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

Roughness — where it appears

The scale of a surface's texture, which matters only when it exceeds the viscous sublayer's thickness. A pipe is hydraulically smooth or rough depending on its Reynolds number rather than on its finish, since the sublayer thins as the flow speeds up.

Named by 9 essays across 3 fields — each of them below, with the objects they name alongside it.

Two drag laws, one derived and one fitted. Flat-plate drag coefficient against Reynolds number, laminar and turbulent, on log axes. The laminar curve is Blasius' similarity solution, solved by shooting; the turbulent one is the 1/7-power correlation, which was fitted to experiment and is drawn dashed-in-kind to keep the difference visible. Their slopes differ — −1/2 against −1/5 — so the gap between them widens rather than staying put.

The number that is not a number

Transition Reynolds numbers are quoted to three figures and vary by two decades. That is not sloppiness in the measurement — it is the honest report of a quantity that depends on the laboratory as much as on the fluid, and knowing which part is which decides what may be designed on it.

turbulence · Transition The chart, with one exact line on it. The friction factor of a pipe against Reynolds number, for five relative roughnesses. Every curve here except one is Colebrook's correlation, solved by iteration rather than read off a chart. The exception is the short straight line at the left: f = 64/Re is the laminar solution and it is exact. The curves flatten to the right because once the roughness pokes out of the viscous layer the Reynolds number has nothing left to change.

The roughness a wall cannot feel

A rough pipe and a polished one carry the same flow for the same pressure over three decades of Reynolds number, and then suddenly they do not. What changed is not the pipe. It is the thickness of the film of fluid at the wall, which is the only part of the flow that can see the roughness at all.

applied · Internal flow The cliff a rough ball reaches sooner. The drag coefficient of a sphere against Reynolds number, on log axes. The smooth curve is Morrison's correlation, which is a fit to measurements and is drawn in the colour this site reserves for a borrowed claim. The other is the same curve shifted along the Reynolds axis by a factor of 6 — a stated model of what roughness does, which is to trip the boundary layer early, and not a measurement of any real ball.

The drag that falls as it speeds up

There is a band of speeds in which a smooth ball experiences less drag the faster it goes. Not a smaller coefficient — a smaller force. Dimples move that band down to where a golf ball actually flies, and they do it by making the friction worse.

applied · Sport ball One number instead of a porous medium. The velocity through the bottom of a channel whose lower wall is a porous block of permeability 1e-4. Inside the block the flow decays over the pore scale √K = 1.0e-2 to Darcy's seepage velocity; above it the channel profile arrives at the interface with a slip velocity rather than at rest. A channel told nothing but u = √K du/dy at a flat wall reproduces that profile to 0.058 per cent of the flow rate, against 3.03 per cent for a wall told to hold the fluid still. The grid solve of the coupled problem agrees with the closed form to 0.0077 per cent.

A wall that is not quite there

A porous surface has structure on every scale below the pore, and no calculation resolves it. The whole of it can be replaced by one length — the square root of the permeability — and the replacement is exact to first order, with what it leaves out identifiable as the flow the wall itself carries.

kinematics · Boundary conditions 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.

turbulence · Roughness 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.

turbulence · Roughness Past transition the cube law gives way to two and a half. The exponent k in Q ∝ r^k obeyed by the cheapest vessel, against that vessel's Reynolds number, for a smooth wall and for a wall with a millimetre of roughness. While the flow is laminar k is 3, Murray's law. Past transition it drops to 2.45 at Re = 2·10⁴ and 2.4 at 1.01·10⁷, near Blasius's 27/11 = 2.455. A fixed absolute roughness barely moves it, because the cheapest pipes for large flows are large and a millimetre is a small fraction of them.

The cheapest turbulent pipe follows two and a half, not three

Murray's cube law comes from Poiseuille's resistance, and it fails the moment the flow in a vessel turns turbulent. The same cost minimised with turbulent friction gives a flow that grows as the radius to the power 27/11 in a smooth pipe and 7/3 in a rough one. Murray's invariant, a wall shear the same in every vessel, goes with it, and so does the local rule that made the law credible as biology: a vessel that holds its shear at a set point is twice too wide.

applied · Branching At one specific speed a bigger pump is a better one, and by less as it grows. Best efficiency against impeller radius for geometrically similar pumps of 40 m head at three specific speeds; the shaft speed falls as the radius grows, so the tip speed is the same at every size. At Ωₛ 0.55 a 25 mm impeller reaches 0.797, a 0.2 m one 0.869 and a 1.6 m one 0.887: 7.17 points for the first factor of eight, 1.84 for the second.

Moody's fifth root is true at one size

A model pump is less efficient than the full-size machine, and since 1925 the gap has been bridged with a fifth root of the size ratio. Built from its losses, a pump's inefficiency scales three ways at once — leakage as the inverse size where a clearance cannot shrink, friction as a fifth root until the passages go rough, and a fixed share not at all — so the exponent runs from a third down to a twentieth, and the fifth root is right only near one size. The same arithmetic says a duty should be split by speed, not by width.

applied · Specific speed 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.

turbulence · Roughness

Named alongside it

The objects these essays reach for when they reach for this one.

Reynolds numberBoundary layerModel limitFriction factorSimilaritySkin frictionTransitionCorrelationMeasurementModel validityPipe flowTurbulence

All concepts