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

Friction factor — where it appears

The dimensionless pressure loss along a pipe per unit length, equal to 64 over the Reynolds number for laminar flow and a fitted correlation for turbulent. The exact half and the correlated half sit on the same chart, which is worth remembering when reading one.

Named by 14 essays across 4 fields — each of them below, with the objects they name alongside it.

The laminar line does not end; the flow leaves it. Friction factor against Reynolds number in a pipe. The laminar law f = 64/Re is exact and is drawn continuing past the transitional Reynolds number, faintly, because it remains a solution there — the flow simply stops taking it. The turbulent branch is Blasius' correlation and begins where experiments find transition, not where any calculation puts it.

The solutions stop being chosen

Hagen and Poiseuille's pipe profile is an exact solution of the Navier–Stokes equations at every Reynolds number, and it is linearly stable at every Reynolds number. Something else happens at 2300 anyway, and it is not that the solution stopped being one.

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 A factor of 1.8 between the extremes. The same numbers as a bar. The spread is exactly 96 over 160/3, which is 1.8 — and it is a spread in a quantity that a single correlation using the hydraulic diameter treats as a constant. A rectangle twenty times as wide as it is deep is 78 per cent more resistant than a circle of the same hydraulic diameter.

A number that is only the shape of the hole

The friction factor times the Reynolds number for laminar flow in a duct is a pure number that depends on the cross-section's shape and on nothing else — not the fluid, not the size, not the flow rate. It is exactly 64 for a circle and exactly 96 for a slot, and the correlation that treats them as the same is wrong by a third.

applied · Internal flow Four wall models, through the buffer layer. Van Driest's damped mixing length, Reichardt's fit, Spalding's implicit law, and a control with no buffer layer at all — the viscous sublayer joined straight to the logarithm where they cross, at y+ = 11.6. The three fitted models agree with each other to a per cent and a half; the control is nineteen per cent above them at y+ = 10.

Three buffer layers, one friction

Four wall models are put through a pipe. The one with no buffer layer at all is nineteen per cent wrong where the turbulence production peaks and two and a half per cent wrong in the friction; the three respectable ones are within two per cent of each other in the buffer layer and spread by seven in the friction. The answer is not where it was expected.

turbulence · Wall law Friction lends the crown 28.4 kPa, and the tank takes it back. The absolute pressure at the crown of a siphon with friction in its hose, through a whole drain, for the crown placed at three positions along the hose, against the frictionless constant of 23.01 kPa. With the crown 0.3 of the way it starts at 63.3 kPa, with the crown 0.5 along it starts at 51.5 kPa and with the crown 0.7 of the way it starts at 40.4 kPa. Every curve is above the constant, every curve falls towards it as the level falls, and every curve reaches it at the end — 23.06 kPa with a centimetre of level left. Friction never brings a siphon nearer to breaking; it lends a margin, and the draining tank returns it pascal by pascal, so the worst the crown ever sees is the frictionless value.

The margin friction lends a siphon

Without friction a draining siphon's crown pressure does not depend on the source level at all, and every real hose has friction. It turns out always to raise the crown pressure, by an amount the draining tank hands back pascal by pascal — and how much it lends is decided by where along the hose the crown sits, not by how rough or how narrow the hose is.

misconceptions · Siphon A 204 m hammer traded for a 8.57 m swing over 299 seconds. The water level in a 10 m surge tank at the end of a 2 km tunnel 3 m across, carrying 2 m/s, after the turbine is shut off at once, with the level measured from the reservoir's. Without friction it rises to V₀√(L Aₜ/g Aₛ) = 8.57 m and swings with a period 2π√(L Aₛ/g Aₜ) = 299.1 s, the integration agreeing with both closed forms. With the tunnel's 5 m of friction the level starts 5 m below the reservoir, peaks at 5.61 m after 98 s, falls to −3.70 m, and decays. The same tunnel shut at its end with no tank would take the Joukowsky rise of 204 m. The tank does not remove the column's momentum; it gives it a free surface to push against, slowly.

A tank that turns a hammer into a swing

Shut a turbine at the end of a two-kilometre tunnel in two seconds and the valve takes a rise of 256 metres of head. Put a shaft open to the air beside it and the rise is 51, the tunnel never carries the closure as a wave at all, and its water slows instead against a level that climbs for a minute and a half — to a height that is a closed form with the tank's area under a square root.

applied · Water hammer Below Thoma's 6.07 m² the governed tank's swing grows; above it, it dies. The tank level after the turbine's power demand drops by two per cent, with a governor holding the power constant, for tanks of 0.7 and 1.3 times Thoma's area of 6.07 m² — a tank 2.78 m across. The smaller tank's swing grows by a factor of 1.47 every 74 s cycle and has reached −12.20 m by 427 s; the larger one's keeps 0.75 of itself every 100 s and is barely visible. Carried on, the smaller tank's run is refused at 794 s, where the head at the turbine has fallen below a quarter of its design value and the governor would be asking for a flow no turbine passes. The instability has nothing to do with the tank's height: it is the governor drawing more water as the level falls, which feeds the swing, against the tunnel's friction, which is the only thing damping it.

