Sand and a commercial pipe part company at six wall units

A second length at the wall put the roughness of a surface into the logarithmic law of the wall. The log law exists because, in the overlap region, the flow can know no length but the distance from the wall; roughness gives it a second one, the roughness height, and the profile shifts down by an amount — the roughness function $\Delta U^+$ — that depends on that height in wall units, $k^+ = k u_\tau/\nu$. The slope does not change. And in the fully rough limit, where the roughness elements stand far out of the viscous sublayer and the viscosity no longer matters to them, the shift becomes $\kappa^{-1}\ln k^+$ plus a constant, and the friction stops depending on the Reynolds number altogether.

Between smooth and fully rough there is a transitional range, and there the essay left a question unasked: what shape the roughness function takes there. It is not universal. The two classic roughnesses of the subject — Nikuradse’s uniform sand of 1933 and the commercial pipes Colebrook fitted in 1939 — take it differently, and the difference is the whole of the disagreement between Nikuradse’s measured friction curves and the Moody chart that every engineering text prints.

Two ways to rise #

Nikuradse glued sieved sand to the inside of brass pipes, at six relative roughnesses from 1/30 to 1/1014, and measured their friction over Reynolds numbers from 500 to a million. His roughness function, as Cebeci and Bradshaw fitted it, is exactly zero until the grains reach 2.25 wall units — grains that small sit entirely inside the viscous sublayer and the flow cannot tell them from a polished wall — then rises along a sine-shaped transition, steeply, to meet the fully rough line at ninety wall units. Colebrook, fitting the friction of commercial iron, steel and concrete pipes, wrote a formula whose roughness function is $\kappa^{-1}\ln(1 + c\,k^+)$: it rises from the very first fraction of a wall unit, smoothly, and approaches the fully rough line from below without ever crossing it.

The two share their fully rough limit, and that fixes Colebrook’s constant: requiring both to reproduce Nikuradse’s fully rough law, $1/\sqrt f = 1.74 + 2\log_{10}(R/k)$, gives $c = 0.303$, the value Colebrook’s own formula uses. The log law in Prandtl’s pipe form, $1/\sqrt f = 2.0\log_{10}(Re\sqrt f) - 0.8 - \Delta U^+/\sqrt8$, then reproduces the Colebrook–White formula to a part in a thousand with Colebrook’s function, and Nikuradse’s fully rough law to a part in four thousand with either. So the two pipes are identical at the smooth end and at the fully rough end, and everything that distinguishes them happens between.

The friction through the transition #

Through the transitional range the two curves part. Colebrook’s pipe leaves the smooth line early and rises steadily towards its plateau; Nikuradse’s stays on the smooth line, dips nowhere below it, and then rises steeply, staying below Colebrook’s curve all the way and meeting it only when fully rough. At a relative roughness of 1/120 the largest difference is 21.9 per cent, at a Reynolds number of 11,300. The famous dip in Nikuradse’s curves — the trough between the smooth line and the rough plateau that Moody’s chart does not show — is this difference, read as a curve that first runs flat and then climbs.

The gap is a single feature slid along the Reynolds number. For every relative roughness the deepest point falls where the grains are about six wall units high — 7.9 for the roughest pipe, whose deepest point is cut off at the chart’s lower edge, and 5.9 to 6.0 for the rest. That is the roughness height at which Nikuradse’s function has barely begun to rise while Colebrook’s is already two wall units of shift into its climb; the pipe’s relative roughness only decides at what Reynolds number its grains reach that height. The gap’s depth falls slowly with the relative roughness, from a quarter to a sixth, because a smoother pipe reaches six wall units at a higher Reynolds number, where the log law’s own friction is lower and a given shift is a smaller share of it.

Where each pipe stops being smooth #

The two functions disagree most plainly about where roughness begins. Nikuradse’s pipe leaves the smooth law by one per cent when its grains reach four wall units, whatever its relative roughness: four wall units is where sand first pokes out of the sublayer far enough to shed its own eddies. Colebrook’s pipe leaves it when its roughness is a fifth of a wall unit high — twenty times smaller — because his function rises from zero at once. A pipe of relative roughness one in ten thousand is rough from a Reynolds number of 25,000 by Colebrook’s reckoning and smooth until a million by Nikuradse’s.

