The roughness a wall cannot feel

A water main is designed without knowing what temperature the water will be. That is a strange thing to be able to do. The pressure drop along a pipe is a viscous effect, viscosity varies by a factor of two between a cold winter and a hot summer, and yet the engineer picks a diameter, quotes a head loss and is right.

The reason is that above a certain speed the friction in a rough pipe stops depending on the Reynolds number at all. Not weakly: not at all. And the reason for that is a competition between two lengths — the height of the bumps on the wall and the thickness of the film of slow fluid that covers them — with the second one shrinking as the flow speeds up.

One exact line, and everything else fitted #

The chart above is the most-consulted diagram in engineering fluid mechanics, and it is worth being precise about what kind of object each part of it is.

The straight line on the left is a theorem. Laminar pipe flow has a closed-form solution — the parabolic profile, and a friction factor of exactly $64/\mathrm{Re}$ — derived on this site as the flow that stays a solution after the flow stops taking it. It contains no fitted constant and no roughness, because a laminar flow does not feel the wall’s texture: the fluid follows the contour, the disturbance decays, and a rough laminar pipe and a smooth one have the same friction to within a per cent.

Everything to the right of Re ≈ 4000 is a correlation. Colebrook’s equation interpolates between two limits that are themselves semi-empirical, it was fitted to commercial pipe in the 1930s, and the roughness that goes into it is an equivalent sand grain — the grain size of Nikuradse’s glued-sand pipes that would give the same friction — rather than anything a stylus measures on a real surface.

And between 2300 and 4000 there is a band where neither branch holds, shaded on the chart. A pipe there can be laminar, turbulent, or intermittently both, and which one depends on the inlet, the vibration and the history. It is the same Reynolds number that is not a number that transition carries everywhere: a property of the laboratory as much as of the fluid.

The distinction matters because the chart presents all three in the same ink at the same weight, and a designer reading a head loss off it has no way to know that one of the three is a derivation and one of them is somebody else’s plumbing.

It matters in a second way that is easy to miss. The exact branch and the fitted branch disagree about what the answer depends on. On the laminar line the friction factor depends on the Reynolds number and on nothing else — not the roughness, not the material, not the history of the pipe. On the fully rough asymptote it depends on the roughness and on nothing else — not the Reynolds number, not the viscosity, not the temperature. Those are two completely different design problems that happen to be plotted on the same axes, and the transition between them is where the chart is doing the least honest work.

A designer who understands only that the chart gives a number will get the number right and will not know which variables to worry about. A designer who has noticed the two asymptotes knows immediately that a laminar duct is temperature-sensitive and a rough main is not, which is the more useful thing to have taken away.

The equation that cannot be rearranged #

Colebrook’s correlation is implicit:

$$\frac{1}{\sqrt{f}} = -2\log_{10}\left(\frac{\varepsilon/D}{3.7} + \frac{2.51}{\mathrm{Re}\sqrt{f}}\right)$$

The friction factor appears on both sides and cannot be extracted, which is the whole reason the Moody chart exists as a chart: before iteration was cheap, reading a curve was the practical way to solve it.

Here it is solved. The fixed-point iteration $x \leftarrow -2\log_{10}(\ldots)$, with $x = 1/\sqrt{f}$, converges in about eight passes from any sensible start, and the solver returns the residual so that an assertion can check the root rather than trusting the loop count. It comes out at zero to machine precision.

The two limits it interpolates between are each worth a line:

• Delete the roughness term and it becomes Prandtl’s smooth-pipe law, which is what a turbulent boundary layer over a flat wall gives. The check is worth a footnote of its own: the constant that falls out is $2\log_{10} 2.51 = 0.79988$, and every textbook prints 0.8. Testing the reduction against the rounded constant would put a floor of $1.2\times10^{-4}$ under an assertion whose job is to catch a solver that has stopped converging, so the exact value is used.
• Delete the Reynolds term and it becomes von Kármán’s fully-rough law, in which $f$ depends on the relative roughness alone.

The length the wall is measured against #

The quantity that decides everything is not the roughness. It is the roughness divided by the thickness of the viscous film at the wall, and that film gets thinner as the flow gets faster.

Everything happens in a thin layer is the site’s oldest motif and this is its sharpest instance. Right against a wall, turbulent fluctuations are suppressed — a fluctuation towards the wall has nowhere to go — and there is a film a few tens of microns thick in which the flow is essentially laminar and the shear is essentially uniform. The natural length there is $\nu/u_\tau$, built from the viscosity and the friction velocity, and measuring anything in units of it is what the law of the wall is written in.

