Concept

Friction — where it appears

A drag proportional to the velocity, acting at every scale, used to represent bottom drag or Ekman pumping in a quasi-two-dimensional flow. It is what arrests an inverse cascade, and the limit in which it vanishes has no steady state at all.

Named by 4 essays across 3 fields — each of them below, with the objects they name alongside it.

Two different physics, one destination. The Fanno and Rayleigh lines, each with temperature up the page and entropy across, each referred to its own sonic state. Neither is a curve anybody drew: the entropy was swept over Mach number and its maximum located, and it sits at Mach 1 on both to within the resolution of the sweep. The subsonic branch runs up to the nose from the left and the supersonic branch runs up from below, so whatever is being done to the flow, it can only be moved towards that point.

Two ways to choke

Friction and heat are different physics acting on different conservation laws, and they drive a duct's flow to exactly the same place. Both have their entropy maximum at Mach 1, so neither can push a subsonic flow past it and both drag a supersonic one down to it — and where their two curves cross is a shock.

compressible · Fanno rayleigh

The condensate, and the limit of no friction which does not exist. In a steady state the friction must remove everything the forcing puts in, so the energy is eps/(2 alpha) and the coherent velocity is sqrt(eps/alpha) — exactly a minus one half power, checked to 10⁻¹². As the friction is weakened the condensate grows without bound: the limit alpha to zero is not a flow with a weak condensate, it is a flow with no steady state at all.

Where the inverse cascade stops

Two-dimensional turbulence sends its energy upward in scale, and the upward direction has an end: the box. Without something to remove the energy before it arrives, it accumulates there in a pair of vortices filling the domain, and the limit of no friction has no steady state at all.

turbulence · Two-dimensional

The Fanno line, which has only one direction on it. The entropy relative to the sonic state, against Mach number, for both branches. Friction raises the entropy, so a duct flow moves to the right along this curve whichever branch it is on — up in Mach number from below and down from above — and it stops at the sonic point.

A duct that cannot be run backwards

Friction drives a compressible duct flow towards the speed of sound from either side, and the entropy rises the whole way. So the state of the gas at a station is an odometer: it records how much duct is behind it, and no amount of further duct can take it back.

compressible · Fanno rayleigh

Along the carry's ridge the run hardly moves the best spin, because every drive on it lands alike. The total distance of a 70 m/s drive, carry and run, at each spin rate launched at the angle that carries furthest for that spin, on a firm, a medium and a soft fairway, with the carry alone. The carry peaks at 234.6 m at 2273 rpm; the total along the ridge peaks at 1728 rpm on the firm turf, 274.3 m, 2052 rpm on the medium turf, 249.3 m, 2375 rpm on the soft turf, 239.9 m. Every drive on the ridge lands between 45.2° and 49.2°.

The run moves the best drive below the ridge that carries furthest

A drive's carry is greatest along a ridge of launches on which more spin trades for a lower angle at almost no cost. Every drive on that ridge lands at about the same steep angle, so the run after landing hardly chooses between them. What the run rewards is leaving the ridge: less spin and a launch several degrees lower, a lower flight that lands flatter, gives up thirteen metres of carry and gets back twenty in the run. On an ordinary fairway the best drive in all lands near thirty-five degrees, at every ball speed; on a firm one the model's ground stops being a turf anybody has measured before the run stops paying.

applied · Sport ball

Named alongside it

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

Choking Entropy Model limit Angular momentum Box scale Condensate Conserved quantity Duct Energy flux Enstrophy Fanno flow Fanno line

All concepts