ME 310 Chap.11 One-Dimensional Compressible Flow
One-Dimensional Compressible Flow
Where it fits
The course catalog description covers compressible high-speed flow alongside incompressible flows, and the content outline frames it as one-dimensional gas dynamics.
The calendar gives it the two lectures just before the final review, so it is examined on the comprehensive final.
Mach number and the incompressible limit
The speed of sound in an ideal gas depends on temperature through the square root of the heat-capacity ratio times the gas constant times absolute temperature. The Mach number compares the flow speed with it.
Below about Mach 0.3, density changes stay within a few percent and the incompressible methods of earlier chapters remain accurate; above that, density, temperature and pressure change together and must be treated as coupled.
Stagnation properties
For flow without friction or heat transfer, bringing a gas to rest raises its temperature, pressure and density according to the isentropic relations, each a function of Mach number and the heat-capacity ratio.
At moderately high subsonic speeds, the incompressible Bernoulli estimate of stagnation pressure falls noticeably short.
Nozzles and choking
Combining mass and momentum conservation gives the area-velocity relation. Subsonic flow speeds up in a converging passage, but supersonic flow speeds up in a diverging one, so reaching supersonic speed requires a throat where the Mach number is exactly one.
Once the throat is sonic the nozzle is choked and lowering the back pressure no longer raises the mass flow. At off-design back pressures, a normal shock appears in the diverging section, an abrupt irreversible return to subsonic flow.
What to memorise and what to derive
Memorise the speed of sound, the Mach-number threshold and the area-velocity relation.
The stagnation ratios and the area ratio are better written on the equation sheet, with a worked value at Mach 1 to check them.
What this chapter covers
- 01
Speed of sound in an ideal gas
- 02
Mach number and the incompressible limit
- 03
Isentropic stagnation relations
- 04
Area-velocity relation
- 05
Converging-diverging nozzles and choking
- 06
Normal shocks in nozzles
Worked example · free
Mach number at cruise altitude
- 2Speed of sound c = √(kRT) = √(1.4 × 287 × 223) = 299 m/s.
- 1Mach number Ma = 250 / 299 = 0.84.
- 2Stagnation temperature T₀ = T(1 + 0.2 Ma²) = 223 × 1.139 = 254 K.
Key terms
- Mach Number
- The Mach number is the ratio of the local flow speed to the local speed of sound.
- Choked Flow
- Choked flow is the condition in which a nozzle throat reaches Mach 1, so further lowering of downstream pressure cannot increase the mass flow.
- Normal Shock
- A normal shock is a thin, irreversible jump from supersonic to subsonic flow across which pressure and temperature rise sharply.
One-Dimensional Compressible Flow FAQ
When must I treat a gas flow as compressible?
When the Mach number exceeds about 0.3, because density then changes by more than a few percent. Air around a car at highway speed is incompressible for practical purposes; air over a jet wing at cruise is not.
Why does a supersonic nozzle get wider?
Above Mach 1 the gas density falls faster than its velocity rises as it expands, so a larger area is needed to carry the same mass flow at higher speed. The area-velocity relation changes sign at Mach 1.
What does it mean for a nozzle to be choked?
Its throat has reached Mach 1. Information cannot travel upstream through sonic flow, so lowering the downstream pressure further no longer changes the flow at the throat, and the mass flow rate is at its maximum for the supply conditions.
Exam move
Keep a one-line table of the isentropic ratios as functions of Mach number on the equation sheet, with the reminder that all temperatures and pressures are absolute.
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