| Basically, it would already work if you left out the diverging part.
However, the exhaust has still excess pressure.
You want to make it do work while expanding to ambient pressure, so you provide a ring around the throat where the exhaust pressure pushes against to direct the exhaust momentum vector for every part of the exhaust plume to be as close to retrograde as possible.
Sideways momentum is wasted momentum. To actually get it to push against your ring, the diameter can't increase too fast, because the sideways flow velocity of the exhaust inside your bell needs to be subsonic. So you first increase steeply, because it's still high pressure and thus hot and thus has a high speed-of-sound, and gradually reduce how fast you increase (gradually reducing the taper). Eventually you have expanded it to a pressure equalling ambient pressure, and won't get more thrust from further expansion (doing more also causes flow separation at the edge where ambient air pushes the plume radially inwards and separates it from the inside surface of the bell. The supersonic shock effects can break your bell if you're not careful.). Also at low pressure you get little thrust per additional nozzle area, while the low temperature requires a low taper that requires a lot of surface for the additional nozzle area. That is why even vacuum engines don't go down to extremely low exhaust pressure. IIRC you can only ever double your momentum with an expanding nozzle, even in the asymptotic limit of an infinite bell in perfect vacuum and compared to a knife edge throat. Fundamentally, rocket nozzles are weird open-cycle heat engines subject to the Carnot limit. Thrust times (exhaust) velocity is power, and the chamber (combustion) temperature your hot side.
If your exhaust has a different molecular mix than the atmosphere, there is some additional energy to theoretically gain from there. Otherwise, you should have IIUC reached outside temperature when you reach outside pressure. Or maybe you would need to expand to outside temperature regardless of pressure to scratch at Carnot-efficiency... |