Compressible flow describes gas motion when changes in density become important, especially at high speed. Engineers use it to design jet engines, rockets, wind tunnels, pipelines, and supersonic aircraft. The key measure is Mach number, which compares the flow speed to the local speed of sound.
When Mach number approaches 1, pressure, temperature, and density can change rapidly and must be included in the analysis.
In a converging-diverging nozzle, subsonic air speeds up in the converging section until it can reach Mach 1 at the throat. Past the throat, a properly expanded flow accelerates further in the diverging section and becomes supersonic. If the pressure conditions are not matched, a shock wave can form downstream or inside the nozzle, causing an abrupt jump in pressure, temperature, and density while slowing the flow.
These effects are central to propulsion, aerodynamic drag, and high-speed testing.
Understanding Engineering: Compressible Flow and Mach Number
A moving gas can carry information only as fast as small pressure disturbances travel through it. Those disturbances are sound waves. At low flow speeds, a pressure change near one part of a pipe can influence the gas upstream.
Near sonic speed, the upstream influence becomes weak. Once the gas is supersonic, disturbances cannot move upstream against the flow. This changes how engineers control a system.
A valve, nozzle shape, or pressure change downstream may no longer affect the earlier supersonic region in the expected way. This limit is called choking. At a choked narrow passage, lowering the downstream pressure further does not increase the mass flow rate unless the upstream conditions or throat area change.
Temperature has a major role because gas molecules move faster when they are hotter. A hotter gas has a higher speed of sound, so the same aircraft speed can correspond to a lower Mach number at one temperature than at another. Engineers therefore distinguish between static properties and total properties.
Static pressure and temperature describe the gas as it moves. Total pressure and total temperature describe the state the gas would reach if it were slowed smoothly to rest. In an ideal smooth slowing process, total pressure is preserved.
Friction, turbulence, heat transfer, and shocks reduce useful total pressure. Jet engines need this pressure to compress air and produce thrust, so losing it reduces performance.
Shock waves are thin regions where a supersonic gas changes state extremely quickly. A normal shock stands nearly across the flow and creates a strong loss of total pressure. Oblique shocks form at angles from sharp corners, wedges, and aircraft noses.
They can turn a flow while keeping part of it supersonic. Curved expansion fans form when a supersonic flow turns around an outward corner. They lower pressure and temperature while increasing speed.
These patterns are visible in schlieren images from wind tunnels, where changes in density bend light. Students should notice that shocks are not solid walls. They are moving changes in gas properties, though they can appear fixed relative to an aircraft or nozzle.
High speed flight shows why geometry matters. A rounded shape can reduce sudden compression, while a sharp inlet ramp may create planned oblique shocks that slow incoming air before it reaches an engine compressor. Rockets use nozzles to turn thermal energy from hot combustion gases into directed exhaust speed.
At high altitude, outside pressure is low, so a nozzle designed for sea level may expand the exhaust differently than intended. In pipelines, rapid valve closure can create pressure waves that damage equipment. When solving problems, track the direction of flow, identify whether it is below or above sonic speed, and state which properties are static or total.
Check units carefully. Most mistakes come from applying an incompressible rule after density changes have become too large to ignore.
Key Facts
- Mach number is M = v/a, where v is flow speed and a is the local speed of sound.
- For an ideal gas, the speed of sound is a = sqrt(gamma R T).
- Compressibility effects usually become important when M > 0.3.
- Subsonic flow has M < 1, sonic flow has M = 1, and supersonic flow has M > 1.
- In a converging-diverging nozzle, subsonic flow accelerates in a decreasing area, while supersonic flow accelerates in an increasing area.
- Across a normal shock, Mach number decreases, static pressure increases, static temperature increases, and total pressure decreases.
Vocabulary
- Compressible flow
- Compressible flow is gas flow in which density changes significantly because of changes in pressure, temperature, or speed.
- Mach number
- Mach number is the ratio of an object's speed or flow speed to the local speed of sound.
- Choked flow
- Choked flow occurs when the Mach number reaches 1 at a restriction, limiting the maximum mass flow rate for given upstream conditions.
- Converging-diverging nozzle
- A converging-diverging nozzle is a duct that narrows to a throat and then widens to accelerate gas to supersonic speed.
- Shock wave
- A shock wave is a very thin region where supersonic flow changes abruptly, causing sudden increases in pressure, temperature, and density.
Common Mistakes to Avoid
- Using one constant speed of sound for all parts of the flow is wrong because the speed of sound depends on local temperature through a = sqrt(gamma R T).
- Assuming a wider duct always slows a gas down is wrong because supersonic flow accelerates in a diverging area while subsonic flow slows in a diverging area.
- Treating Mach 1 as just another speed is wrong because sonic conditions can choke a nozzle and prevent further increases in mass flow rate by lowering downstream pressure alone.
- Ignoring total pressure loss across a shock is wrong because shocks are irreversible and reduce the useful pressure available for producing thrust or doing work.
Practice Questions
- 1 Air flows at 255 m/s where the local speed of sound is 340 m/s. Calculate the Mach number and classify the flow as subsonic, sonic, or supersonic.
- 2 In hot exhaust gas, gamma = 1.33, R = 287 J/(kg K), and T = 900 K. Calculate the local speed of sound using a = sqrt(gamma R T).
- 3 A converging-diverging nozzle has Mach 1 at its throat and a shock wave in the diverging section. Explain qualitatively what happens to Mach number, pressure, temperature, and total pressure as the flow crosses the shock.