The better tunnel needs the bigger tank

A turbine governed to hold its power opens further when the head at it falls, and draws the tank down harder. That makes it a negative resistance, the tunnel's friction is the only thing damping the swing against it, and so the smallest stable tank grows as the friction shrinks — 2.78 metres across for five metres of friction, 6.09 for one.

applied · Water hammer With friction the flow goes sonic after the throat, and leaves slower. The Mach number along a convergent–divergent nozzle of exit area ratio 2.5, for friction lengths 4fL/Dₜ of 0, 0.2 and 1, each solution passing smoothly through Mach 1 at the point where the sonic condition holds. Without friction that point is the throat, at x = 0.42. With friction it moves downstream — to 0.4357 and 0.4996 — and the throat itself is subsonic, at Mach 0.965 and 0.852. The exit Mach number falls from 2.443 to 2.247 and 1.747. The throat is where the area is least; the sonic point is where the widening has caught up with the friction, and the two coincide only when there is none.

Friction moves the sonic point past the throat

A choked nozzle is sonic at its throat — in a nozzle with frictionless walls. With friction the flow reaches Mach one where the section's widening rate has caught up with the friction, which is downstream of the throat, and the throat itself is subsonic. The solution through that point is a saddle that can only be found from the inside, and the mass flow it passes is less than the throat's area allows.

compressible · Area mach A pipe at the gas's temperature does not keep the gas at it. Left, the static temperature of the gas along a pipe, as a fraction of the wall's, against the local Mach number, entered at Mach 0.1: with no heat transfer, with the heat transfer Reynolds' analogy gives, and with ten thousand times that. Right, the stagnation temperature. The isothermal model holds the static temperature at the wall's and needs the stagnation temperature to climb by 14 per cent; every real case keeps it within two per cent, cools, and runs on to Mach one.

A pipe cannot hold its gas at the wall's temperature

The textbook model of a long gas pipe in contact with the ground holds the gas at the ground's temperature and has it choke at 0.845 of the speed of sound. No pipe does either. A wall at the gas's temperature draws heat out of it rather than putting heat in, and with any strength of heat transfer at all the flow runs on to Mach one, within a tenth of a per cent of the length a perfectly insulated pipe would need.

compressible · Fanno rayleigh A long line empties from the break backwards. The pressure along a gas line three hundred friction lengths long — fL/D = 300 — at five times after it breaks at its far end into still air at a twentieth of its pressure, the break on the right. Half a transit time in, the rarefaction has crossed half the line and the gas beyond it has not moved. Friction then holds the pressure in a slope that lengthens back towards the closed end; at six transit times the closed end is at 0.55 of its starting pressure, at fifteen 0.21, and the break has long since stopped being choked.

A broken line empties at the pace of its friction

When a gas line breaks, its open end chokes and the textbook stops there: a choked outlet passes gas at its own speed of sound and nothing downstream can change that. On a long line the choke is the least of it. The break stays choked only while friction lets enough gas reach it, which on a line a thousand friction lengths long is the first eighth of the time to half pressure. The pressure falls at a pace friction sets, the half-time grows as the square root of the line's length in friction lengths, and the wall's heat — negligible in the steady line — makes the blowdown a third slower.

compressible · Fanno rayleigh The largest nucleus decides how tall a siphon can be. The tallest crown a siphon of degassed water can run over, above the upper reservoir's surface, against the radius of the largest gas nucleus the water carries, for outlets 0.2, 1 and 3 m below the upper reservoir. Nuclei of ten microns and more leave the ordinary limit of about ten metres, where the crown reaches the vapour pressure. A largest nucleus of one micron lets a slow siphon stand 14.4 m tall; of three-tenths of a micron, 27.3 m. The water's own strength is never the limit: its dirt is.

A degassed siphon is as tall as its largest nucleus allows

The textbook says a siphon cannot lift water more than about ten metres, because above that the pressure at its crown would fall below zero. Water can hold a pressure below zero — a tension — and siphons of degassed water have run over crowns taller than the barometric height. How much taller is not set by the water's strength, which is enormous, but by the largest speck of gas it carries. A nucleus of a micron lets a slow siphon stand fourteen metres tall; three-tenths of a micron, twenty-seven. And the flow's own speed takes metres off every limit.

misconceptions · Siphon 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 the cheapest fork each daughter runs at the parent's velocity along it, so the split loses only the sideways part. The power lost at a symmetric split, as a multiple of the flow times the parent's dynamic pressure, against the daughters' radius as a fraction of the parent's, at each of three cheapest angles. For laminar, k = 3 the angle is 37.5°, the curve's minimum is at a ratio of 0.7937 — the cheapest ratio — and its value is sin²θ = 0.37; for smooth turbulent, 27/11 the angle is 28.4°, the curve's minimum is at a ratio of 0.754 — the cheapest ratio — and its value is sin²θ = 0.226; for rough, 7/3 the angle is 25.1°, the curve's minimum is at a ratio of 0.743 — the cheapest ratio — and its value is sin²θ = 0.18.

The cheapest fork loses only its sideways speed, and pricing it closes the fork

A turbulent network built at least cost forks at 28.4°, with daughters three quarters of the parent's radius, and those numbers priced the pipes and never the split. Price the split as Borda priced a sudden widening and something exact appears: at the cheapest fork each daughter already runs at the parent's velocity component along it, so the junction loses nothing but the sideways part of the parent's speed, and its loss cannot be reduced by resizing the daughters. It can be reduced by narrowing the fork. A split costing under two per cent of the bill moves the cheapest angle by seven degrees, because the pipes' bill is flat at its best angle and the split's loss is not.

applied · Branching 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.

Model limitPipe flowTurbulenceChokingReynolds numberRoughnessLaminar flowLinear stabilityMach numberModel validityMurray's lawOptimisation

All concepts