Isentropic Flow Calculator
Type into any box — Mach, a stagnation ratio, A/A* — and both roots follow, at any γ.
Every quantity here fixes the Mach number outright except A/A*, which occurs once each side of the throat. That is why there are two columns — the two flows that share your area ratio.
Results
| Quantity | Subsonic before the throat | Supersonic after the throat |
|---|
What this is computing
Isentropic means adiabatic and reversible: no shock, no friction, no heat crossing the boundary. The stagnation state is then constant along the flow, and every local property follows from the Mach number alone — which is why one number in any box on this page fixes all the others.
The three stagnation ratios each fall monotonically with M, so any one of them identifies the state outright. A/A* does not — it is the reason this page has two columns.
Why the area ratio has two roots
A/A* falls to 1 at the throat and rises on both sides of it, so a given duct area occurs once subsonically and once supersonically: A/A* = 2 is either M = 0.306 or M = 2.197. The geometry does not choose between them — the back pressure does.
Sonic conditions, and choking
At M = 1 in air, p/p₀ = 0.5283, T/T₀ = 0.8333 and ρ/ρ₀ = 0.6339. Once the pressure ratio across a nozzle exceeds 1/0.5283 = 1.893 the throat is sonic, and lowering the back pressure further cannot increase the mass flow.
Where it stops being true
A real nozzle follows these relations until it does not: a shock in the diverging section, separation when it is over-expanded, heat loss through the wall. And γ is not a detail — an area ratio of 4 gives M = 2.94 in air but M = 2.77 in combustion products at γ = 1.3. Same nozzle, different exit Mach number.