Counting what matters

The drag on a sphere in a steady stream depends on five things: the force $F$, the density $\rho$, the speed $U$, the diameter $d$ and the viscosity $\mu$.

Measuring a function of four variables properly means a four-dimensional table — ten values of each gives ten thousand experiments. The measurement that actually exists is one curve, and every sphere that has ever been dropped, blown at or towed lies on it.

The reduction from four dimensions to one is not a matter of luck or of physical insight. It is the rank of a matrix.

The theorem, as arithmetic #

Write each quantity’s dimensions as a column of exponents in mass, length and time. Any product of powers $\prod q_i^{a_i}$ has dimensions given by the matrix acting on the exponent vector $\mathbf{a}$, so the product is dimensionless exactly when $\mathbf{a}$ lies in the null space of the matrix.

That is the whole theorem. The number of independent dimensionless groups is the dimension of that null space, which by rank–nullity is

$$(\text{number of quantities}) - (\text{rank of the dimension matrix})$$

For the drag list the matrix has three rows and five columns, its rank is three, and the nullity is two.

No physics has been used. The theorem does not know that $\mu$ is a viscosity or that $F$ is a force; it knows their dimensions. Every step above would run identically on a list of quantities from acoustics, from finance if finance had dimensions, or from nothing at all.

What the solver computed, and how it was checked #

The matrix is reduced by Gaussian elimination with partial pivoting, its pivot columns are read off, and the null-space basis is constructed from the free columns. Three assertions then run.

Rank plus nullity is the number of variables. Computed both ways rather than assumed.

Every returned group is dimensionless. Each exponent vector is fed back through the dimension matrix and every component of the result must vanish, to $10^{-9}$. That check is worth having because a dimensionless number is not distinguishable from a dimensional one by looking at it: both are just numbers on a figure.

The groups are independent. The matrix of exponent vectors is itself reduced and must have full rank, or the “basis” contains a redundancy and the count is wrong.

The basis is not unique, and the theorem does not care #

The null space has a basis; it does not have the basis. What comes out of the elimination above is a pair of perfectly good groups that no textbook uses, and the familiar pair —

$$C_D = \frac{F}{\rho U^2 d^2}, \qquad \mathrm{Re} = \frac{\rho U d}{\mu}$$

— are combinations of them. The check assertGroupIsInSpan solves for those combinations and finds that $C_D$ is the first basis vector inverted, and $\mathrm{Re}$ is a half-power of one over a half-power of the other, both to a part in $10^{16}$.

That is not a formality. It is where a reader’s suspicion should land, because “the Reynolds number is the dimensionless group” is a claim with a hidden step in it: what the theorem says is that some two independent groups exist, and choosing $\mathrm{Re}$ rather than $1/\mathrm{Re}$ or $\mathrm{Re}^2$ is a convention chosen for convenience. The physics that makes the Reynolds number the useful choice — that it is the ratio of two terms in the momentum equation — comes from the equation rather than from the counting.

The doctored case makes the same point from the other side. A group with a length accidentally left in it — $F/\rho U^2 d$ rather than $F/\rho U^2 d^2$ — is refused by the dimensional check, which reports it as $\mathrm{L}^{1}$. Nothing about its appearance would have given it away.

What that buys, in experiments not done #

The consequence for the laboratory is the reason anybody cares.

$C_D = f(\mathrm{Re})$ is a function of one variable. Every sphere in every fluid at every speed and every size lies on that single curve, so the experiment is a curve rather than a table, and a measurement made on a 5 mm bead in glycerine is directly useful for a 5 m sphere in air, provided the two are at the same Reynolds number.

Adding physics, one dimension at a time #

Widen the problem and the count grows, and each new group is another number a model test must match.

Add the speed of sound, because the flow is fast enough for compressibility: six quantities, still rank three, three groups — and the new one is the Mach number.

Add gravity, because there is a free surface to make waves on: seven quantities, rank three, four groups, and the new one is the Froude number.

That is the counting behind the model that cannot match. A half-scale model at the same speed has half the Reynolds number and the same Mach number; correcting the Reynolds number by doubling the speed doubles the Mach number too, and there is no way out without changing the fluid. The impossibility is arithmetic before it is engineering, and this figure is where it comes from.

Two lengths, and the meaning of “geometrically similar” #

A body with two lengths in it — a chord and a thickness, a diameter and a length — adds a variable without adding a dimension, so it adds a group. That group is a pure ratio of lengths.

