When Air Stops Behaving Like Water
At low speeds, air can be treated as incompressible — its density barely changes. But as speeds approach and exceed the speed of sound, density variations become significant, and the flow behaves dramatically differently. This is the realm of compressible flow, essential for high-speed aircraft, jet engines, and rockets.
Mach Number and the Speed of Sound
The governing parameter is the Mach number, the ratio of flow speed to the local speed of sound:
M = V / a
The speed of sound in an ideal gas is:
a = √(γRT)
where γ is the ratio of specific heats (1.4 for air), R is the specific gas constant, and T is absolute temperature. Notably, the speed of sound depends only on temperature — about 340 m/s at sea level, falling to roughly 295 m/s in the cold upper atmosphere. The same true airspeed therefore corresponds to a higher Mach number at altitude.
Flow Regimes
| Regime | Mach range | Character |
|---|---|---|
| Subsonic | M < 0.8 | Smooth, no shocks; compressibility minor below ~0.3 |
| Transonic | 0.8 – 1.2 | Mixed subsonic and supersonic pockets; local shocks form |
| Supersonic | 1.2 – 5 | Flow everywhere above Mach 1; shock and expansion waves dominate |
| Hypersonic | M > 5 | Intense heating, chemical effects, thin shock layers |
The transonic regime is the most challenging: even when the aircraft flies below Mach 1, air accelerating over the wing can locally exceed Mach 1, forming shock waves that cause a sharp rise in drag known as drag divergence.
Stagnation Properties and Isentropic Relations
When a moving gas is brought to rest without losses, its kinetic energy converts to pressure and temperature, defining stagnation (total) properties. For reversible, adiabatic (isentropic) flow, the ratios of stagnation to static properties depend only on Mach number:
T₀/T = 1 + ½(γ−1)M²
Similar relations give the pressure ratio p₀/p and density ratio ρ₀/ρ. These isentropic relations let engineers compute how temperature and pressure change as a gas accelerates through a nozzle or decelerates in an inlet — provided no shocks are present. The stagnation temperature rise also explains the intense aerodynamic heating of fast vehicles.
Shock Waves
A shock wave is an extraordinarily thin region across which flow properties change almost discontinuously. Shocks form when supersonic flow must decelerate or turn faster than pressure signals (moving at the speed of sound) can smoothly adjust it.
Normal Shocks
A normal shock stands perpendicular to the flow. Across it:
- The flow decelerates from supersonic to subsonic.
- Pressure, temperature, and density rise sharply.
- Stagnation pressure drops — the process is irreversible and raises entropy.
Normal shocks appear in engine inlets, on transonic wings, and at the exit of over-expanded nozzles. The stagnation-pressure loss represents wasted energy, so designers work to minimize shock strength.
Oblique Shocks
When a supersonic flow is turned by a wedge or compression corner, an oblique shock forms at an angle to the flow. Oblique shocks turn and compress the flow more gently and, unlike normal shocks, often leave the flow still supersonic. A series of weak oblique shocks (as in a supersonic inlet) compresses air far more efficiently than a single strong normal shock. At sharp enough turning angles, the oblique shock detaches and becomes a curved bow shock.
The Area-Mach Relation and Nozzles
How a duct's cross-sectional area changes determines whether a compressible flow accelerates or decelerates — and the rule flips at Mach 1:
| Area change | Subsonic (M < 1) | Supersonic (M > 1) |
|---|---|---|
| Converging (area ↓) | Flow accelerates | Flow decelerates |
| Diverging (area ↑) | Flow decelerates | Flow accelerates |
This reversal explains the converging-diverging (de Laval) nozzle. Subsonic gas accelerates through the converging section, reaches exactly Mach 1 at the throat (where the flow is choked), and then continues accelerating to supersonic speed in the diverging section. Such nozzles produce the high-speed exhaust of rockets and supersonic wind tunnels.
Choked Flow
When the throat reaches Mach 1, the flow is choked: the mass flow rate is maximized and cannot increase no matter how much the downstream pressure is lowered. Choking sets the limiting mass flow through rocket nozzles, jet-engine sections, and flow-control orifices, making it one of the most practically important results in compressible flow.
Why Compressibility Matters
From the drag rise that limits airliner cruise speed, to the shock-controlled inlets of supersonic jets, to the choked nozzles that generate rocket thrust, compressible-flow physics governs all high-speed flight. Mastering the Mach number, isentropic relations, shock waves, and the area-Mach rule is essential for anyone designing vehicles or engines that operate near or beyond the speed of sound.