Aviation Calcs

Isentropic flow

Isentropic Flow Calculator

One-dimensional isentropic flow of a perfect gas. Enter a Mach number, a stagnation ratio, or an area ratio A/A* (subsonic or supersonic branch) and get the full NACA Report 1135 set: T/T0, P/P0, ρ/ρ0, A/A*, and M*. Optionally attach a static temperature or pressure to see the dimensional stagnation state. Default γ = 1.4.

I want to enter

Air at ordinary temperatures is 1.4. Hot combustion products are often taken near 1.33. The NACA 1135 tables are for γ = 1.4.

Mach number

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Isentropic relations (perfect gas)

Along an isentropic streamtube,

T0/T = 1 + ((γ−1)/2) M²
P0/P = (T0/T)γ/(γ−1)
ρ0/ρ = (T0/T)1/(γ−1)

T0, P0, and ρ0 are the stagnation (total) temperature, pressure, and density — the state the gas would reach if it were brought to rest isentropically. For γ = 1.4, T0/T = 1 + 0.2 M².

Area ratio and the two roots

The isentropic area–Mach relation is

A/A* = (1/M) [ (2/(γ+1)) (1 + ((γ−1)/2) M²) ](γ+1)/(2(γ−1))

A* is the throat area at which M = 1. For any A/A* > 1 there are two Mach numbers, one subsonic and one supersonic. At M = 2, γ = 1.4, A/A* = 1.6875 exactly, P/P0 = 0.12780, T/T0 = 0.55556, and ρ/ρ0 = 0.23005 — the NACA 1135 table values this page is tested against.

Sources

NACA Report 1135, isentropic-flow tables (γ = 1.4); NASA Glenn Beginner’s Guide; Anderson, Modern Compressible Flow. For a normal shock, which is not isentropic, see the normal-shock calculator.

Frequently asked questions

What is stagnation temperature?

It is the total temperature T0 = T (1 + ((γ−1)/2) M²). For air at M = 2 and a static temperature of 216.65 K, T0 = 389.97 K. Recovery on a real probe is slightly less than ideal.

Why are there two Mach numbers for one A/A*?

A converging–diverging nozzle can pass the same area ratio on the subsonic side of the throat or on the supersonic side. You have to choose the branch. At A/A* = 1 the flow is sonic and there is only one Mach number, M = 1.

Are the NACA 1135 numbers exact?

For γ = 1.4 they come from closed-form identities, so the usual table entries (M = 1, 2, 3, …) can be reproduced to the printed number of digits. This page uses those same identities.