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 8 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

Named alongside it

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

Model limitReynolds numberLaminar flowLinear stabilityModel validityOscillationPipe flowPoiseuille flowSkin frictionSurge tankTurbulenceViscous sublayer

All concepts