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To anyone still curious about this, sorry for disappearing for so long, life gets in the way sometimes lol. I have been reading through the various articles and books ive found online, and from what I can tell online it seems pretty unclear. From what I can tell, it seems like regardless of pressure (assuming the engine is still working properly), the fuel flow has *some* degree of a hyperbolic streamline (generally the hyperbolic part is just so unbelievably small that its beyond negligible), which bends towards a more parabolic shape in the direction of the engine exhaust. The amount this streamline bends (and therefore shape of the FLOW) seems to really depend on both the engine itself, and the surrounding pressure. Now the fun part though: what shape would it look like? Well, the unfortunate answer is.... I still have no idea. I am working my way through a few articles about light emissions from gasses, but all I have found so far is grey body emissions, which wont really tell me how much visible light is being emitted. That being said, I ran a super janky fluid sim, and I did actually see a temperature curve similar to what we see in the game and in the other books, so unless im misunderstanding this I think its pretty safe to say that I think I was wrong on my initial thoughts! I am happy to say that more likely than not, it should be hyperbolic like it is in the game! If I find anything else I will make sure to report it here, but thank you all for discussing this with me, I really appreciate being exposed to all of this new knowledge, and I really enjoyed sharing my knowledge with you all, so thank you!!

I am definitely gonna be reading into that book now, thank you so much for the reference!! To be honest, I really do hope I am wrong about the way the plume looks now, its a very unique shape that ive never really seen before, plus it means theres even more stuff for to learn lol. Thanks for taking the time to answer this question!

Interesting, yeah you are right. In the fourth one though the exposure is actually high enough to see a very faint outline of the greater bell, so I am curious as to what's actually happening. I wonder if maybe the plume does still grow in the bell shape, but the hottest and densest part grows in a much smaller hyperbolic shape like that which is why it looks the way it does? I hope @Nertea can answer this.... that is my best guess for those images,

Thats a really good question, but as strange as it sounds, there is a point where the radial pressure is more of a dominating force than the axial, and that is immediately after it leaves the engine bell. The reason this is the case is because its not just about total pressure, its also about the differential on either side of each particle. At the very beginning of the post-nozzle flow, the flow is not expanding much, and so the axial pressure change (and therefore axial acceleration) is extremely small. On the flip side, the radial pressure drop is VERY high, as it goes from the high pressure exhaust to the vacuum of space, which causes the initial expansion of the plume. As it expands though, the radial pressure drop quickly becomes almost zero, and so the radial acceleration of the fuel slows to a crawl, while the axial acceleration starts to increase with the decreasing plume pressure (hence the bell shape). You can sort of visualize it in the image I attached here! Hopefully this helps!

definitely plan to, I 100% plan to try and figure out how to run a sim to test (I have literally no idea how to do optics in this situation so no promises but I will try), so if I do get interesting results I will definitely post em! Yeah, honestly it blows my mind how such a seemingly simple game can be as thought provoking and brilliant as KSP is. squad was a pioneer, and I really cant wait to see what intercept does with the amazing groundwork theyve laid. If there is anything I can do to clear anything up please let me know! I would love to make this as comprehensive as possible, if for no other reason than to make this a resource people can find in the future if they are curious. And yeah I hope they do, I am really curious to hear their feedback on it, because i know they've done a ton of research and id love to know the results of that. Honestly, this topic is especially interesting because afaik this is kind of a new discussion. From what ive seen, engineers dont care one way or another what the plume *looks* like, we're way more concerned with the actual parameters like thrust and pressure, so im really curious to find out the answer to this.

Ah I see what you are talking about, yes it does seem to be somewhat hyperbolic, but this sim is actually of a plume contacting a surface. Dont worry about missing it though, reading these papers is a hassle sometimes and i do the same thing all the time lol

Do I have good news for you: Unfortunately, due to the fuel used in the ascent module and the massive bell of the engine, the fuel isnt really hot enough to see it with the naked eye, save for that split frame where it absolutely BLASTS the descent module. realistically, its very rare to see any plume in space, (heck, falcon 9 launches are barely hot enough to see once they are in vacuum) but thats no fun for a game like KSP, so I am totally happy with them making that change That being said, there is still hope to see that one day soon! I believe there are a few fuels that do burn visibly in space, and I think hydrolox is one of them. if we ever do begin manufacturing fuel on the moon, there is a good chance we can actually see it P.S. iirc the reason you see it so clearly on earth is actually not because the fuel is glowing, but its because its heating up the surrounding air enough that it glows!

Yeah, they have really been doing a phenomenal amount of research and its super impressive, honestly all the kids who grow up on this are gonna be a different breed when they hit the industry haha. I am definitely going to need to do more research into this, (hell, maybe ill do a sim of my own haha), but if I am wrong I will be happy to own up to it. That being said, streamlines can be misleading, especially when youre dealing with supersonics and visuals. Any time a supersonic flow turns, even if it is 0.0001 degrees, there are shocks, which lead to a massive pressure drop and deceleration of the flow, which leads me to believe that these streamlines are likely not representative of what it would actually look like, but I cant say for sure so I wont say anything concrete. All of the information I have given so far is just based on my current understanding of supersonic flow, but I am still not convinced it would not be parabolic. Another potential hole in that thread you sent is the strange behavior from transitional -> Vacuum. Think about it, if the plume is able to expand at the exit due to internal pressure in transitional: Why would it not also expand in vacuum at the exit, where it has even LESS resistance to its expansion? If what that person said was true, it would likely be closer to: Unfortunately I do not know much about ION engine, but AFAIK they are very low pressure, so it makes sense that it wouldn't have a substantial initial expansion. I have no idea though, that is a total shot in the dark cuz I know very little about those electric magic machines. ooooh more fun things to read, well guess I know what im doing instead of going to sleep tonight.... Glad I could help! Its a really complicated topic so i was honestly expecting to do a terrible job haha. But yeah, that's really the best answer, it depends on SOOO much, but generally its a bell that turns conical, with the size of the bell being dependent on the exit pressure of the flow Quick question: I didnt see any that fanned out in that paper you sent, would you mind telling me what figure you are talking about? The only ones I see that actually seem to "fan out" are ones that simulate a collision, like a lander near the surface or an engine right after stage separation where the flow is hitting the lower stage