A fifth of a wall unit is not a physical roughness. No element that small can affect a flow whose sublayer is five wall units thick; it would sit within the region where viscosity damps every disturbance. Colebrook’s early rise is a property of the population of roughness in commercial pipes: a few tall elements among many short ones, the tall ones poking out of the sublayer long before the equivalent sand height does. The equivalent sand roughness is a single number matched in the fully rough limit, where the tall elements dominate; in the transition the population matters, and Colebrook’s function is the average over a population, which rises early because its tallest members are tall.

Why six wall units #

Six is not an accident of the fits. The viscous sublayer of a turbulent wall flow is about five wall units thick, and the buffer layer above it, where the turbulence is produced most intensely, peaks at about twelve. A grain of sand four wall units high is just reaching the top of the sublayer, where it first sheds eddies of its own; at six it is well into the region where the near-wall streaks and vortices that carry the smooth wall’s turbulence live, and its eddies begin to replace theirs. Nikuradse’s function is rising steeply there, but it starts from zero at four, so at six it has accumulated little shift. Colebrook’s population has been shifting the profile since a fifth of a wall unit, through its tallest members, and by six it has accumulated two units of shift. The gap is largest where uniform sand is starting and a population is well under way, and that is where a grain meets the buffer layer.

The same reasoning says where the gap closes. By thirty wall units the sand grains stand well above the buffer layer and dominate it; Nikuradse’s function has caught up most of the way, and by ninety it is fully rough, as Colebrook’s is asymptotically. The transitional range is the range of roughness heights that share the near-wall region with the smooth wall’s own turbulence — between the sublayer’s top and the buffer layer’s — and the shape of the roughness function there is a statement about how the two kinds of eddy share it. A uniform roughness hands the region over all at once, when its single height arrives; a population hands it over gradually, one height at a time, and its function is the record of that handover read in wall units.

What the Moody chart is #

The Moody chart is Colebrook’s formula drawn on logarithmic axes, and it is therefore right for pipes whose roughness is a population like Colebrook’s and conservative for pipes whose roughness is uniform like Nikuradse’s. The difference is not small: a designer sizing a pump for a honed or uniformly roughened pipe in its transitional range from the Moody chart overestimates the friction by up to a fifth. The Princeton superpipe measurements of honed steel pipes in the 2000s found inflectional behaviour of Nikuradse’s kind rather than Colebrook’s, as do measurements on sand-blasted and machined surfaces; aged commercial pipes, with corrosion and scale of every size, behave as Colebrook fitted them.

The refutation is aimed at the common reading of the equivalent sand roughness — a pipe’s $k_s$ looked up in a table and read off the chart — as a complete description of the pipe. It is a complete description of the fully rough pipe. Through the transitional range the chart is a statement about the population of roughness that commercial pipes happened to have in the 1930s, and the same $k_s$ can belong to pipes whose transitional friction differs by a quarter.

That is the same lesson the roughness a wall cannot feel drew from the smooth end of the chart: what decides whether a wall is rough is the roughness against the sublayer’s thickness, not the roughness alone. This essay adds that what decides how it becomes rough is the shape of the roughness, and that the familiar chart encodes one particular shape.

A test that tells them apart #

A pipe of unknown roughness can be told apart by the slope of its friction against the Reynolds number in the transitional range, without knowing its equivalent sand roughness at all. For a relative roughness of 1/120 the Colebrook curve falls steadily, from 0.0457 at a Reynolds number of 5,000 to 0.0373 at 40,000 and 0.0363 at 100,000: its friction always decreases as the flow speeds up, approaching the plateau from above. Nikuradse’s falls along the smooth line from 0.0372 at 5,000 to 0.0309 at 20,000 and then rises again, to 0.0319 at 40,000 and 0.0349 at 100,000, on towards the same plateau. A friction curve that turns upward with increasing Reynolds number is the signature of uniform roughness; one that only falls is the signature of a population. Two measurements, a decade apart in Reynolds number through the transition, are enough to see which.

That test is what the superpipe and later experiments did, and it is why the question of which function a surface has is now answered surface by surface rather than assumed. Honed and sand-blasted surfaces show the upturn; aged, corroded and scaled pipes do not, and a new commercial pipe sits somewhere between, closer to the sand the smoother its manufacture.

What a fifth of the friction costs #

The gap is worth real power. Water at a Reynolds number of 11,000 in a pipe of relative roughness 1/120 — a 25-millimetre pipe at about 0.45 metres a second, with roughness a fifth of a millimetre — loses 22 per cent less pressure per metre on Nikuradse’s curve than on Colebrook’s, and a pump sized from the Moody chart for such a line is sized for a fifth more head than the line needs. In the other direction, a heat exchanger whose tubes are deliberately roughened to raise their heat transfer — which follows the friction closely in this range — gains less than the chart promises if its roughness is uniform, because the uniform roughness has not begun to work at the Reynolds numbers where the chart says it already is.