So the roughness height gets measured the same way:

$$k^{+} = \frac{k\,u_\tau}{\nu} = \frac{\varepsilon}{D}\,\mathrm{Re}\,\sqrt{\frac{f}{8}}$$

and the three regimes are three bands of that one number.

Hydraulically smooth, $k^{+} < 5$: the roughness is inside the film. The turbulent flow above never touches it, the friction is the smooth-wall friction, and polishing the pipe buys nothing.

Transitional, $5 < k^{+} < 70$: the bumps are beginning to poke through. Both the viscosity and the roughness matter, and this is the only region where the interpolation in Colebrook’s equation is doing real work rather than reproducing a limit.

Fully rough, $k^{+} > 70$: the bumps are large compared with the film, which has effectively been destroyed. The drag on the wall is form drag on individual roughness elements, and form drag on a bluff obstacle at high Reynolds number is proportional to the dynamic pressure and independent of the viscosity — which is exactly why $f$ stops depending on $\mathrm{Re}$.

What was computed, and what the assertions catch #

Three checks run before any curve is drawn, and each catches a different failure.

The root is a root. The residual of the implicit equation is computed at the returned value and required to be below $10^{-10}$. An iteration that had stopped early would produce a friction factor a per cent or two out, which is invisible on a logarithmic chart and would be wrong in every number an essay quoted from it.

It reduces to the smooth law. With the roughness set to zero the result must satisfy Prandtl’s equation, which it does to $10^{-9}$ against the exact constant.

It becomes Reynolds-independent. At a relative roughness of 0.02, the friction factor at $\mathrm{Re} = 10^7$ and at $10^8$ are compared: they differ by 0.007 per cent across a whole decade, with $k^{+}$ at 15,595. That is the claim the essay’s refutation turns on, and a correlation whose roughness term had been implemented wrongly would still fall with Reynolds number there and would still look entirely convincing.

The solver also refuses two things rather than computing them. It refuses a Reynolds number below 4000, because there is an exact solution in that range and replacing a closed form with a fit to somebody else’s pipes is a strictly worse answer that looks identical. And it refuses a relative roughness above 0.05, which is outside the range the correlation was fitted over — at that point the “pipe” is closer to a channel full of obstacles than to a pipe with a texture.

The arithmetic, on a pipe somebody has to buy #

Take a 300 mm water main carrying 100 litres per second. The mean velocity is 1.4 m/s, the Reynolds number is about $4\times10^{5}$, and the question is whether the pipe’s material matters.

Commercial steel has a handbook roughness of about 0.045 mm, so $\varepsilon/D = 1.5\times10^{-4}$. Colebrook gives $f = 0.0165$ and $k^{+} = 3.4$ — hydraulically smooth, just. The friction factor is within a per cent of a polished pipe’s, and the difference between an expensive smooth liner and plain steel is not measurable with a pressure gauge on the ends.

Now leave the same main in service for thirty years. Tuberculation raises the effective roughness by an order of magnitude to 0.5 mm, so $\varepsilon/D = 1.7\times10^{-3}$, $f$ rises to 0.0227 and $k^{+}$ to 45 — deep into the transitional band. The pumping power for the same flow has gone up by 38 per cent, and the pipe looks the same from the outside. That, rather than any question about material selection, is what the chart is used for in practice.

And take the same pipe at night, at a tenth of the flow. The Reynolds number falls to $4\times10^{4}$, $k^{+}$ falls with it to roughly 5, and the aged main is behaving as a smooth one again. The roughness has not changed by a micron. The pipe is in a different regime because the demand changed, which is the site’s oldest refrain in a place nobody expects it: the same object, in the same fluid, obeying different physics at different speeds.

The duct that is not round, and the substitution everybody makes #

The chart is for a circular pipe and a great many ducts are not. The standard remedy is one substitution — replace the diameter everywhere by the hydraulic diameter, four times the cross-sectional area over the wetted perimeter — and read the chart as before. It is worth knowing how well that works, because the answer is the reverse of what the rest of this essay would suggest.

