Where a pure number comes from

Counting what matters is the collection’s account of what dimensional analysis does: it takes a list of quantities and a matrix of dimensions and returns the number of independent dimensionless groups the answer can possibly depend on. That is a real and useful result, and it has a hard boundary.

It never produces a value. It says the friction factor of a pipe is a function of the Reynolds number alone; it cannot produce the 64. This essay is about where the 64 comes from, and about the three other places numbers like it come from, and about the fact that one of the four is systematically worse known than the rest.

Fourteen of them #

Every entry is dimensionless, exact, and quoted somewhere as a fact about fluids. They fall into four kinds.

Algebra — a ratio falling straight out of the equations. The escape speed $2/(\gamma-1)$, which is exactly five for air; the exactly $-2U$ slip of Stokes’ paradox; the cube in a weak shock’s entropy. These have no computation behind them at all beyond rearrangement.

An integral — the mean of a solved field. The Poiseuille numbers: exactly 64 for a circle, 96 for parallel plates, 160/3 for an equilateral triangle. A field is solved and averaged, and the average is the number.

A root — where a computed function crosses zero. Blasius’ wall shear $f''(0) = 0.3320573$, found by shooting until the far-field slope comes out at one; the jet’s half-width $2\sqrt{\sqrt2-1}$, the solution of an algebraic equation.

An optimum — where a computed function is stationary. Betz’s $16/27$; the bearing’s convergence ratio 2.1887048; the critical Rayleigh number 1707.76, which is a minimum over the wavenumber.

The worst check in the table is the critical Rayleigh number at three parts in a million, which is a Richardson extrapolation of a grid eigenvalue solve. The rest are at round-off or at the accuracy of a shooting integration.

What each kind costs to produce #

The four kinds differ in more than how well they are known; they differ in what it took to get them, and that is worth setting out because it explains why some of these numbers are two centuries old and some are recent.

Algebra is free. The escape speed is one line from an invariant; the $-2U$ of Stokes’ paradox is four constants and two boundary conditions. Nothing has to be integrated and nothing converged, which is why these were all known by 1850.

An integral needs a solved field, and the field is usually the hard part. The 64 requires the Poiseuille profile, which is a two-line solve; the 160/3 requires the triangle’s, which is not obvious; and a shape without a closed form requires a Poisson solve on a mesh, which is where a number that is only the shape of the hole spends its effort.

A root needs a function that can be evaluated and a search. Blasius’ 0.332 is a shooting problem — guess, integrate, correct — and although the method is elementary it is not something that can be done by hand to seven figures, which is why the accepted value moved in the fourth decimal place for several decades after 1908.

And an optimum needs the same function and one more derivative, or a search that is ill-conditioned in exactly the way the rest of this essay is about.

So the four kinds are also a rough chronology of the subject, and the numbers that were available earliest are the ones that are exact and the ones that came last are the ones with the fewest reliable digits.

A root and an optimum are not equally knowable #

Here is the finding.

Blasius’ wall shear is located by a residual — the far-field slope minus one — that crosses zero transversally, with a slope of 2.008. So if the residual is known to $\delta$, the root is known to $\delta/2.008$. Every digit of the objective buys a digit of the answer.

Betz’s coefficient $4a(1-a)^2$ has its maximum at exactly $a = 1/3$, and the curve is flat there — the second derivative is $-8$. If the objective is known to $\delta$, the location is known only to $\sqrt{2\delta/c}$.

A square root. Half the significant figures are gone before anybody starts.

The bearing’s is worse. Its load bracket’s curvature relative to its own value is 0.41, so the same precision locates the ratio thirty times less well again — which is why the number printed almost everywhere, 2.1889, is not the maximum, why nobody noticed for decades, and why it does not matter.

At a relative precision of $10^{-6}$: the two roots are located to five and seventeen parts in ten million, and the two optima to about one part in a thousand. A ratio of 676.

Over ten decades of precision, a root’s uncertainty is proportional to the precision — exponent 1.0000000000000002 — and an optimum’s goes as the square root — exponent 0.5. Neither is fitted. Both are the leading term of a Taylor expansion, and the half is the reason.

