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

Stagnation temperature — where it appears

The temperature a compressible flow would reach if brought to rest without heat loss, exceeding the static value by the kinetic energy per unit heat capacity. It is conserved through a shock even though almost nothing else is.

Named by 11 essays across one field — each of them below, with the objects they name alongside it.

The energy is all there the whole way: enthalpy spent to buy speed. At each Mach number, the fraction of the stagnation enthalpy still held as heat and the fraction converted into directed motion. The two always sum to the total, which is the compressible replacement for Bernoulli's equation — an energy statement rather than a pressure one. At Mach 2 more than four fifths of the heat has become speed.

Energy instead of pressure

Bernoulli's equation is a statement that pressure and speed trade against each other at fixed density. Take the fixed density away and the trade is between heat and speed instead — and what survives is not a pressure at all.

compressible · Bernoulli's equation A normal shock at Mach 2.00, and what crosses it unchanged. The state in front of the shock and the state behind it. Every ratio was computed from the standard jump relations and then substituted back into mass, momentum and energy, which is an independent route — a mistyped exponent in the total-pressure expression cannot survive a momentum balance it never appeared in. The residuals are printed below because a check nobody can see is a check nobody can audit.

The jump the equations allow

A shock is a discontinuity in a fluid, which sounds like a breakdown of the description rather than a solution of it. It is a solution: mass, momentum and energy can all be satisfied across a jump, and every ratio across one follows from that alone.

compressible · Shock Everything a normal shock does, against the Mach number in front of it. Four quantities across a normal shock, each scaled to fit one axis. Pressure and density rise without limit and without bound as the Mach number grows; the Mach number behind falls towards a floor it never passes; and the total pressure — the flow's ability to be turned back into speed — collapses. That last curve is why a supersonic intake is designed around avoiding a single strong shock.

What a shock costs

All the heat survives a shock and none of it is lost. What is lost is the ability to turn that heat back into speed — 27.9 per cent of it at Mach 2 and 93.8 per cent at Mach 5 — and every supersonic intake ever built is a scheme for paying less.

compressible · Shock Past 23.0° at Mach 2.00 there is no attached shock. The same wedge at two half-angles. On the left the θ–β–M relation has a root and the shock sits on the nose. On the right it has none, and the solver throws rather than returning the nearest thing — which matters, because a solver that quietly clamped to the maximum would draw a neat attached shock on a body that cannot carry one. The bow shock on the right is indicative: its shape is not solved here.

When the wedge is too blunt

Every curve of shock angle against deflection has a maximum. Past it there is no attached shock at any angle — the solver has no root to return and must say so, rather than quietly handing back the nearest thing and drawing a picture that cannot exist.

compressible · Oblique shock 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 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 curved shock, and the entropy each streamline picks up crossing it. A parabolic bow shock ahead of a blunt nose at Mach six, with the streamlines drawn arriving horizontally and a marker at each crossing whose size is the total pressure lost there. The streamline through the nose crosses a normal shock and keeps three per cent of its total pressure; one four nose radii out crosses at fourteen degrees and keeps ninety-four per cent. Every streamline gets a different entropy, and the stagnation enthalpy is the same on all of them.

The spin a shock leaves behind

A curved shock gives every streamline a different entropy rise and the same stagnation enthalpy. Crocco's theorem then forces vorticity into a flow with no viscosity anywhere — and it scales as the inverse of the shock's radius of curvature, exactly, so a straight shock makes none.

compressible · Crocco 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. Every compressible result on this site has used 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 compressible result on this site has used 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 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

Named alongside it

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

EntropyMach numberTotal pressureRecovery factorAdiabatic wallBoundary conditionEnergy equationHeat transferMeasurementModel limitThe second law of thermodynamicsShock wave

All concepts