A limit nothing reaches

Counting what matters works Buckingham’s theorem rather than quoting it: a set of variables whose dimension matrix has rank $r$ gives exactly $n - r$ independent dimensionless groups, and which groups they are is the null space of the matrix. The theorem is exact and it is about counting.

What it cannot do is say what happens to the answer when one of those groups becomes large or small. That is a question about the solution, and there are two possible answers, and they are almost never distinguished.

The group drops out. The answer approaches a finite limit that does not contain it, and the limit is reached quickly enough to be useful. Barenblatt calls that complete similarity, and it is what everybody assumes.

The group does not drop out. The answer approaches a limit that the group leaves only logarithmically, or approaches nothing at all, or approaches a limit with a power of the group still attached. That is incomplete similarity, and this collection has one of each.

The case where it works #

The Blasius boundary layer is the clean case, and it is worth being precise about what is meant.

Solve the laminar layer on a flat plate and the velocity profile depends on $x$, $y$, $U$ and $\nu$ — four quantities, two dimensions, so two groups. It could depend on both. It does not: it depends on one combination, $\eta = y\sqrt{U/\nu x}$, and the shape $f'(\eta)$ is a single universal curve. The wall gradient is a pure number,

$$f''(0) = 0.332057,$$

and it is that number at $\mathrm{Re}_x = 10^3$ and at $10^6$ and at $10^{20}$.

That is what a successful dimensional argument looks like. The variable is gone, it was gone at the first Reynolds number anybody tried, and there is no approach to speak of.

The case where it does not #

Now the overlap layer of a turbulent channel. Millikan’s argument is dimensional: in a region far enough from the wall that the viscosity has stopped mattering and near enough to it that the channel’s half-width has not started, the velocity gradient can depend only on the friction velocity and the distance. Then $y\,du/dy$ is a pure number, and integrating gives the logarithmic law.

The layer with no length in it computes that, and finds the constant coming out at 0.511 before the limit and 0.412 after it. The essay’s own conclusion is that the log law is a limit rather than a fact about any particular Reynolds number.

Barenblatt’s objection is sharper: what if the viscosity does not stop mattering — if it enters weakly, through a power? Then instead of a logarithm the profile is a power law,

$$u^+ = C(\mathrm{Re})\,(y^+)^{\alpha(\mathrm{Re})},$$

with an exponent that shrinks with Reynolds number rather than vanishing at any of them. A logarithm is the $\alpha \to 0$ member of that family, so the two pictures agree in the limit and disagree everywhere else.

What this site’s own profile says #

The collection has a mixing-length channel solve, and fitting a power law across the plateau its diagnostic function finds gives an exponent at each Reynolds number.

It falls, monotonically, and it falls slowly. A factor of five hundred in Reynolds number buys a reduction of twenty-two per cent in the exponent.

Regressing against $1/\ln \mathrm{Re}_\tau$ — the form Barenblatt’s argument predicts, fitted through the origin with one parameter — gives $a = 1.140$, against his own 3/2. Inverting: an exponent of 0.01 needs $\ln\mathrm{Re}_\tau = 114$, that is $\mathrm{Re}_\tau \approx 3\times10^{49}$.

The limit exists. Nothing ever gets there.

The sequence does not even fall that fast #

Being honest about the fit makes the conclusion stronger rather than weaker.

The $1/\ln\mathrm{Re}$ form describes the sequence to within about twenty per cent over the range computed, which is why it is used above. It is not the best description: regressing $\ln\alpha$ against $\ln\ln\mathrm{Re}_\tau$ gives a straight line with slope $-0.427$, fitting far better, and implying $\alpha \propto (\ln\mathrm{Re}_\tau)^{-0.43}$.

On that fit, an exponent of a hundredth needs $\ln\mathrm{Re}_\tau \approx 3{,}270$. The Reynolds number itself overflows every floating-point format there is, and the only way to write it down is as its own logarithm.

That number is a property of this site’s closure rather than of turbulence, and the figure that draws it says so. What is not closure-dependent is the direction: the sequence falls more slowly than $1/\ln\mathrm{Re}$, so whatever the true behaviour is, the limit is further away than Barenblatt’s own estimate rather than nearer.

Why it matters that the two look alike #

The practical difficulty is that a complete and an incomplete similarity are almost impossible to tell apart from data over a decade.