The same choice on a hull #

A ship’s hull is a rough wall in the transitional range for much of its length. Its painted surface has roughness of tens to hundreds of micrometres, and at a ship’s speeds and lengths the friction velocity puts that at a few to tens of wall units — straddling the region where the two functions differ most. Naval architects have long argued over which roughness function a hull coating has, because the choice moves the predicted frictional resistance of a new hull by several per cent, and a ship’s fuel bill by the same. The log law and the roughness function are the same for a boundary layer as for a pipe; what changes is that the friction velocity, and so the roughness in wall units, varies along the hull, so a single hull can be smooth near its stern, transitional along most of its length and fully rough near its bow, each part on its own branch of the function.

Why a fit, and not a theory #

Neither function is derived. Nikuradse’s is a fit to one experiment and Colebrook’s to a class of pipes, and the log law into which both are put is itself an asymptotic statement that says nothing about how the shift should rise — only that, at each roughness height in wall units, there is one shift for each kind of roughness. That is the content of the earlier essay’s second length: roughness enters only through $k^+$. The transitional range is where that statement is still true and no longer enough, because $k^+$ is one number and a rough surface’s geometry is many. Two surfaces with the same equivalent sand roughness and different distributions of height, spacing and shape have the same $k^+$ at every Reynolds number and different shifts in the transition, and no single-parameter law can reconcile them. The two curves drawn here are the extremes that the history of the subject happened to measure first; most real surfaces lie between, and a surface’s place between them is a property of its geometry that has to be measured, or computed from a resolved simulation, rather than looked up.

The convention, and what the fits leave out #

The roughness function is the downward shift of the log law, $\Delta U^+$, at a given roughness height in wall units $k^+$, with $\kappa = \ln 10/(2\sqrt8) = 0.4071$ and $B = 5.57$ — Prandtl’s pipe constants, which make the log law’s pipe form exactly $2.0\log_{10}(Re\sqrt f) - 0.8$ and its fully rough limit Nikuradse’s. The friction factor is Darcy’s, $f = 8(u_\tau/U_b)^2$. Nikuradse’s function is Cebeci and Bradshaw’s fit to his data, with $B - 8.5$ adjusted to these constants; Colebrook’s is the roughness function his formula implies. The log law is used down to Reynolds numbers of a few thousand, where it is approximate: below about 4,000 a pipe’s flow is transitional between laminar and turbulent, and the curves there are the log law’s extrapolation, not a description of a real pipe.

Who measured the dip #

Nikuradse’s sand-grain experiments were published in 1933; Colebrook and White’s experiments on artificially roughened pipes with mixed grain sizes in 1937 showed that a roughness population rises early, and Colebrook’s formula of 1939 fitted commercial pipes on that basis. Moody’s chart of 1944 drew Colebrook’s formula. Cebeci and Bradshaw’s fit of Nikuradse’s roughness function is of 1977. Shockling, Allen and Smits measured a honed pipe’s inflectional transition in the Princeton superpipe in 2006, and Flack and Schultz have reviewed which surfaces behave which way.

How it was checked #

The log-law pipe with Colebrook’s function matches the Colebrook–White formula to a part in a thousand over Reynolds numbers from $10^4$ to $10^7$ and relative roughnesses from zero to one in a hundred. At a Reynolds number of $10^9$ and a relative roughness of 1/30 the two functions give the same friction to five parts in ten million, and $1/\sqrt f$ matches Nikuradse’s fully rough law to $10^{-3}$. Colebrook’s constant from the shared asymptote is 0.303, the value in his formula. The model refuses a negative roughness, an unknown roughness function and a laminar Reynolds number.

Still open: a roughness function from the population #

Colebrook’s early rise was explained here as a population of roughness heights, the tall ones arriving first. That can be made a calculation. The next calculation would take a distribution of element heights — a narrow one like sand, a broad one like corrosion — give each element Nikuradse’s own transition at its own height, add their effects on the shift in proportion to the area each occupies, and ask what roughness function the population has: whether a log-normal spread of heights with a standard deviation of half its mean reproduces Colebrook’s curve, and so whether the Moody chart’s shape is a measurement of what commercial pipes’ roughness looked like under a microscope.