When air stops being incompressible

Air is a gas. Squeeze it and it compresses; that is what makes a bicycle pump work. And yet most of aerodynamics proceeds by treating its density as a constant, which sounds like an obvious error and is an excellent approximation.

The question is not whether air can be compressed but whether, in this flow, it is being compressed enough to matter.

Why a threshold exists at all #

Pressure changes travel through a fluid at the speed of sound. That is what sound is: a pressure disturbance propagating.

So when a body moves through air, the air ahead of it finds out that it is coming — the pressure disturbance runs ahead at the speed of sound, and the air begins to move aside before the body arrives. It gets out of the way smoothly.

If the body moves slowly compared with that signal speed, the air has plenty of time. Density changes are small because the fluid can accommodate the body by moving rather than by being squeezed.

As the body’s speed approaches the signal speed, the warning arrives later and later. The air has less time to move aside, so it has to be compressed instead. Density variations grow, and above the signal speed the warning never arrives at all.

The number, and the threshold #

The relevant ratio is the Mach number:

$$M = \frac{U}{a}$$

Flow speed divided by the speed of sound. Like the Reynolds number, it is dimensionless, and like the Reynolds number it emerges from the equations rather than being invented.

The density variation in a flow scales roughly as $M^2/2$. At Mach 0.3 that is about 4.5 percent, which is where the conventional threshold comes from — below it, treating density as constant introduces an error smaller than most other approximations in the problem.

The threshold is a convention rather than a physical boundary. Nothing happens at Mach 0.3; the error simply grows, and somewhere around there it stops being negligible.

What changes above it #

Several things at once, which is why compressible flow is a separate subject rather than a correction.

Density becomes an unknown. The continuity equation has an extra term, an equation of state is needed, and an energy equation joins the system. Three equations become five.

Bernoulli’s equation changes form. The incompressible version assumes constant density; the compressible one carries enthalpy instead of the simple pressure term.

Nozzles behave backwards. Below the speed of sound a narrowing duct accelerates flow, as continuity says it must. Above it, a narrowing duct decelerates flow and a widening one accelerates it, because density is falling faster than area. That is why a rocket nozzle has a throat and then flares out.

Shocks become possible. Above Mach 1 a disturbance cannot propagate upstream, so the flow can arrive at a condition it has no way of anticipating. The adjustment happens in a region a few molecular mean free paths thick, across which pressure, density and temperature jump discontinuously.

The temperature nobody asked for #

The list above says density becomes an unknown and an energy equation joins the system. That energy equation has a consequence which arrives whether or not anybody wanted a thermodynamic problem: air brought to rest gets hot.

Bringing a stream to rest converts its kinetic energy into enthalpy, so the temperature at a stagnation point exceeds the free stream’s by the flow’s kinetic energy per unit mass divided by the specific heat. In the perfect-gas form that is a ratio of $1 + \tfrac{\gamma-1}{2}M^2$, and it depends on the Mach number and on nothing else — not on the shape, not on the pressure, not on the altitude except through what the temperature there already was.

Read off some numbers and the character of high-speed flight follows. At Mach 0.3 the rise is under two degrees, which is why nothing in ordinary experience makes this a subject. At an airliner’s Mach 0.85 it is about thirty degrees, enough that the wing’s leading edge is meaningfully warmer than the air it is flying through and enough to matter for ice. At Mach 2 it is a hundred and seventy, and at Mach 3 the free stream’s 220 kelvin becomes over 600 — which is not a detail of the aerodynamics but a choice of metal.

That is why Concorde’s nose ran at over a hundred degrees Celsius at cruise, why the airframe grew several inches in flight and had to be built with a joint that closed as it heated, and why the aircraft that cruised at Mach 3 was made of titanium rather than aluminium.

The wall does not quite reach the full stagnation temperature — some heat conducts away down the layer, and the fraction recovered is a little under nine-tenths in air. Which is a boundary-layer question, so the heating a supersonic aircraft actually feels is decided in the thin layer like everything else.

The transonic trouble #

The hardest range is not supersonic. It is the region around Mach 0.8 to 1.2, where the flow is subsonic in some places and supersonic in others at the same time.

An aircraft cruising at Mach 0.85 has subsonic flow almost everywhere and a pocket of supersonic flow over the upper surface of the wing, where the acceleration round the curvature has pushed the local speed past Mach 1. That pocket has to be closed by a shock wave, and the shock imposes a sudden adverse pressure rise on the boundary layer beneath it.

