The expansion ratio, throat diameter, shape, gas and chamber conditions are the controls above; these set the rest of the contour. Changing any of them makes this your own design rather than the engine you picked.
Everything above, plotted from the injector face to the exit, with the nozzle itself drawn to scale on top. Watch the pressure fall off a cliff at the throat once it chokes — and jump straight back up wherever a shock is standing.
Two ideas carry the whole of this page. Choking: once the throat reaches Mach 1, the gas leaves faster than any pressure signal can travel back against it, so lowering the pressure downstream stops drawing more flow. Shocks: when a supersonic flow has to meet a higher pressure than it has, it cannot ease into it; it jumps, in a layer a few molecules thick, and pays for the jump in total pressure.
Start with the back pressure equal to the chamber (right) and nothing flows. Lower it and flow rises, until the throat reaches Mach 1 at the dashed line. From there the curve is flat: all the way to a vacuum, not one more gram a second gets through. That flat line is choking, and it is why a choked orifice is a flow meter and a rocket’s mass flow is set at the throat.
The textbook picture, drawn for this nozzle. High back pressure: the flow speeds into the throat and slows again (green). Lower it and the throat chokes (blue). Lower still and the flow carries on accelerating past the throat, then jumps back up at a shock, and the jump walks down the bell as the back pressure falls (pink, violet), reaching the exit (orange). Below that, nothing inside the nozzle changes at all; the curve is the design curve (amber), and the mismatch is settled outside, in the plume. Yours is white.
It cannot stand in a converging duct in steady flow, and it cannot form at the throat of a choked nozzle running at design. It can only make supersonic flow subsonic (normal) or turn it and slow it a little (oblique). And an oblique shock can only turn a flow so far: past the limit on chart 4 it detaches and becomes a curved, partly normal shock. That limit is exactly what decides whether a plume’s shocks reflect cleanly into diamonds or break into a Mach disc.
The losses are small for weak shocks and brutal for strong ones: a Mach 1.5 shock keeps 93% of the total pressure, a Mach 3 shock keeps 33%. Supersonic inlets use several weak oblique shocks rather than one strong normal one for exactly this reason.
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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Releases in which this page changed, newest last. Derived from the archived copy of every release, not from notes written afterwards — so it reflects what actually shipped. Site-wide passes are left out; they are in the full changelog.
Mass cannot pile up, so ρAV is the same everywhere. Below Mach 1 a narrowing duct speeds the flow; above it, it slows it. To keep accelerating, the duct must open out.
Once the throat reaches Mach 1 the news of lower downstream pressure never arrives: mass flow stops responding. For air, below 52.8% of chamber pressure.
The throat meters the flow; the expansion ratio (exit ÷ throat area) alone sets exit Mach. Throat size is the throttle, bell size the gearing.
Over-expand far enough and the outside air wins and tears the flow off the wall, once exit pressure falls below about 40% of ambient (Summerfield).
The most misleading word in the subject. Nothing is stopping up: the flow is at its maximum and cannot be increased by lowering the pressure downstream further. The throat has gone deaf, not shut, which is why a choked orifice makes such a dependable flow meter, and why a rocket’s thrust is set at the throat and not at the exit.
Once choked, mass flow is set by throat area, chamber pressure and chamber temperature only. Nothing done to the diverging half changes how much gas leaves per second; the bell decides how much of its heat becomes speed.
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. First stages fly undersized bells and knowingly waste exhaust energy; upper stages fly enormous ones.
Between choking and perfect expansion the nozzle reconciles supersonic flow inside with a fixed pressure outside using a shock: a discontinuity a few molecular free paths thick where 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 thrown away. Sweep the back pressure down and you pass in order: subsonic venturi, first choke, a shock marching down the bell, the shock reaching the exit plane, oblique shocks outside, perfect expansion, and an under-expanded jet.
The glowing beads mean the jet left at the wrong pressure and is oscillating its way to agreement with the air: a shock or expansion fan reflects off the jet’s free edge, converts to the opposite kind, crosses, reflects again, and each crossing compresses and heats the gas enough to glow. A perfectly matched nozzle gives a plain, clean plume and no diamonds, which for a first stage is almost never.
Mach number: speed divided by the local speed of sound: a ratio, not a speed; Mach 1 is much slower in cold air than hot. Throat: the narrowest point, the only place flow can be exactly sonic, because it is the only place the area stops changing. Choked: the throat has reached Mach 1 and cannot hear the pressure downstream. Expansion ratio: exit area over throat area: a bigger bell means faster, colder, lower-pressure exhaust. Total pressure: the pressure the gas would have brought smoothly to rest: the currency of thrust, which a shock spends irreversibly.
Normal shock: a near-instant jump from supersonic to subsonic; pressure and temperature rise, total pressure falls, and the loss is never recovered. Over-expanded: exit pressure below the surrounding air: the atmosphere squeezes the plume inward and, if bad enough, the flow leaves the wall. Under-expanded: exit pressure above the air: the plume balloons outward, normal for an upper stage in near-vacuum. Separation: the flow letting go of the wall, unevenly, shaking the engine sideways: why a vacuum bell cannot be lit at sea level. Mach diamonds: bright beads in a mismatched plume where reflected shocks cross.
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 constant. Separation uses a single pressure ratio in place of a boundary-layer calculation. Outside, the plume is placed, not solved. Use it to understand the behaviour, not to cut metal.
Not blocked. The throat has reached Mach 1, and news that you lowered the pressure downstream travels upstream at the speed of sound — which is exactly how fast the gas is already leaving. The message never arrives, and mass flow stops responding. For air it happens once the downstream pressure falls below 52.8% of the chamber pressure.
Because of the sign of 1 − M² in dV/V = −(dA/A) / (1 − M²). Below Mach 1 a narrowing duct speeds the flow up; above it, a narrowing duct slows it down, so to keep accelerating the duct must open out. At exactly Mach 1 the equation only allows dA = 0, which is why the sonic point is always at the narrowest section.
Over-expansion. A big bell drives the exit pressure far below sea-level air, and once it drops below about 40% of ambient the outside air forces its way back up the wall and the flow separates — usually unevenly, hammering the nozzle sideways. So first stages fly deliberately small bells and upper stages fly enormous ones. No single nozzle is right for the whole climb.
A symptom, not a triumph. They mean the jet left at the wrong pressure and is oscillating its way to agreement with the air, overshooting each time; every crossing compresses and heats the gas enough to glow. A perfectly matched nozzle produces a plain plume and no diamonds at all.