Isentropic Flow Relations Calculator

The isentropic flow relations are formulas relating the stagnation (total) and static properties of a calorically perfect gas to its Mach number: T0/T = 1 + (γ-1)/2·M², with p0/p and ρ0/ρ derived from T0/T using the isentropic exponents γ/(γ-1) and 1/(γ-1), plus the area ratio A/A* from the area-Mach relation.

T0/T = 1 + (γ-1)/2·M², where T0 is the stagnation temperature, T is the static temperature, γ is the ratio of specific heats (1.4 for air), and M is the Mach number. This ratio depends only on M and γ, not on the actual pressure or temperature.

A/A* is the ratio of the local flow area A to the area A* at the sonic throat where M = 1 exactly. By definition, the throat is where A = A*, so A/A* = 1 there. For any other Mach number, whether subsonic or supersonic, A/A* is greater than 1, meaning the duct must be wider than the throat both upstream and downstream of it.

This is a general-purpose gas-dynamics reference tool, the same isentropic ratios and area-Mach relation taught in every compressible-flow course and used for Pitot-static instrumentation, wind tunnel design, and general coursework. The site's rocketry section has separate calculators that apply this same underlying theory narrowly to size a specific rocket nozzle's throat and expansion ratio for a chosen thrust and propellant, a different, more specialized workflow for a different audience.

Any calorically perfect gas works if you set the correct γ (ratio of specific heats): air and diatomic gases use γ≈1.4, combustion products typically use γ≈1.30-1.33, and monatomic gases like helium or argon use γ≈1.67. The formulas are identical, only the exponents change.

p0/p = (T0/T)^(γ/(γ-1)), and for air γ/(γ-1) = 3.5, so p0/p is T0/T raised to the 3.5 power. Small increases in the temperature ratio compound into much larger increases in the pressure ratio, which is why stagnation pressure rises so sharply at high Mach numbers, the same effect behind Pitot-tube overpressure warnings on high-speed aircraft.

A calorically perfect gas has constant specific heats (cp, cv) that do not vary with temperature. This holds well for air up to roughly 1000 K. At higher temperatures, such as behind strong shocks or in hypersonic flow, vibrational and dissociation effects make the gas thermally perfect but not calorically perfect, and these simple closed-form isentropic relations become approximations rather than exact results.

For a wind tunnel, the test-section Mach number sets A/A* relative to the nozzle throat area: multiply the throat area by the A/A* value at your target Mach number to size the test section. The stagnation pressure and temperature ratios then tell you how much the settling chamber conditions must exceed the desired test-section static conditions.

No, the isentropic relations are valid for any Mach number from just above 0 up to hypersonic speeds, as long as the flow remains shock-free and isentropic. Subsonic diffusers, sonic throats, and supersonic nozzle sections can all be analyzed with the same formulas, only A/A* behaves differently on each side of M = 1 (two solutions exist for A/A* > 1, one subsonic and one supersonic).

At low Mach number (M < 0.3 or so), p0/p ≈ 1 + γ/2·M², which recovers the incompressible Bernoulli result p0 - p ≈ ½ρv². As Mach number rises, the compressible isentropic formula p0/p = (1+(γ-1)/2·M²)^(γ/(γ-1)) diverges from the incompressible approximation, which is why compressibility corrections matter above roughly Mach 0.3.

The area ratio A/A* = (1/M)·[...] has a 1/M term that grows without bound as M approaches 0, so A/A* is mathematically undefined at exactly M = 0. This calculator requires M ≥ 0.01 to keep the area ratio finite and physically meaningful; extremely low Mach numbers correspond to a duct area many times larger than the throat.