Shock-induced separation follows, and with it a sharp rise in drag, buffeting, and control problems. That is the drag rise that made early jet aircraft difficult and killed several test pilots, and the supercritical aerofoil — flatter on top, with the curvature moved aft — is the response to it.

Mathematically the trouble is that the equations change type. Subsonic flow is elliptic, meaning every point influences every other; supersonic flow is hyperbolic, with influence travelling along characteristics. A transonic flow is both at once in different regions, with the boundary between them part of the unknown.

A shock is not a sound wave #

The distinction is worth drawing because the two are often conflated, and the difference explains why supersonic flow is genuinely different rather than merely fast.

A sound wave is a small disturbance. It travels at the speed of sound, it does not change the state of the gas it passes through in any lasting way, and it is reversible.

A shock is a large one. Pressure, density and temperature jump across it, entropy rises, and the process is irreversible. Total pressure is lost, permanently — which is why Bernoulli’s equation does not survive a shock even though it survives a sound wave.

The relation between them is that a shock is what a sound wave becomes when it is strong enough. The speed of sound rises with temperature, and a compression raises the temperature, so the back of a compression wave travels faster than the front. It steepens as it propagates, and eventually the front becomes a near-discontinuity.

That steepening is why shocks form spontaneously in supersonic flow rather than needing to be created: any compression, given enough distance, becomes one.

What supersonic flow cannot do #

The property that makes supersonic flow behave so differently is easy to state and its consequences are not.

Below the speed of sound, disturbances travel upstream. The air ahead of a body knows it is coming, and every point in the flow is influenced by every other. That is what makes subsonic flow elliptic, and it is why a subsonic aerofoil’s whole shape matters everywhere.

Above the speed of sound, disturbances cannot travel upstream. Information propagates only downstream, within a cone. A supersonic body meets air that has no warning at all, and the air’s first news of the body is the shock.

Practically: a supersonic wing’s leading edge cannot influence the flow ahead of it, disturbances stay confined to Mach cones rather than spreading, and the flow can be computed by marching downstream rather than solving everywhere at once — which is a genuine simplification and one of the few things that is easier about supersonic flow.

What the solver computed #

Nothing on this site is compressible, and the figures say so.

This site’s solver handles incompressible flow only. Every field here assumes constant density, and the model note under each figure states it. That is an honest restriction rather than an oversight — a compressible solver is a substantially larger undertaking, and the essays that would need one are not written yet.

What can be said within the assumption is the threshold itself, and the reasoning for it: density variation scales as $M^2/2$, which is arithmetic rather than simulation. The regime axis is drawn from the literature and the site says so.

The one thing worth being careful about is not letting the incompressible figures imply more than they can. A pressure coefficient computed here is valid for low-speed flow; the same section at Mach 0.8 would have a quite different distribution, and nothing in these figures anticipates it.

The everyday cases #

Where the threshold falls in practice is worth a moment, because it is higher than intuition suggests.

Mach 0.3 at sea level is about 100 metres per second, or 370 kilometres an hour. So virtually everything in ordinary experience — cars, cyclists, birds, wind, most model aircraft, all sailing — is comfortably incompressible, and the constant-density treatment is not an approximation anybody needs to apologise for.

What crosses it: airliners in cruise, propeller and helicopter blade tips (which can be transonic while the aircraft is slow), turbine blades, and anything ballistic.

Helicopter rotors are the interesting case, because the advancing blade tip approaches the compressible range while the retreating one is slow and close to stall. The two limits arrive from opposite directions and together they cap the speed of every helicopter ever built.

Why the sound speed is the right comparison #

One more step of explanation, since “the speed at which pressure travels” can sound like a definition rather than a reason.

The speed of sound in a gas is set by how stiff it is — how much pressure rises for a given compression — and by its density. A stiff, light gas carries sound quickly.

That same stiffness is what resists being compressed by a passing body. So the sound speed is not merely a signalling speed that happens to be relevant; it is the direct measure of the gas’s resistance to the thing being asked of it.

Which is why the Mach number, like the Reynolds number, is a ratio of the effect being applied to the property that resists it — and why both emerge from the equations rather than being chosen.

Two numbers, one aircraft #

The interaction between the two dimensionless numbers is worth stating, because an aircraft has both and they are not independent in practice.

