Isentropic Flow Calculator

Determine the properties of fluid flow during an isentropic process using the Isentropic flow calculator.

Stagnation temperature518.67 K245.52 °C — what the gas reaches if brought to rest. At Mach 2 that is 230.52 K of stagnation heating, which is why fast aircraft get hot without any friction being involved

Stagnation temperature ratio T₀/T1.8

Stagnation pressure792.8123 kPa7.82445× the static pressure

Stagnation density ratio ρ₀/ρ4.346916

Area ratio A/A*1.6875this is DOUBLE-VALUED: the same area ratio has one subsonic and one supersonic solution, which is exactly why a converging nozzle can never exceed Mach 1 and a converging–diverging one can

Flow regimesupersonicdensity changes matter here

Speed of sound at this temperature340.294 m/s

Flow speed680.588 m/s

Critical pressure ratio p*/p₀0.5282820.5283 for air — squeeze a reservoir below this fraction and the throat CHOKES at Mach 1. Lowering the downstream pressure further adds no more mass flow at all

Exit area for this Mach number16.875 cm²a throat of 10 cm² needs this much exit area to reach Mach 2

Mach number after a normal shock0.57735ALWAYS subsonic, however fast the flow arrives — that is a theorem, not a coincidence

Static pressure jump across it4.5×

Density jump2.66667×capped at 6× however strong the shock, because the gas cannot be compressed further by a single shock

Temperature jump1.6875×

Stagnation pressure retained72.0874%a shock is irreversible, so stagnation pressure is LOST — 27.913% of it here. Stagnation temperature, by contrast, is unchanged: no energy left the flow, only its usefulness

The formula

T₀/T = 1 + (γ−1)M²/2; A/A* = (1/M)[2(1+(γ−1)M²/2)/(γ+1)]^((γ+1)/2(γ−1))

Stagnation heating is not friction

Bringing a fast-moving gas to rest converts its kinetic energy into internal energy, raising the temperature by (γ−1)M²/2 times the static value. At Mach 2 that is already 80% hotter; at Mach 6 it is more than sevenfold. This happens in an ideal, frictionless, reversible flow — it is compression, not rubbing, and it is why hypersonic vehicles have a thermal problem no amount of surface smoothness can fix.

Why nozzles have a throat

The area ratio A/A* has a minimum of exactly 1 at Mach 1 and rises on both sides, so every area ratio corresponds to two Mach numbers — one subsonic, one supersonic. A purely converging nozzle can therefore accelerate flow only up to Mach 1 at its exit, no further. Going supersonic requires the passage to converge to a throat and then DIVERGE, which reverses the usual intuition that narrowing speeds a flow up.

Once the pressure ratio falls below about 0.528 for air the throat chokes at Mach 1 and the mass flow stops responding to further reductions downstream. That is why a compressed-air line delivers a fixed flow regardless of what is downstream of the orifice.

Shocks destroy pressure, not energy

Across a normal shock the stagnation temperature is unchanged — no energy leaves the flow. Stagnation pressure, however, always falls, because the shock is irreversible. What is lost is not energy but the ability to do work with it, and recovering pressure efficiently is the entire design problem of a supersonic inlet.

The downstream flow is always subsonic, no matter how fast the upstream flow, and the density ratio saturates at (γ+1)/(γ−1) — six for air — however strong the shock becomes.