Which is a rule about how many figures to print #

A number that is algebra is exact and may be printed to any length. One that is an integral or a root is worth as many digits as its computation is converged to. An optimum’s location is worth half of that, though its value is worth all of it — which is a distinction the literature does not usually make and is why $16/27$ is quoted to four figures while the induction factor that achieves it is quoted as “a third”.

It also explains a set of results in this collection that looked like separate curiosities. The elliptic wing’s optimum is flat — a band of taper ratios from 0.31 to 0.42 within a tenth of a per cent. The bearing’s is flat. Kelvin’s minimum energy theorem cannot identify a wrong flow, because the energy error is the square of the velocity error. Every one of those is this square root, and none of them is a fact about wings or bearings or energy.

And the flutter boundary is the exception that proves it. It is a root, so it should be in the good class — and it is not, because its slope is small: the damping changes by 0.0975 per unit speed ratio, so the divisor is a tenth rather than a one and a half-per-cent damping error becomes a five-per-cent speed error. A root with a small slope behaves like a stationary point without being one, and the quantity to look at is always the derivative rather than the classification.

A note on checking a number computed elsewhere #

The catalogue is checked against values it is “supposed to be”, which is a circular-sounding procedure and is not, provided the check is arranged properly.

For the entries that are algebra, the check is that two routes to the same expression agree — the escape speed from the invariant and from the energy budget, for instance. For the integrals, the check is a quadrature against a closed form, and each closed form was derived independently of the quadrature. For the roots, the check is that the defining residual is small at the accepted value, and that residual can be evaluated without knowing the answer. For the optima, the check is that the derivative vanishes there.

In every case the check is against a condition rather than against a number. That distinction is what makes the exercise worth doing: comparing a computed value with a published one tests only whether they agree, and agreeing with a published value that is wrong is the most common way for a mistake to survive. Comparing a computed value with the equation it is supposed to satisfy tests something.

The one entry that could not be checked that way is the critical Rayleigh number, whose defining condition is an eigenvalue problem that has to be solved to state it — and it is also the entry with the worst agreement, at three parts in a million.

Why the square root, in two lines #

The derivation is short enough to have in full, and having it makes the rule stick.

Near a root, $g(x) \approx g'(x^*)(x - x^*)$. If $g$ is known only to $\delta$, the set of $x$ consistent with $g = 0$ has width $2\delta/|g'|$. Linear in $\delta$.

Near an optimum, $f(x) \approx f^* - \tfrac12 c (x - x^*)^2$, because the first derivative vanishes — that is what makes it an optimum. If $f$ is known only to $\delta$, the set of $x$ consistent with $f = f^*$ has width $2\sqrt{2\delta/c}$. Square root in $\delta$.

The whole content is that at a stationary point the first term of the expansion is missing, so the uncertainty has to be carried by the second, and a second-order relation inverts to a square root.

Two consequences worth carrying beyond fluid mechanics. A quantity measured at its own maximum is badly located — which is why a resonance’s frequency is found from its phase crossing rather than from its amplitude peak, and why an optical fringe is centred on a zero rather than on a bright band. And an optimisation’s answer is more robust than its argument, which is usually the good news: a design at the wrong point of a flat optimum performs almost as well, and that is the practical content of every flat optimum in this collection.

How to locate an optimum anyway #

The square root is a property of the objective, and it is not a property of the procedure. There is a standard escape, and this essay has already used it once without saying so.

Differentiate, and find a root. A stationary point is a zero crossing of the derivative, and a zero crossing is in the good class: located linearly in the precision rather than as its square root. The bearing’s 2.1887048 above was obtained that way — by bisecting the derivative — precisely because searching the load itself cannot distinguish neighbours that agree to parts in $10^{13}$. Converting a maximum-finding problem into a root-finding one recovers the digits the stationary point threw away.

The same move in a laboratory is a symmetric measurement: instead of hunting for the peak, measure at two settings either side and null the difference. That is a zero crossing too, and it is why a resonance is located by its phase rather than by its amplitude and why a null instrument beats a deflecting one.

The catch is where the objective is noisy. Differentiating amplifies noise, so the trick works when the objective can be evaluated to high precision — a computation — and fails on a measurement. There the right move is the opposite: fit a parabola over a wide span either side of the peak, which averages the noise down and uses the curvature that made the problem hard as the thing that solves it.

