Mach Number and Stagnation Pressure

Mach number M = V/a, with sound speed a = √(γRT) for a calorically perfect ideal gas. Use absolute temperature and a mass-specific gas constant. Stagnation quantities describe bringing that local state to rest isentropically. These relations assume constant R and γ. See NASA Glenn, equations 1–8.

Original calculated example: set R = 287.05 J/(kg·K), γ = 1.4 and static pressure to 101325 Pa. At 100 m/s and 288.15 K, sound speed is 340.292 m/s and Mach is 0.2939. Change only temperature to 400 K: sound speed becomes 400.934 m/s and Mach becomes 0.2494. These are illustrative constant-property inputs, not an atmosphere or gas-composition lookup.

Enter these states in the gas-state calculator. The unit converter handles fixed speed units; it links to this calculation for Mach. Enter static pressure as an absolute pressure, not a gauge reading.

Define F = 1 + (γ − 1)M²/2. Then T₀/T = F and p₀/p = F raised to γ/(γ − 1). Dynamic pressure q = ρV²/2 = γpM²/2. Thus p₀ − p approaches q at low Mach but differs as compressibility increases. Subscript 0 denotes stagnation here, not a second measured static station. These are NASA Glenn's isentropic and dynamic-pressure relations.

Original comparison: keep the preceding pressure, 288.15 K, R and γ fixed and vary speed to give each Mach number. Percentage difference below means 100[(p₀ − p)/q − 1], using q as the denominator. It is not a universal instrument-error estimate.

Try doubling static pressure with all other inputs fixed: both pressure columns double, while Mach and the percentage difference stay unchanged. At zero speed both pressure rises are zero, so this percentage has a zero denominator and is omitted.

A normal shock raises static pressure, temperature and density, reduces Mach from supersonic to subsonic, and loses total pressure. For the adiabatic, no-work model, total temperature remains constant. It is not an isentropic compression. See NASA Glenn's normal-shock relations.

The gas-state tool does not calculate that shock or the pressure behind it. Use the normal-shock calculator for a stationary, one-dimensional shock with constant gas properties. In particular, its ideal upstream stagnation pressure is not a supersonic Pitot reading after shock loss. It also does not solve nozzle geometry, friction, heat transfer, variable heat capacity or real-gas behavior. Supplying a high speed does not establish that constant γ remains appropriate.