Everything above, plotted from the chamber to the exit. Watch the pressure fall off a cliff at the throat once it chokes — and jump straight back up wherever a shock is standing.
This is the part that sounds wrong. A garden hose speeds up when you squeeze the end — and it does, because that water is nowhere near the speed of sound. Above Mach 1 the arithmetic changes sign and squeezing slows the flow down.
Mass must keep flowing, so ρAV stays constant. Below Mach 1 the gas barely compresses, so squeezing the area forces the speed up. Above Mach 1 the density falls faster than the area does — so to keep ρAV constant, the area has to grow. And exactly at Mach 1 the only place the equation allows is where the area stops changing at all. That is the throat, and it is the only place sonic flow can ever sit.
A bell is cut for one air pressure and is wrong everywhere else. Too small and you leave energy in the exhaust; too big and the atmosphere pushes back up the bell until the flow tears off the wall. This is thrust against altitude for the nozzle above, and for two others.
| Type | Where you see it | What it is good at | What it costs |
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No changes to this tool’s own behaviour since the earliest archived release (v3.73). The full history for the site is in the changelog.
It is the most misleading word in the subject. Nothing is stopping up; the flow is at its maximum and simply cannot be increased by lowering the pressure downstream any further. The throat has gone deaf, not shut. That is precisely why a choked orifice makes such a dependable flow meter — whatever happens on the far side, the rate does not move.
Once choked, mass flow is set by three things only: throat area, chamber pressure and chamber temperature. Nothing you do to the diverging half changes how much gas leaves per second. The bell only decides how much of that gas’s heat becomes speed — which is why throat size is the engine’s throttle and bell size is its gearing.
A big bell wins in vacuum and can tear itself apart at sea level, because outside air forces its way back up the wall and the flow separates — roughly once exit pressure drops below 40% of ambient. So first stages fly deliberately undersized bells and knowingly waste exhaust energy, and upper stages fly enormous ones. There is no nozzle that is right for the whole climb.
Those glowing beads mean the jet left at the wrong pressure and is oscillating its way to agreement with the air, overshooting each time. A perfectly matched nozzle produces a plain, clean plume and no diamonds at all. They look like power and they are really a small, photogenic tax.
Mass cannot pile up in a pipe, so ρAV is the same everywhere. Put that next to the momentum equation and you get a single relation that runs the whole of nozzle design:
dV/V = − (dA/A) · 1/(1 − M²)
Look at the sign of 1 − M². Below Mach 1 it is positive, so shrinking the area speeds the flow up — the hose-pipe intuition, and it is correct. Above Mach 1 it is negative, and shrinking the area slows the flow down. To keep accelerating past the speed of sound the duct has to open out.
And exactly at Mach 1 the equation only permits dA = 0. Sonic flow cannot happen anywhere except where the area stops changing. That is why every supersonic nozzle has a waist, and why the sonic point is always exactly at it.
Drop the pressure downstream and more gas flows — up to a point. Once the throat reaches Mach 1, the news that you have lowered the pressure travels upstream at the speed of sound, and the gas at the throat is already leaving that fast. The message never arrives. The throat cannot know what you did, and the mass flow stops responding entirely.
For air that happens once the downstream pressure falls below 52.8% of the chamber pressure. After that, mass flow depends only on throat area, chamber pressure and chamber temperature — which is exactly why a choked orifice makes such a good flow meter, and why a rocket's thrust is set at the throat and not at the exit.
Between choking and perfect expansion the nozzle has to reconcile two things that do not agree: supersonic flow inside, and a fixed pressure outside. It does it with a shock — a discontinuity a few molecular free paths thick where the flow slams from supersonic to subsonic, pressure and temperature jump, and total pressure is permanently lost. Total pressure is thrust, so every shock is thrust you have thrown away.
Sweep the back pressure down and you pass through the whole sequence in order: subsonic venturi, first choke, a shock appearing at the throat and marching down the bell as you keep lowering it, the shock reaching the exit plane, oblique shocks outside, perfect expansion, and finally an under-expanded jet still pushing outward as it leaves.
Over-expand far enough and the outside air wins. It pushes back up the inside of the bell until the boundary layer can no longer hold on and the flow tears away from the wall, usually unevenly, hammering the nozzle sideways. The rough guide is that separation begins once the exit pressure falls below about 40% of ambient — the Summerfield criterion, which is a rule of thumb standing in for a boundary-layer calculation and is quoted anywhere between 30 and 40 per cent.
That is the entire reason vacuum engines cannot simply be lit on the pad. A vacuum bell with an area ratio of 165 has an exit pressure far below sea-level air; it would separate violently and quite possibly destroy itself. So first stages fly small bells and waste some exhaust energy, and upper stages fly enormous ones.
Those bright beads in a rocket or afterburner plume are the jet correcting itself and overshooting. If the exit pressure does not match the air, the plume must adjust — and it does so with a shock or an expansion fan that reflects off the free edge of the jet, converts to the opposite kind, crosses, reflects again, and repeats. Each crossing point compresses and heats the gas enough to glow. Regular diamonds mean the nozzle is not matched to the air it is flying in, which for a first stage is almost always.
Quasi-one-dimensional isentropic flow of a perfect gas, with normal shocks. Real nozzles are two-dimensional: they lose thrust to flow divergence, grow boundary layers, transfer heat into the wall, and in a rocket the chemistry is often still finishing partway down the bell, so the ratio of specific heats is not really constant. Separation uses a single pressure ratio in place of a boundary-layer calculation. The plume outside is drawn schematically — the oblique shocks and expansion fans are named and positioned, not solved. Use it to understand the behaviour, not to cut metal.