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

Recovery factor — where it appears

The fraction of a flow's kinetic energy that reappears as a temperature rise at a surface told nothing about its temperature. It is exactly one where the Prandtl number is one, about 0.84 for a laminar layer in air, and it is what a heat flux is driven from.

Named by 7 essays across 2 fields — each of them below, with the objects they name alongside it.

What the skin settles at, before anything is done to it. The adiabatic wall temperature against Mach number, in air at 216.7 K, with the stagnation temperature above it. The gap between the two is the recovery factor, which is 0.8417 here and stays there at every Mach number — it is a property of the Prandtl number and not of the speed. At Mach 2 the skin sits at 363 K, at Mach 3 at 545 K, and at Mach 5 at 1128 K, which is past what aluminium will do. Nothing has been burnt and nothing has been rubbed: the air was brought to rest, and this is where its kinetic energy went.

The wall that heats itself

A surface told nothing about its temperature does not settle at the air's. It settles most of the way to the stagnation temperature, and the heat flux is driven from that invented temperature rather than from the free stream's — so a wall hotter than the air can be being heated by it.

compressible · Recovery A skin lags the air by a time its own thickness sets. A skin held at one flight condition, relaxing towards the adiabatic wall temperature. The approach is exponential with a time constant ρc τ / h — 8.37 s at 2 mm, 25.1 s at 6 mm, 50.2 s at 12 mm — so the temperature a steady recovery calculation gives is reached after several minutes rather than at once. The fluid supplies one number to this calculation, the adiabatic wall temperature, and the structure supplies everything else.

The skin that lags the flight

A wall can be told its temperature or told nothing, and both are solved problems. A real skin is told neither. It has heat capacity, so its temperature is a transient whose time constant is its own thickness divided by what the layer delivers — and the number the steady calculation returns is an upper bound a short exposure never collects.

compressible · Recovery Measuring the air temperature in flight is not measuring the air temperature. Three temperatures against Mach number in a stream at 220 K: the free stream itself, the stagnation temperature a perfect probe would read, and what a probe recovering 0.98 of the rise actually reads. At Mach 0.85 the rise is 31.79 K and the probe misses 0.64 K of it. The instrument is a small stagnation region with a thermocouple in it, so it obeys the same recovery arithmetic a wall does — a probe is a wall told nothing about its temperature, made small and put on a stalk, and its recovery factor is a calibration constant rather than a one.

The thermometer that heats itself

A total-temperature probe is a wall told nothing about its temperature, made small and put on a stalk, so it obeys the same recovery arithmetic an aircraft skin does. Its recovery factor is a calibration constant near 0.98 rather than a one — and the static temperature inferred from its reading amplifies that shortfall rather than inheriting it.

compressible · Recovery Every output of the air-data reduction, and how hard it leans on each reading. The logarithmic sensitivity of each derived quantity to each of the three readings — the total pressure, the static pressure and the indicated total temperature — at Mach 0.3 and Mach 0.85, at 11 km with a probe recovering 0.98: a one per cent error in a reading times the number is the per cent error it puts into the output. The Mach number leans on the two pressures by 8.08 and −8.08 at Mach 0.3 and by 1.13 and −1.13 at Mach 0.85, and not at all on the temperature. The static temperature leans on its probe by exactly one and on the pressures by −0.280 at Mach 0.3 and −0.281 at Mach 0.85 — almost the same. The true airspeed leans on the probe by exactly one half and the density by exactly minus one, at every speed.

Three readings, and the one each answer leans on

An aircraft works out the air it flies through from three readings — a static pressure, a total pressure and a probe's temperature — and every derived number inherits their errors through one small table of sensitivities. Written out, the table says the airspeed is afraid of the pressure sensors almost to Mach one at cruise altitude and only to Mach 0.52 at sea level, and that the density belongs to the thermometer at every speed.

compressible · Recovery 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 Where a carburettor's cold comes from. The temperature changes in a carburettor's throat, in kelvin: the cooling of the air as it expands to a hundred metres a second, the far smaller cooling of the metal it flows over, the cooling as petrol evaporates into it, and the warming as the water in humid air condenses back out. The fuel does five times what the venturi does to the air and fifty times what it does to the metal.

Carburettor ice is made by the fuel, not the venturi

Pilots are taught that a carburettor ices because air cools as it expands through the venturi. The expansion does cool the air, by five kelvin at a hundred metres a second, but the metal the ice grows on barely feels it. The cold comes from the petrol evaporating, and whether it makes ice is decided by the water in the air, which warms the mixture as it condenses. The worst day is not a cold one.

misconceptions · Venturi A fuel-wetted plate sits far below the stream that cools it. The temperature of a throttle plate wetted by petrol, against the day's temperature at 80 per cent humidity, beside the temperature of the stream flowing over it — the mixture after half the fuel has evaporated into it, at the plate's recovery temperature. A plate kept wet with fresh fuel sits nine to twenty-three kelvin below the stream, more on warmer days, and below freezing on any day up to about 34 °C. A plate whose film has lost its light ends sits one to six kelvin below it. The dashed line is freezing.

The throttle plate is a wet bulb for petrol

A carburettor's icing is usually worked out for the mixture — the fuel's latent heat spread through all the air — and that puts the warmest icing day near 15 °C, well short of the 30 °C the charts pilots use allow for. The ice does not grow in the mixture. It grows on the throttle plate, and a plate wet with evaporating petrol is a wet-bulb thermometer for fuel: nine to twenty-three kelvin colder than the stream flowing over it. Kept wet with fresh fuel it ices on a 30 °C day at full humidity and a 40 °C day at half — which is what the training handbook warns of. Once its light ends have gone it does not, and that difference is most of the story.

misconceptions · Venturi

Named alongside it

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

Heat transferStagnation temperatureAdiabatic wallMach numberModel limitBoundary conditionAerodynamic heatingCorrelationEnergy equationInstrumentMeasurementMisconception

All concepts