather than the fluid: two free surfaces give 657.5 and two rigid walls give 1707.8, a factor of 2.6 from nothing but what the top and bottom are permitted to do.

A discriminant #

Two more, and these are sharp. The threshold is where the character of a solution changes: real roots becoming complex, or a quadratic losing its roots altogether. There is no tolerance anywhere in such a statement and the answer comes out as an exact algebraic number.

A body in a stream collects nothing at all below a critical Stokes number, because near the front stagnation point the particle equation is linear and its roots stop being real at $4A\,\mathrm{St} = 1$; for a cylinder $A$ is exactly two, so the threshold is exactly one eighth. And an anticyclone’s balanced wind is the root of a quadratic whose discriminant vanishes at a geostrophic Rossby number of exactly a quarter, past which no balanced flow exists — which is why highs are broad and gentle while lows can be small and violent.

A characteristic condition #

And two where the folklore is right — for a reason that has nothing to do with term sizes.

At a Froude number of one the flow speed equals the speed of a surface wave. Past that, a disturbance cannot travel upstream at all, so the flow stops knowing what is ahead of it and the governing equations change type. Nothing is being compared; a signal is being cut off. The same argument with sound in place of a surface wave gives Mach one.

These are the only two numbers on the site whose threshold is exactly the value they are named for, and what makes them different in kind is that no tolerance appears anywhere in the statement.

The test that separates them #

Four kinds is a classification, and a classification that cannot be checked is a taxonomy of opinions. The check is one line: ask for the threshold at a different tolerance and see whether it moves.

A term ratio’s threshold is proportional to the tolerance. The other three do not move at all. That is not a heuristic — it is a property of how each threshold is defined, and it is why the sorting by ratio and the sorting by kind agree.

The sweep is bounded at both ends by what the rows can honestly answer rather than by the width of the picture. Below a tolerance of about $10^{-2.7}$ the drop-shape row is reading its own numerical noise, and above $10^{-0.7}$ the unsteady amplitude error has saturated at a half, so a threshold asked for at twenty per cent no longer exists. Both bounds belong to the rows and are declared with them. That is itself a small instance of the essay’s point: even the range over which a threshold can be quoted is a property of the solution rather than of the group.

What this does to a rule of thumb #

The practical consequence is not that rules of thumb should be discarded. It is that a rule of thumb has a tolerance in it, and the tolerance is the part that is never quoted.

The most quoted threshold in the whole subject is “air is incompressible below Mach 0.3”. Working it out: the density at a stagnation point rises by five per cent at $M = 0.314$. So the number 0.3 is a five per cent tolerance on the density at a stagnation point, and it has been passed down for eighty years with the tolerance stripped off — which is why nobody can say whether it applies to the shoulder of a wing that is working, where the density change is twice as large and the same tolerance falls at $M = 0.22$.

That number is worth following, because the two speeds it can be formed on are far apart. The nose of any body reaches five per cent at $M = 0.314$; a section with a suction peak of $c_{p0} = -2$ reaches the same five per cent at $M = 0.22$, and is already at Mach 0.55 over its own shoulder while the free stream reads 0.3. The crossover between the two conventions is exact and sits at $c_{p0} = -1$, which is the pressure coefficient of a stagnation point read backwards — so a body with any suction at all is on the wrong side of it.

The same is true of the thermal entry length, quoted universally as $x^+ = 0.05$. Computing what that delivers: a Nusselt number 1.45 per cent above the developed value. A perfectly sensible tolerance, and nobody says it is one — so nobody can tell whether their own problem needs a longer duct or a shorter one. Asked for at a tenth rather than a thousandth the required length moves by a factor of three and a half, and the quoted 0.05 is one unlabelled point on that curve.

Where the folklore came from, and why it survived #

It is worth being fair to the rule. For a term ratio the two numbers are within a decade whenever the tolerance is around ten per cent, and ten per cent is roughly where a difference becomes visible in a sketch. The folklore is calibrated to the eye, and for as long as the answer was a curve on graph paper it was calibrated well enough.

What changed is that the answers stopped being sketches. A structural load calculation wants three figures, a heat exchanger design wants two, and a flutter margin is decided by a phase angle whose first degree of error appears at a reduced frequency of 0.003 — three hundred times below the value at which the two terms of that group are equal.

What to report instead of a threshold #

The essay so far is a diagnosis, and a diagnosis with no prescription is half a job. The prescription falls out of the arithmetic in the second section and is short enough to state as a rule.

An observable that departs smoothly from its limiting theory does so as

$$Q = Q_0\left(1 + c\,\Pi^{\,p} + \dots\right),$$

with two numbers in it. Report the pair $(c, p)$, and every threshold anybody could want follows by division. A threshold is one point on that curve with its tolerance stripped off, which is exactly one number where two were available and is why the folklore is unrecoverable once written down. The pair is not harder to compute — it is what the computation produced before somebody solved for a value.