Over one decade of $y^+$, a logarithm and a power law with $\alpha = 0.13$ differ by about two per cent — which is inside the scatter of every measurement ever made in a pipe, and inside the uncertainty in the friction velocity that both are normalised by. The controversy over whether the log law or a power law is right has run for thirty years and cannot be settled by better measurements in the same range, because the question is about behaviour at Reynolds numbers nobody can reach.

That is the general hazard of an incomplete similarity, and it is not confined to turbulence. It applies whenever a limit is approached logarithmically: the data look converged, the extrapolation is unconstrained, and the difference shows up only where nothing can be measured.

Where else the same thing happens #

Three cases from elsewhere in this collection have the same structure and are usually described as though they did not.

The drag crisis. A sphere’s drag coefficient — the one a ball’s flight depends on — is quoted as constant at about 0.5 across the subcritical range, and it is not: it drifts by tens of per cent over three decades and then falls by a factor of five at the crisis. The plateau is not a limit, it is a slow variation being read as one.

Fully rough pipe flow. The friction factor is said to become independent of Reynolds number, which is the roughness a wall cannot feel read from the other side, and it does — but the criterion is not a Reynolds number at all; it is the roughness Reynolds number $k^+ = k u_\tau/\nu$ exceeding about 70, which mixes a geometric ratio with the flow. A pipe can be at $\mathrm{Re} = 10^8$ and not be fully rough.

Dissipation. The limit that is not the value is the purest case: the dissipation rate becomes independent of viscosity as the viscosity vanishes, which is an experimental fact of ninety years’ standing and is not a theorem, and the approach to it is slow enough that low-Reynolds-number simulations still show a residual dependence.

The test that separates them, and it is cheap #

There is a practical test, and it costs one extra calculation rather than an argument.

Compute the quantity at two Reynolds numbers a decade apart and see whether it moved. If it did not, the similarity is complete and the limit is already there. If it did, the similarity is incomplete and the only question left is how fast the movement decays — which needs a third point, and which the movement between the first two cannot answer.

Applied to the two cases here: Blasius’ $f''(0)$ is 0.332057 at every Reynolds number, to as many figures as the solver carries. The overlap exponent moves by five per cent per decade at the top of the range computed and by two per cent per decade at the bottom, which is small, unmistakable and inconvenient.

The test is cheap and it is almost never run, for a reason worth naming: a quantity that has been non-dimensionalised looks like a constant. Writing $f''(0)$ or $\kappa$ or $C_D$ signals that the Reynolds number has been dealt with, and the notation itself discourages asking whether it has. A plot against Reynolds number is the whole of the test, and the habit of plotting non-dimensional quantities against each other rather than against the group they were supposed to eliminate is what hides it.

What to do about it, which is a change of what gets reported #

The test in the previous section decides which kind of similarity is in front of somebody, and it is worth saying what follows from each answer, because the two lead to entirely different programmes of work.

If the similarity is complete, a single measurement suffices. Take it at whatever Reynolds number is convenient, report the number, and use it anywhere. That is the assumption every handbook coefficient is published under, and where it holds it is enormously valuable — it is the whole reason a model test means anything.

If the similarity is incomplete, a single measurement is worth much less, and the difficulty is not that it is inaccurate. It is exact at its own Reynolds number and says nothing about any other, so the quantity has to be measured at several and the form of the variation fitted before anything can be extrapolated. And the extrapolation is then only as good as the form — which is this essay’s own arithmetic in miniature, where two fits that both describe three decades of computed exponents to within twenty per cent gave limiting Reynolds numbers of $e^{114}$ and $e^{3270}$.

Two forms that agree over the measured range can disagree by any amount outside it, and no amount of care within the range distinguishes them.

The aeronautical case where this bites hardest is the maximum lift coefficient. It rises with Reynolds number across the whole span from a small tunnel to flight, by tens of per cent, with no plateau anywhere in the range anybody can measure — the layer is thinner, it survives more of the pressure recovery, and the separation moves aft. A landing speed inferred from a model test is therefore an extrapolation along a curve whose form is fitted, not a reading of a converged coefficient, and that is the honest reason certification requires the aeroplane to be flown rather than the model to be trusted. It is the same difficulty a model in a tunnel meets in general, stated in the vocabulary of similarity rather than of matching.

