De Laval Nozzle Calculator

A converging–diverging nozzle — a de Laval nozzle — is the only way to take a gas past Mach 1 in steady flow. The area has to fall to a throat and then rise again, because area and velocity swap their relationship at M = 1: below sonic a narrowing duct speeds the flow up, above sonic a widening one does.

Two numbers decide everything. The geometry contributes one, the exit-to-throat area ratio, which fixes the two Mach numbers the exit can have — a subsonic root and a supersonic one, both from the same area relation:

The back pressure contributes the other, and it chooses between them. Raise it high enough and the throat never reaches M = 1 at all: the nozzle behaves as a venturi and the flow is subsonic everywhere. Lower it to the first critical and the throat chokes. Below that the diverging section goes supersonic, and the flow has to get back up to the back pressure somehow.

It does that through a normal shock standing in the diverging section, and where it stands is what this page solves. Downstream of the shock the flow is subsonic again, on a new and larger sonic reference area, because losing stagnation pressure means a notional throat has to grow to pass the same mass:

Push the back pressure lower still and the shock reaches the exit plane, and then leaves it. From that point the nozzle stops responding: nothing downstream can send a signal back up through supersonic flow, so the flow inside is identical for every back pressure below it. The adjustment happens outside instead — oblique shocks when the exit pressure is below the back pressure, an expansion fan when it is above. Those are the overexpanded and underexpanded regimes, and between them sits the design point where the two match exactly.

The mass flow follows the same logic. Once the throat is sonic the nozzle passes a flow that depends only on γ and the throat area, and lowering the back pressure further does not increase it. That is choking, and it is why the number stops moving.