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.
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.