Nozzles

After the turbine's tax, the gas still holds a fortune in pressure and heat — and one exit. The nozzle is where all of it is finally converted into what lesson 1 demanded: velocity. It looks like the simplest part of the engine — a shaped pipe, no moving parts (usually) — and it is where the thrust equation actually gets paid.

Squeeze to speed

The subsonic rule is lesson 2 of Aerodynamics, working for a living: pinch a flow and it accelerates. A converging duct trades the gas's pressure for speed — pressure falls, velocity soars, and the momentum of that fast exhaust is the thrust. But the trick has a ceiling: a converging nozzle can accelerate gas only up to the speed of sound at its throat, and no further — the flow is then choked. Downstream news (like the low pressure outside) travels at sound speed, so once the throat runs sonic, nothing outside can coax the flow to hurry any more. Airliner nozzles live happily at or below this limit.

Past the ceiling: the hourglass

Rockets and supersonic jets need exhaust much faster than sound — and the escape from the choke is one of the loveliest reversals in fluid dynamics. Once a flow is supersonic, the rules invert: expanding the duct makes it go faster, not slower (the gas trades its remaining pressure and heat for speed as it spreads out). So the answer is an hourglass: converge to sonic at the throat, then diverge — the convergent–divergent (de Laval) nozzle, accelerating exhaust to Mach 2, 3, 5 beyond the pinch. Every rocket bell you have ever seen is this shape; fighter jets carry an adjustable version, flexing petals to reshape the hourglass for each flight condition.

Thrust's fine print

Two honest footnotes complete the picture. First, if the gas leaves the nozzle still above ambient pressure (common in choked engines), the leftover pressure pushing on the exit area contributes extra thrust — the full equation reads $F = \dot{m}\,\Delta v + (p_e - p_0)A_e$. Second, the nozzle is a precision instrument disguised as a pipe: too small an exit and the engine backs up (pressure, temperature and surge margin all suffer); too large and thrust is thrown away. On fighters the variable petals visibly breathe with the throttle — the engine literally changing the shape of its final argument with the atmosphere. And in the next lesson, the biggest thrust trick of all turns out to involve barely heating most of the air at all.