Fluid Flow & Aerodynamics
why things fly, and why the usual explanation is wrong
Series

Roughness — the series

3 essays on one idea, from the one that introduces it to the one that assumes the rest.
  1. The roughness function, and the asymptote in which the viscosity has gone. The whole effect of a rough wall on a turbulent boundary layer is one number: the downward shift of the logarithmic profile. It vanishes on a smooth wall, rises through a transitional band, and becomes (1/kappa)ln(k+) + B − 8.5 — at which point substituting it back leaves u+ = (1/kappa)ln(y/k) + 8.5, with the fluid's own length gone from the answer entirely.

    A second length at the wall

    The logarithm in a turbulent wall profile exists because a region of the flow is not allowed to know about any length except the distance to the wall. Roughen the surface and there is one it does know about, which belongs neither to the fluid nor to the flow — and the slope does not change at all.

    part 1 · turbulence
  2. The layer that grows from a change of surface. The internal boundary layer's height against distance downwind of a change in roughness, with the sublayer inside it that is genuinely in equilibrium with the new surface. The layer grows as the fetch to the four-fifths power and the equilibrium sublayer is a tenth of it.

    How far downwind a surface is remembered

    Walk from a field into a wood and the wind ten metres above your head is still the field's wind. It takes about a kilometre of trees before a ten-metre measurement is measuring the trees — a hundred times the height it is made at, and a great deal more than most masts are given.

    part 2 · turbulence
  3. Through the transitional range Nikuradse's sand pipe runs below Colebrook's curve by up to a fifth. The friction factor against the Reynolds number for a pipe of relative roughness 1/120, with Nikuradse's roughness function and with Colebrook's, beside the smooth pipe. Nikuradse's falls furthest below Colebrook's, 21.9 per cent, at a Reynolds number of 11318, where the roughness is 6 wall units; both reach 0.0356 when fully rough.

    Sand and a commercial pipe part company at six wall units

    A rough wall shifts the logarithmic velocity profile down by an amount that depends on the roughness height in wall units, and once the wall is fully rough every kind of roughness shifts it the same way. Before that they disagree. Nikuradse's uniform sand stays smooth until its grains reach four wall units and then rises steeply; Colebrook's commercial pipes feel grains a fifth of a wall unit high and rise smoothly. Put both into one log law and the friction factor of a pipe through its transitional range differs by up to a quarter, always at the same roughness in wall units, about six. The Moody chart, drawn from Colebrook, is the upper of the two.

    part 3 · turbulence

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