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

Temperature — where it appears

The thermodynamic state variable that fixes a fluid's transport properties and, in a compressible flow, its density and speed of sound. A flow that dissipates raises it, and a flow whose viscosity depends on it is coupled to its own heating.

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

A source at Mach 0.55, and the sound it has already made. Each circle is one pressure pulse, expanding at the speed of sound from the point the source was at when it left. Below the speed of sound every circle still contains the source, so the pressure disturbance reaches every point of the fluid before the source does. Above it the circles have an envelope, and outside that envelope nothing has been heard at all.

What a signal travels at

Air finds out that an aeroplane is coming, and it finds out at a definite speed. That speed does not depend on how hard the air is squeezed or on how much of it there is — only on how hot it is — and every result in compressible flow is downstream of that one fact.

compressible · Speed of sound How hot seven films get. The peak temperature rise inside the film itself, on a logarithmic axis, for seven lubricated contacts with ordinary engineering numbers. A journal bearing runs tens of kelvin above its own housing, which is why lubrication systems are designed around heat rather than around load; a water-lubricated bush runs at a hundredth of a kelvin, because water conducts six hundred times better per unit of viscosity. The air bearing is marked: its gap is smaller than air's mean free path, so its film is not a continuum at all.

The film that heats itself

The oil in a bearing is not at the temperature of the metal around it. It is tens of kelvin hotter, the rise contains no length whatever, and whether the heating runs away or settles down depends on something that is not a property of the oil at all — whether the machine driving it holds the speed or holds the force.

viscous · Viscous heating γ for air, against temperature. The ratio of specific heats for air as a mixture of nitrogen and oxygen, with the vibrational mode filling according to the Einstein function. It is 1.400 at room temperature, where only translation and rotation are available; 1.337 at a thousand kelvin; and 1.288 at six thousand. The usual compressible-flow tables use 1.4, and that is the value for a gas that is not hot — which, behind any shock worth drawing, it is not.

When gamma stops being a number

Every perfect-gas result uses γ = 1.4, which counts the ways a nitrogen molecule can hold energy at room temperature. Behind a Mach 10 shock the gas is at 3,800 kelvin and the count is different — and the pressure barely moves while the temperature falls by fifteen per cent.

compressible · Real gas The temperature behind a shock, which is not the jump condition's. The static temperature along the flow behind a Mach 6 normal shock, with the frozen value the jump conditions give and the equilibrium value they give a long way behind. The gas arrives at 2382 K and settles at 2059, over about four tenths of a millimetre.

A gas that has not finished being shocked

The jump conditions give the state a long way behind a shock. Immediately behind it the molecules have not started vibrating yet, so the temperature is 2,382 K where the equilibrium answer is 2,059 — and the gas takes four tenths of a millimetre to get from one to the other.

compressible · Real gas A degree of temperature is worth 0.34 per cent of pressure. The error in the pressure a paint reports, against how much warmer the surface is at the test condition than at the reference, at four pressures. At 0.8 of the reference pressure the sensitivity is -0.34 per cent of pressure per kelvin, so 10 degrees is -3.45 per cent. That is not a small number against what the technique is used to measure, and a model's surface temperature is not uniform: it is warmer where the flow has been brought to rest and cooler where it has accelerated, which means the temperature error is largest exactly where the pressure gradients are. The standard remedy is a second, temperature-sensitive paint measured at the same time — an instrument added to correct an instrument.

The paint that measures the wrong field

Dye, seeded particles and the optical methods all look through the flow or at something put in it. Pressure-sensitive paint looks at the surface, reports a scalar rather than a derivative, and turns a row of taps into a field. What quenches its luminescence is oxygen, which is what makes it a pressure gauge — and temperature, which is the other field a flow is guaranteed to produce.

misconceptions · Visualisation A film just past its fold lingers where the fold was. The temperature excess at the middle of a sheared film, scaled by the temperature over which the viscosity falls by e, against time in thermal diffusion times, switched on from rest at five stresses either side of the fold. Below the fold the film settles. Just above it the film climbs to the fold's own centre temperature, 1.19, lingers there as if it had found a steady state, and only then runs away — the longer the closer it is to the fold.

A lost steady state still holds the film

Push a self-heating oil film one per cent past the stress at which it can no longer settle and it does not run away at once. It warms to the temperature the vanished steady state would have had, sits there as if nothing were wrong, and only then goes — after a delay that grows as one over the square root of the overload, and during nearly all of which the load could still be taken back.

viscous · Viscous heating 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 Where a motor's line crosses the film's own characteristic. The stress a self-heating film carries against the speed it lets one wall slide past the other, both scaled, for every steady state (solid): it rises while the film is cool, peaks at the fold and falls as the film heats and thins. A motor's torque falls along a straight line as its speed rises (dashed), and the film runs where the two cross. A stiff drive crosses once. A soft drive, shallower than the falling limb's steepest slope, 0.0738, can cross three times. A drive held at a fixed stress is a flat line, and at the fold's stress it only touches.

A motor turns the runaway into a jump

A self-heating oil film has a fold at fixed stress and none at fixed speed, and a real motor is neither: its torque falls along a line as its speed rises. Put that line across the film's own torque–speed curve and the runaway disappears for any motor at all. What replaces it depends on the line's slope. A drive softer than one fourteenth of the cold oil's resistance jumps to a hot state and keeps a memory of the load; a stiffer one does neither. And the boundary between the two falls exactly where the motor is turning at a quarter of its no-load speed, which no motor near its rated speed can reach.

viscous · Viscous heating 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.

Model limitBifurcationHeat transferLubrication filmMeasurementMisconceptionStabilityThermal runawayViscous heatingDensityDissipationEnergy equation

All concepts