That is the formal content of the phrase geometrically similar: the shape ratios are themselves dimensionless groups, and a model test is only comparable when every one of them matches. It is why a model must be the same shape and not merely the same size class, and why a wing tested with a thickened root is not testing the wing that will be built.

The recipe, and why it is the same thing #

The version taught in laboratories does not mention rank. It says: pick $r$ repeating variables that between them contain all the dimensions and are not themselves dimensionless; then combine each remaining variable with them to make one group each.

For the drag list, pick $\rho$, $U$ and $d$. Then

$$\pi_1 = \frac{F}{\rho^a U^b d^c} \Rightarrow C_D, \qquad \pi_2 = \frac{\mu}{\rho^a U^b d^c} \Rightarrow \frac{1}{\mathrm{Re}}$$

with the exponents in each case fixed by requiring the result to be dimensionless — three equations, three unknowns, one solve each.

That recipe is the elimination above, performed by hand. The repeating variables are a choice of pivot columns; requiring each combination to be dimensionless is the null-space condition; and the number of groups is the number of non-repeating variables, which is $n - r$. The recipe fails exactly when the elimination would: if the repeating set does not span the dimensions, its columns are not independent, and the little linear systems have no solution.

Doing it as a rank computation rather than as a recipe has one practical advantage, which is that the machine cannot forget to check. assertBuckingham verifies the count, the dimensionlessness and the independence every time a figure is drawn — three things a hurried hand calculation routinely assumes.

The groups this site already runs on #

Every regime number in this collection is one of these, and each is a ratio of two terms in an equation:

• Reynolds, inertia against viscosity, which decides whether a wake exists at all;
• Mach, speed against signal speed, which decides whether the density can be treated as constant;
• Froude, speed against wave speed, which decides the angle of a ship’s wake and whether a channel flow is sub- or supercritical;
• Rayleigh, buoyancy against diffusion, which decides whether a heated layer convects;
• Strouhal, shedding frequency against convection, which is the one this site cannot compute;
• the cavitation number, ambient margin against dynamic pressure, which decides whether a liquid tears.

They are all the same kind of object and they were all found the same way. What distinguishes them is which physics was on the list.

The count is a ceiling, and engineering lives underneath it #

The theorem says the answer can depend on two groups. It does not say it does, and the cases where it depends on fewer are the ones that make the subject usable.

The sphere is the example already on the page. Over three decades of Reynolds number its drag coefficient sits near a half and barely moves, so in that band $C_D = f(\mathrm{Re})$ has collapsed to $C_D = \text{constant}$ and the two-group problem has become a one-group problem. Nothing in the counting predicted that; it is the statement that the dissipation does not follow the viscosity to zero, and it is a fact about the equations rather than about the dimensions.

That collapse is why an engineering drag coefficient is quotable as a single number. A car has a $C_D$ printed in its brochure, and the number is meaningful only because the vehicle spends its life in a range where the Reynolds dependence has flattened; the same number for the same shape at a tenth of the speed in glycerine would be wrong by orders of magnitude. A rough pipe does the same thing at its own end — past a Reynolds number set by the roughness, the friction factor stops depending on Reynolds number and depends on the roughness ratio alone, which is the flat right-hand side of the Moody chart and the reason a plumber can size a pipe from a table.

The general name for it is complete similarity in a group: the dependence on that group tends to a finite non-zero limit, so the group can be dropped and the problem loses a dimension. It is what every correlation, every handbook chart and every extrapolation from a model test relies on, and it is worth being explicit that the counting argument on this page has nothing to say about whether it holds.

Three ways it fails, and each of them is a place where a confidently quoted coefficient becomes wrong.

A group can re-enter abruptly. The sphere’s flat curve ends at the drag crisis, where the coefficient falls by a factor of four across a factor of two in speed. Anything designed on the flat part and operated across the crisis has been designed on a collapse that stopped.

The limit can be zero or infinite rather than finite. Then the dependence does not flatten; it turns into a power law whose exponent dimensional analysis cannot supply, and the problem has incomplete similarity — which this collection treats as an exponent dimensions cannot give.

And the collapse can hold for one observable and not another. A sphere’s drag coefficient is flat where its shedding frequency is not, so the same experiment supports dropping the Reynolds number from one answer and not from another. Whether a group may be neglected is a property of the question, not of the flow, which is the same conclusion this collection reaches about every threshold it computes.