The remedy is a small change in what a measurement reports, and it is the practical conclusion of this whole ladder. A coefficient should be published with its Reynolds-number derivative, or with the two or three Reynolds numbers it was measured at, rather than as a single number. A quantity quoted alone has thrown away the one piece of information that says whether quoting it alone was legitimate — and it costs nothing to keep, because any programme that measured the number at all measured it at some Reynolds number and usually at several.

That is the same discipline this collection applies to a threshold, which is quoted with its tolerance, and to a figure, which is quoted with its regime. A number without the variable it was supposed to have eliminated is a number that cannot be checked, and the habit of eliminating the variable in the notation rather than in the physics is what this essay is about.

What the theorem does and does not promise #

It is worth returning to Buckingham, because none of the above is a criticism of the theorem.

The theorem says how many groups there are, and working it rather than quoting it is what makes that clear. It says nothing about the function of them, and in particular it does not say the function has a finite limit as any group goes to infinity, nor that the limit is approached quickly, nor that it is approached at all. Every one of those is an additional assumption, made silently, whenever a group is described as large enough not to matter.

The assumption is usually right, which is why it is invisible. Complete similarity is the common case: it is why model testing works, why non-dimensional plots collapse, and why most of this collection’s figures can be drawn at all. The failures are rare and they are concentrated in exactly the places the subject finds hardest.

Two constants, and only one of them is one #

The comparison worth carrying away is between two numbers this collection prints often.

$f''(0) = 0.332057$ is the solution of an ordinary differential equation with no parameters in it. It is a mathematical constant in the same sense that $\pi$ is: computable to any precision, identical for every fluid and every speed, and not measured.

$\kappa = 0.41$ is not that. It is the limiting value of a quantity which, on every profile anybody has measured or computed, is measurably different at every Reynolds number — 0.511 before the limit and 0.412 after it on this collection’s own diagnostic. Published values range from 0.38 to 0.43 and the spread is not measurement error; it is the Reynolds numbers of the facilities.

Printing both to three figures in the same paragraph, as textbooks do, is the notational habit above made concrete. One is a constant and the other is an asymptote, and nothing in the way either is written distinguishes them.

What the picture cannot show #

These exponents are the closure’s. They are measured on a mixing-length profile, and a mixing length is a model with a constant in it. A direct simulation gives a different sequence and the extrapolations would move; what would not move is that the exponent is positive at every Reynolds number computed and falling more slowly than any power of it.

The fit is a fit. Regressing against $1/\ln\mathrm{Re}$ tests a form rather than discovering one, and reporting both that fit and the better one is the honest way to say that the sequence does not determine its own asymptotics over three decades.

Below Re_τ = 2,000 there is no overlap to fit. The plateau the diagnostic function finds is under half a decade wide there, so the “power law across the overlap” would be a power law across the buffer layer and the exponent is not monotone at all. The ladder begins where the overlap does, and says so.

And nothing here settles the controversy. Whether real turbulence has a log law or a power law in the limit is not decided by a closure that was built to produce a log law. What the closure does establish is the shape of the difficulty: even a model designed to give a logarithm takes an unreachable Reynolds number to give one.

Who found it, and when #

Buckingham’s theorem is 1914, though Vaschy had it in 1892 and Rayleigh was using the method without stating it decades earlier. Millikan’s overlap argument is 1938. Barenblatt’s work on incomplete similarity runs from the 1970s onward and the specific proposal about the overlap layer from 1993, which provoked an exchange with Zagarola, Perry and Smits that has not concluded.

The controversy is genuinely open and this essay takes no side in it. What is not open is the general point, which predates the argument about turbulence by a century: a limit’s existence and its accessibility are different questions, and dimensional analysis answers neither.

The surprising connection is with a number this collection quotes constantly. The Blasius wall gradient, 0.332057, is a pure number because a similarity worked; the von Kármán constant, quoted as 0.41, is not a pure number in the same sense — it is the limiting value of a quantity that is measurably different at every Reynolds number anybody has built. Two constants of the same subject, printed to the same number of figures, and only one of them is a constant.

Where the ladder goes next #

Above this rung is the matched asymptotics itself: the machinery for handling a problem with two regions and no single expansion, which Oseen’s correction to Stokes’ law needed for the same reason at the other end of the Reynolds axis, and which produces a logarithm there too.

Beside it sits the layer with no length in it, where the overlap’s own constant is measured, and the dissipation anomaly, which is the same kind of limit taken seriously. Below it is the theorem that counts the groups and promises nothing about them.