Of the two, the exponent is the one that decides how the group behaves, and it is the one never quoted. Knudsen’s channel has $p = 1$ with $c = 6$: a linear departure with a large coefficient, so the onset arrives at a five-hundred-and-ninety-fourth and moves by a full decade for every decade of tolerance. The Womersley number’s flow amplitude has $p = 4$: a fourth power climbs so steeply that a hundredfold change of tolerance moves its threshold by only a factor of three, which is why that group’s folklore threshold is very nearly honest while Knudsen’s is out by a factor of six hundred. A steeply-varying observable makes a well-behaved threshold and a gently-varying one makes a threshold that is mostly a report of how carefully somebody looked.

That also puts the four kinds on one axis instead of four. A term ratio has a finite $p$ and its threshold moves with the tolerance as $\varepsilon^{1/p}$. An eigenvalue, a discriminant and a characteristic condition are the limit $p \to \infty$ — the departure is not smooth at all, so no tolerance can move the answer, and the flat lines in the tolerance figure are what infinite exponents look like when they are drawn.

And which quantity was held fixed while the group moved #

One further clause belongs on every threshold and is almost never present. A group is a combination of several physical quantities, so changing the group is ambiguous: the Reynolds number can be raised by flying faster, by building bigger, or by cooling the air, and those are three different paths through the parameter space.

It does not matter when the observable depends on the group alone, which is the case the whole essay has assumed. It matters as soon as a second group is present and is not held fixed along the path. A Knudsen number can be raised by narrowing the channel at fixed pressure or by lowering the pressure at fixed width; the first leaves the Reynolds number falling and the second leaves it falling faster, and an experiment that reports “the departure began at $\mathrm{Kn} = 0.01$” has reported a point on whichever curve its apparatus could draw.

So a threshold is a triple rather than a number: the observable, the tolerance, and the path. The table in this essay fixes the first two explicitly and the third implicitly, by varying one group with every other held at the value its own solver was built for — which is the right convention and is a convention rather than a fact about the fluid.

Reporting the pair also makes a threshold auditable. Two authors quoting different onsets for the same group can be checked against each other in one line — the ratio of their thresholds should be the ratio of their tolerances raised to the power the exponent sets — and a disagreement that survives that test is a disagreement about the physics rather than about how carefully each of them looked.

What the picture cannot show #

Every “onset” here is an onset of one stated observable. The Knudsen row is about a flow rate; a heat-transfer coefficient in the same channel departs at a different Knudsen number, and a concentration profile at another. There is no such thing as the value at which a model fails, only the value at which it fails to do a stated job to a stated accuracy — which is the whole content of this essay stated the other way round.

The four kinds are not exhaustive, and nothing here proves they are. They are the four that fourteen groups on this site fall into. A fifth kind would be a group whose threshold is set by a bifurcation with hysteresis, where there are two thresholds and which one applies depends on which way the parameter is being moved; pipe transition is such a case, and it is left out of the table for exactly that reason.

And the balance is written as one rather than computed. For a well-formed ratio that is what “well-formed” means, and a row whose balance came out as anything else would be a row whose group had been assembled wrongly. Only the reduced frequency’s balance is computed, because the term comparison there involves the transfer function itself.

Who found it, and when #

Nobody found it, which is part of why it persists. Buckingham’s theorem is from 1914 and says nothing about thresholds; the intuition that a group of order one marks a transition is folklore in the strict sense — universally held, nowhere written down as a claim, and therefore never tested.

Where it has been tested it is treated as a local surprise rather than a general one. Knudsen measured the minimum in his tube’s flow rate in 1909 and it is still called a paradox. Barenblatt spent a career on the cases where a limit exists and nothing ever reaches it, and those are filed under “intermediate asymptotics” rather than under “the number was never the threshold”. Miles and Howard proved a sufficient condition for stability at a Richardson number of a quarter, and the quarter is quoted as a transition by people who would not defend the claim if asked.

The surprising connection is with the way this collection is organised. Its own index is by regime, and the axis of that index is the dimensionless group — so a table of fourteen groups whose named value is not their threshold is a table of fourteen places where the index is filed under a number that decides less than it appears to. The index stays, because the number is still the right way to sort a collection this size. What changes is that every one of them now has to say which of the two numbers it means.

Where the ladder goes next #

Above this rung is the harder question the four kinds sharpen rather than answer: given a new group, which kind is it, before the answer is known? The tolerance test settles it afterwards, and what would settle it beforehand is an argument about the structure of the equation — whether the term being dropped is regular or singular, whether it raises the order, whether it changes the type. That is the classification perturbation theory already has, under other names, and connecting the two properly is a rung this collection has not written.

Beside it sit the thirteen essays that supply the rows, each computing one of them from its own solver. Below it lie the theorem that says how many groups there are and the first number this collection introduced — which is a term ratio, and is one per cent honest at Reynolds number 0.054.