So the honest reading of the theorem is that it draws the largest box the answer can live in. Where the answer actually lives is smaller, it is decided by the equations, and finding out how much smaller is most of what an experimental programme is for.

The sharpest thing it does #

Add a temperature difference to the drag list and something more interesting happens: the count does not change at all.

The theorem is saying: either the temperature cannot matter, or the list is missing whatever else carries its dimension. A conductivity, a specific heat, an expansion coefficient — anything that would let the temperature combine with the rest.

That is a real result about an experiment, obtained from a rank computation. If a measurement shows a dependence on temperature that the list cannot accommodate, the list is wrong, and the theorem has found the omission before any data was taken. Used this way, dimensional analysis is a check on whether a problem has been posed properly rather than a way of guessing answers.

An argument that predicted an atomic bomb’s yield #

The most famous single use of the method is worth stating because it shows how much can come from how little.

A large explosion in air produces a roughly spherical blast wave. The quantities that could matter are the radius $R$, the time $t$, the energy released $E$ and the ambient density $\rho$: four quantities, three dimensions, rank three, so one dimensionless group,

$$\Pi = \frac{R^5\rho}{E t^2}$$

and since the answer can only be that this group is a constant,

$$R \propto \left(\frac{E t^2}{\rho}\right)^{1/5}$$

The radius goes as the two-fifths power of time, and the constant of proportionality is of order one. G. I. Taylor took a sequence of published photographs of the first atomic test, each with a scale and a time on it, plotted $\log R$ against $\log t$, found the slope of two fifths, and read the energy off the intercept. The yield was classified; the photographs were not.

Everything in that argument is on this page. The rank fixed the number of groups at one, one group means an equation rather than a function, and a fitted intercept then gives a number nobody had published. It is also a fair demonstration of the method’s limits: the shape of the constant needed a full similarity solution, and the choice of the four-quantity list needed the physical judgement that viscosity, ambient pressure and the details of the device could all be neglected during the early expansion.

What the picture cannot show #

The function. The theorem gives the arguments and never the function. $C_D = f(\mathrm{Re})$ is as far as counting goes; the shape of $f$ — including the drag crisis, which is a band where the drag force falls as the speed rises — requires either the equations or a measurement.

Which variables belong. Leave the viscosity out of the drag list and the theorem returns one group and a confident wrong answer, that $C_D$ is a constant. Include something irrelevant and it returns a spurious group, and the data will show no dependence on it. Choosing the list is physics, and it is the only step in the procedure that can be wrong.

Whether a group is small enough to drop. The counting says the answer depends on both $C_D$’s arguments; it does not say that at $\mathrm{Re} = 10^7$ the dependence is weak. That judgement is what makes an approximate theory possible and it comes from elsewhere.

Where the model stops #

Dimensional analysis assumes the system of units is complete and independent — that mass, length and time really are separate. Choose a different system and the count changes: in a subject where temperature is measured in energy units, the temperature above stops being a lone dimension, and the group count goes up by one.

That freedom is not a flaw, and it is worth understanding rather than worrying about: the physics is what fixes which quantities are independent. Rayleigh’s original examples occasionally got this wrong, and the resulting arguments in the pages of Nature in 1915 are still instructive.

The theorem also assumes there is a complete list of relevant quantities. In a problem with a large range of scales — turbulence, most obviously — there may be quantities that matter in one part of the range and not in another, and a single set of groups then describes the whole problem badly. That is the situation Kolmogorov’s argument navigates by applying the counting separately in each range.

Who found it, and when #

Vaschy stated the theorem in 1892 and was largely ignored. Rayleigh used the method throughout his career and set it out plainly in “The Principle of Similitude” in 1915, which is the paper that made it standard practice — and which provoked a famous exchange when Riabouchinsky pointed out that counting molecular degrees of freedom differently changes the answer, an objection Rayleigh answered by insisting that the physics decides the list.

Buckingham’s 1914 paper is where the modern statement and the $\pi$ notation come from, and his name is attached to it in English-speaking practice. Bridgman’s Dimensional Analysis of 1922 is the careful treatment of what it does and does not license.

Where the ladder goes next #

Counting gives the arguments of the answer and never the answer. The next step is a case where a single dimensionless group does something sharper than merely being an argument: a threshold, below which the answer is not small but exactly zero.