On the fitted branch it works well. For turbulent flow the friction is set by what happens in a film a few tens of microns thick against the wall, and that film does not know what shape the duct is: it sees a flat surface, a local shear and a local viscous length, exactly as it would in a pipe. So the whole cross-section enters only through how much perimeter there is per unit area — which is what the hydraulic diameter measures — and the substitution is good to something like ten per cent for squares, rectangles and most practical sections.

On the exact branch it fails. In laminar flow there is no thin film and no locality: the velocity profile fills the whole section, and its shape is a property of the boundary’s shape. Solving Poiseuille flow in each geometry and expressing the answer as $f\,\mathrm{Re}$ on the hydraulic diameter gives

Against the circle’s 64, a duct between parallel plates is fifty per cent out and a triangular one is seventeen per cent the other way. Nothing about the hydraulic diameter repairs it, because the disagreement is about the shape of a profile rather than about a ratio of area to perimeter.

That is a neat inversion of the essay’s own contrast, and it is worth stating as the general moral. The exact branch is exact for one geometry and has no universality at all: change the section and the constant changes, and there is a different closed form for every shape. The fitted branch is approximate everywhere and is nearly universal, because the physics it summarises is local to the wall and therefore indifferent to the shape the wall encloses.

So a designer’s confidence should run opposite to the way the chart’s ink suggests. The line that is a theorem may not be transferred; the curves that are somebody’s plumbing may. And the one case where the substitution fails in turbulent flow is the one where the near-wall assumption fails too — a section with a sharp corner, where the fluid in the corner is nearly stagnant and the local film argument has nothing to stand on.

Where the correlation is weakest, and it is not where it looks weakest #

The chart looks least trustworthy in the shaded transition band, and it is least trustworthy in the smooth curve just to the right of it.

Colebrook’s equation was fitted to commercial pipes, and its transitional region — the gentle S-shaped approach to the fully-rough asymptote — does not match Nikuradse’s uniform sand at all. Nikuradse’s data dip below the smooth-pipe line before rising to the rough asymptote; Colebrook’s curve approaches monotonically from above. Both are real measurements of real surfaces. The difference is that a uniformly glued sand grain and a commercially drawn pipe have different distributions of roughness height, and a single equivalent sand grain cannot represent both.

This is the honest limit of the whole apparatus. The equivalent sand roughness of a real surface is not measurable except by measuring its friction, which makes the roughness in the correlation a fitted parameter dressed as a physical dimension. Handbook tables giving $\varepsilon = 0.045$ mm for commercial steel are summarising friction measurements, not surface profiles, and a pipe of the same stated roughness that has been in service for twenty years is a different pipe.

Two further absences:

• Nothing here is a solution. This site computes no turbulent flow at any Reynolds number, and the reason is arithmetic rather than effort. Every curve on the Moody chart is drawn in the colour reserved for a claim that came from somewhere else.
• The pipe is straight, circular, long and fully developed. Bends, fittings, valves and entrances add losses that this chart does not contain — some of which, unexpectedly, can be computed exactly while the friction cannot.

Who found it, and when #

Johann Nikuradse published the sand-grain experiments in 1933, at Göttingen: pipes coated inside with graded sand of known size, measured over four decades of Reynolds number, producing the family of curves that every later correlation is fitted to.

Cyril Colebrook and Cedric White published the interpolation in 1937 in London, explicitly to fit commercial pipes rather than sand — the difference described above. Lewis Moody drew the chart in 1944 at Princeton, and the reason his name is on it rather than Colebrook’s is that he drew it: the chart solved an implicit equation for a generation of engineers who had no other way to solve it.

There is a pleasing irony in that. The chart exists because the equation cannot be rearranged; the equation can now be solved in eight lines in under a microsecond; and the chart is still what is taught, because what it shows — the flat asymptotes, the collapse onto the laminar line, the shape of the transition — is an argument rather than an answer.

Where the ladder goes #

This rung is about the loss along a straight pipe, which is fitted. The next is about the loss at a change in a pipe, which is exact.

That is not the order anybody expects. Friction on a wall feels like the tractable part and a sudden enlargement feels like the messy one — there is a recirculating eddy in the corner, a reattaching shear layer, and nothing anybody would want to solve. And yet the loss at the enlargement can be written down in closed form from momentum and energy alone, without a viscosity appearing anywhere in the derivation, and it has been known since 1766.

The reason is the field’s method again: a control volume does not need to know what is inside it. It cannot help with friction along a wall, because there the wall is the boundary and what happens at it is the whole question. At a step it can help completely, because the step is inside the box.