The one number here that changed a design decision #

Most of the numbers in the catalogue are quoted and used, and one of them was quietly wrong for long enough to be worth a paragraph.

The bearing’s optimum convergence ratio is printed as 2.1889 in the standard references. The maximum of the load bracket is at 2.1887048, found by bisecting the derivative rather than by searching the load — which is flat to parts in $10^{13}$ over that range and cannot locate its own maximum, so a search on the load returns whichever neighbour the arithmetic happened to favour.

The difference costs eight parts in a hundred billion of load. Nothing whatever depends on it, and no bearing has ever been mis-designed because of it.

That is exactly the point. The error survived because it did not matter, and it did not matter for the same reason it happened: the objective is stationary there, so its maximum is nearly impossible to locate and nearly irrelevant to hit. A number quoted to five figures in a quantity whose fourth figure changes nothing is a number nobody has an incentive to check, and the square-root rule predicts both halves of that.

What dimensional analysis is actually for #

Nothing above diminishes it. Three things it does that nothing else does:

It bounds the experiment. Five quantities decide the drag on a sphere and the experiment that measures it has one curve rather than a five-dimensional table, which is the content of counting what matters and is worth more than any single number.

It says which experiments are the same experiment. Two flows with the same dimensionless groups are one flow, which is what makes a model test worth doing at all — and which is exactly where the model that cannot be matched records what happens when two groups cannot be matched at once.

It says when a variable has dropped out, which is a physical statement. That the Blasius profile’s shape contains no Reynolds number at any Reynolds number is a fact about the flow, discovered by counting.

And it says what kind of thing the answer is. An exponent that dimensional analysis produces is exact and rational; one it cannot produce is an eigenvalue that has to be computed and is generally irrational and gas-dependent. Knowing which of the two is in hand beforehand is most of knowing what to expect.

What it cannot do is produce a value, and the four kinds above are the four ways values are actually produced.

The numbers that are not on the list #

Two categories are absent from the catalogue and their absence says something.

Fitted constants. The von Kármán constant, the Kolmogorov constant, the eddy-viscosity coefficient in a jet, the closure constant $C_\mu$ — all dimensionless, all quoted to three figures, and none of them the answer to a problem anybody solved. They are measured, they differ between experiments by more than their quoted precision, and treating them as pure numbers of the kind in the table is the mistake that a guess with a constant in it is about.

The test that separates the two is simple: can the number be computed from a stated problem without looking at any data? Every entry in the catalogue can. None of the four above can.

And numbers that are exact in a limit nobody reaches. Kolmogorov’s $-5/3$, the log law’s slope, the enstrophy range’s $-3$ — all exact, all correct, and all approached logarithmically, which is the three that never converge. Those belong in a third category: a value that is exact, computable, and never observed, whose relationship to a measurement is governed by a rate rather than by a precision.

Sorted that way there are six kinds of dimensionless number in this subject rather than four. The four in the catalogue are the ones that are both exact and reachable, and being clear about which of the six a number is is worth more than any amount of care in quoting it.

What is not claimed #

Fourteen is not a census. They were chosen to span the four kinds and to be ones this collection has already computed. There are certainly others, and there is no claim about the relative frequency of the four kinds in the subject at large.

The classification is about how a number is found, not about what it is. The critical Rayleigh number is an eigenvalue of a boundary-value problem and also a minimum over a wavenumber; it is filed as an optimum because that is the step that locates it, and the eigenvalue solve is what supplies the objective.

The uncertainty formulae are leading-order. $\delta/|s|$ and $\sqrt{2\delta/c}$ are the first terms of expansions, valid when the errors are small enough that the function is well approximated by its tangent or its parabola. Where they are not, both are optimistic.

The precision is treated as a single number. In practice an objective’s error has structure — it is larger in some places than others, and correlated — and both formulae assume it is a uniform bound. That is the right first approximation and it is not the whole story for a computation whose error grows with the parameter being swept.

And an optimum’s value is known as well as anything. The square root is about its location. Betz’s $16/27$ is exact and is worth every digit; what is worth half as many is the induction factor at which it is reached, and confusing the two is the mistake this essay is arguing against in the other direction.