At altitude the air is thinner and colder. Thinner air lowers the density and therefore the Reynolds number; colder air lowers the speed of sound and therefore raises the Mach number at the same true airspeed.

So climbing does two things at once: it moves the aircraft towards the compressible regime and away from the high-Reynolds one. Both trends matter, and cruise altitude is partly a negotiation between them — high enough that drag is low, not so high that the wing runs into transonic trouble or the Reynolds number falls far enough to change the boundary layer’s behaviour.

A wind tunnel matching one number will generally miss the other, which is why transonic tunnels are pressurised and cryogenic facilities exist at all. Matching both at once for a full-size aircraft is possible in only a handful of facilities in the world, and they are national-scale investments.

Where the model stops #

This site is incompressible throughout. No figure here is valid above about Mach 0.3, and the model notes say so.

The threshold is a convention. Nothing happens at 0.3; the error grows smoothly and that is where it stops being ignorable for typical purposes.

Temperature is described and not computed. Compressible flow couples pressure, density and temperature; the stagnation-temperature section above says what that coupling does, and no figure here contains any of it.

Shocks are described, not computed. Their structure requires the compressible equations and, at the finest scale, the kinetic theory that continuum mechanics assumes away.

Why sweep works #

The most visible consequence of all this is on every airliner, and it has a one-line explanation.

A swept wing meets the flow at an angle, so the component of velocity perpendicular to the leading edge — which is the component that matters for the pressure distribution over the section — is smaller than the flight speed by the cosine of the sweep angle.

An aircraft flying at Mach 0.85 with thirty degrees of sweep presents its wing sections with an effective Mach number of about 0.74. That is enough to push the local supersonic pocket and its shock back into a manageable range.

Sweep is not aerodynamic styling and it is not about reducing frontal area. It is a device for lying to the wing about how fast the aircraft is going.

It costs something, as everything does: swept wings have poorer low-speed behaviour, a tendency to stall at the tips first, and structural penalties. All of which are accepted because the transonic drag rise they avoid is worse.

The nozzle reversal, worked #

The claim that supersonic nozzles must widen is counter-intuitive enough to deserve its derivation, and it is short.

Continuity says $\rho A U$ is constant along the duct, so proportional changes must sum to zero. In subsonic flow the density changes slowly, so a reduction in area must be matched by an increase in speed — the ordinary result.

Above Mach 1, density falls faster than speed rises. The proportional density drop exceeds the proportional speed increase, so to keep the product constant the area must increase.

The consequence is the shape of every rocket nozzle and every supersonic wind tunnel: a converging section that accelerates the flow to Mach 1 exactly at the narrowest point, and a diverging section that accelerates it beyond. The throat is where the transition happens, and the flow there is choked — no further increase in mass flow is possible however hard it is pushed.

Who worked it out, and when #

Ernst Mach photographed the shock waves round a supersonic bullet in 1887, using a spark and a schlieren system, and the pictures are still reproduced. The number was named after him much later, in the 1920s.

The transonic problem became urgent in the 1940s when propeller tips and diving fighters began meeting it and aircraft began coming apart. The phrase “sound barrier” belongs to that period and to the mistaken belief that drag rose without limit; it does not, and Chuck Yeager’s flight in 1947 confirmed what the equations already suggested.

The supercritical aerofoil is Richard Whitcomb’s, in the 1960s, along with the area rule — both of them solutions to transonic drag rise, and both visible on every airliner flying.

What incompressible really buys #

A closing accounting of the assumption, since this whole site rests on it.

Dropping density as a variable removes an unknown, and with it an equation of state and an energy equation. Five equations become three. It also removes the coupling between the thermodynamics and the mechanics entirely — temperature stops mattering, and the flow becomes a purely mechanical problem.

That is why ideal flow can be linear and solvable in closed form, why the stream function exists, and why the whole construction kit of sources, doublets and vortices works. None of it survives compressibility: the compressible equations are nonlinear even without viscosity, and superposition fails immediately.

So the constant-density assumption is not one simplification among several. It is the one that makes the classical theory possible, and everything elegant in this subject is downstream of it.

Which is worth remembering when it is described as an approximation. It is a very good approximation below Mach 0.3, and it is also the reason there is a theory to approximate with.

The ladder from here #

Nearby: the speed of sound derived; shock waves and the Rankine–Hugoniot relations; the converging–diverging nozzle; and the supercritical aerofoil and the area rule.

Then across to the other dimensionless number, and to the continuity equation whose simple form this assumption buys.