Stagnation Temperature (Compressible Flow)
Total (stagnation) temperature from static temperature and Mach number for a calorically perfect gas.
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The engineering
Stagnation (total) temperature is what the flow reaches when brought to rest adiabatically — the sum of static thermal energy and the kinetic energy of the bulk motion. Reach for it whenever you need the true energy state of a moving gas: sizing turbine inlet limits, predicting probe recovery temperature, or estimating aerodynamic heating of a fast airframe.
For a calorically perfect gas T₀ depends only on Mach and γ, not pressure — so it stays constant across a nozzle or shock (as long as no heat is added or extracted). Sanity check: at M = 1 with γ = 1.4 the total temperature is exactly 20% above static. Kinetic heating gets brutal fast — a Mach 3 stream at 220 K standard-atmosphere static sits near 620 K total.
Where this math comes from
The relation falls straight out of the steady adiabatic energy equation, cₚT + V²/2 = cₚT₀, which took shape as thermodynamics met fluid flow in the late 19th century. It became a daily aerospace tool once wind tunnels and jet engines forced engineers to distinguish the temperature a thermometer reads in a fast stream from the temperature it would read at rest.
John D. Anderson's gas-dynamics texts codified the modern isentropic-flow form used here, packaging T₀/T = 1 + (γ−1)/2·M² alongside its pressure and density cousins as the everyday compressible-flow relations. The same equation underlies the recovery-factor corrections aerodynamicists apply to real temperature probes.
- 1850Rudolf ClausiusFormalizes the energy equation that links thermal and kinetic energy in a flow.
- 1903Ludwig PrandtlFounds modern compressible aerodynamics, making stagnation properties practical design quantities.
- 1953Ames Research Staff (NACA Report 1135)Tabulates isentropic T₀/T for compressible-flow work.
- 1990John D. AndersonStandardizes the textbook isentropic relations used in aerospace curricula.
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