[Toutouxc]

Can someone ELI5 to me the physics behind the divergent part of a de Laval nozzle? I know that in a straight pipe subsonic flow tends to accelerate towards M=1 and supersonic flow tends to slow down towards M=1, I know what choked flow is and generally how the convergent-divergent design works, but for the life of me I can't find anywhere an understandable, non-hand-wavy explanation of WHY supersonic flow in a divergent nozzle does what it does.

[namibj]

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...

[bombcar]

This made something I never considered thinking about perfectly clear - thank you!

[Cerium]

I have never really looked into it before myself, but this explanation seems to make sense: https://www.grc.nasa.gov/www/k-12/airplane/nozzled.html

Quote: On the other hand, if the converging section is small enough so that the flow chokes in the throat, then a slight increase in area causes the flow to go supersonic. For a supersonic flow (M > 1) the term multiplying velocity change is negative (1 - M^2 < 0). Then an increase in the area (dA > 0) produces an increase in the velocity (dV > 0). This effect is exactly the opposite of what happens subsonically. Why the big difference? Because, to conserve mass in a supersonic (compressible) flow, both the density and the velocity are changing as we change the area. For subsonic (incompressible) flows, the density remains fairly constant, so the increase in area produces only a change in velocity. But in supersonic flows, there are two changes; the velocity and the density. The equation:

- (M^2) * dV / V = dr / r

tells us that for M > 1, the change in density is much greater than the change in velocity. To conserve both mass and momentum in a supersonic flow, the velocity increases and the density decreases as the area is increased.

[Toutouxc]

That's why I was asking for an ELI5 explanation. I know that the equation holds and that compressibility turns everything upside down, I just haven't been able to figure out an intuitive explanation.

I have this idea that in supersonic flow, a pressure wave can't move backwards against the flow, right? Which would mean that any single molecule inside the flow has no way of knowing what's in front of it (because the information simply can't get there), but it feels the pressure of the molecules behind it, so it accelerates towards the void.

The "Fanno flow" article on Wikipedia says that "... For a flow with an upstream Mach number greater than 1.0 in a sufficiently long enough duct, deceleration occurs and the flow can become choked ... Conversely, the Mach number of a supersonic flow will decrease until the flow is choked.", which means that supersonic flow behaves differently in a diverging nozzle than in a simple straight pipe. This is the part that I don't understand. Is the friction inside the nozzle somehow inhibited by the walls of the nozzle gradually moving out of the flow's way or something?

[namibj]

Well, the slow-down won't happen until after it exits the nozzle (c.f. Mach diamonds in rocket exhaust... they are the shocks where the supersonic flow interacts with the ambient air).