Sorry for the confusion everyone, when I initially posted I used the term "parabola" to generally describe the shape, but as far as I know there really isnt a "true" classic shape that the fuel flows in, mainly because of the non-homogenous pressure in the exhaust plume: As you can see here, it actually starts as mostly straight out, then quickly curves outwards before beginning to curve back inwards. I say "parabola" because its a fairly common shape that people know, but really that's not it at all, nor is there really any common geometric shape. Since the flow is supersonic, having a low/zero pressure boundary causes the flow to turn, causing all sorts of annoying-to-model oblique shocks. To make it worse, this changes drastically based on the fuel mix and exit characteristics, which makes answering all of these questions in one go tricky, but I will do my best. Immediately after the fuel leaves the diverging portion of the nozzle, we have extremely high pressure gas, moving super fast. Since the surroundings are much lower pressure, the edges of this flow are bend outwards, causing an oblique shock and large pressure & velocity losses. These ripple out as we move further from the exit, and pretty quickly that high pressure region starts to drop off to (what I am going to call) a medium pressure region shown in red/yellow/green. As more and more of this flow turns outwards, it reaches a point where there is almost no pressure in the center, which causes the flow to turn back in on itself, creating a new (smaller) high pressure region. This continues until the flow is no longer moving supersonic and is as a steady equilibrium, but I honestly don't know how long it takes to reach that point, and that would be on an engine by engine basis. It is important to note that while it is expanding and supersonic, the exhaust velocity is actually still increasing, since the "upstream" flow has a higher pressure than the "downstream" flow. This is where that magic acceleration comes from that causes the classic curve, basically converting high Pressure, Temperature, and Density into velocity. That being said, as it approaches equilibrium, the outer shape of the flow at & beyond will slow to almost nothing, and will be functionally linear (or conical if you'd prefer it in 3-d). While it will still be slightly changing in trajectory, it will be extremely small, practically unnoticeable, so its totally fair to call it linear/Conical as y-> infinity. In regards to the internal pressure pushing the flow outwards, you are absolutely right, and technically the flow can even bend backwards during initial ejection: (Note, this is just the initial expansion, in the 0-0.1D distance from the engine, this is not representative of the entire shape) That being said, the more the flow bends, the higher the shock, and so the slower the flow actually will be travelling in these extreme backflow regions. in some situations, its so sharp that the flow literally cannot turn that far without having an absurd number of oblique shocks, and so it's practically stopped by the time it does so. To make it worse, those edge cases are sporadic, and fluctuate a ton with the pressure, so while yes they are technically there, they are so slow and so small that theyre basically non-existent. If I had to guess, the reason that high altitude test shows such a perfectly straight line is twofold: 1) I suspect it has been sized to *not* have a plume, just to fit within the test bed without burning it, and 2) it likely does expand like this a bit further down, and we just cant see it because the camera view is too short. That is a really cool picture though, I've never actually looked into high-altitude test facilities so I guess thats a new rabbit hole I am about to dive down lol

the point of that second post was to show that no, the expected behavior for an underexpanded engine is *not* what we see in the game. While yes, it does expand the lower the surrounding pressure gets, the geometry/shape should not change entirely: as you can see here, the plume shape IRL is parabolic, and while yes it does expand as it gets further from the engine, this "rate" of expansion should slows down to create that parabolic arch (shown in the bottom left). Instead, it seems to expand at an increasing rate, creating that wide spread hyperbolic shape, shown below on the right (Hopefully that makes sense) Interestingly, the new shape is not far off when youre talking about landing a rocket, as the flow would probably behave in a similar way if the engine bell is close enough to the surface for ground effect to become an issue. Unfortunately that is not what we are seeing here, as this same hyperbolic effect can be seen high in orbit too: - Matt Lowne's most recent video, since I dont want to open the game rn the "point" I guess of my original post was to ask what their reasoning behind this was. I personally find this new plume beautiful, but it gives me a bit of an uncanny valley vibe as its just not quite right

Hey there! Yes, you are (almost) correct! Plume expansion is almost solely a function of nozzle geometry (fixed) and ambient pressure (dynamic), and so for any given engine the plume will expand and change its efficiency strictly as a function of pressure, assuming ideal flow and no reverse flow shenanigans. That being said, I think you may have your terminology wrong, and not quite understand the fundamentals, but dw its a super common mistake so I'll explain it here! The term "Underexpanded", "Overexpanded" or "perfectly expanded" is actually somewhat deceptive compared to the actual shape you see: When a fluid flows through a bell nozzle like a rocket engine, as the ExitArea(AE)/ThroatArea (AT) increases the velocity of the gas rises with it. This is why sea level engines tend to have smaller AE/AT than vacuum optimized engines: for a given throat area, a bigger exit area means a faster exhaust which means higher efficiency. Unfortunately, as this velocity rises, the internal pressure of the gas actually begins to drop off rapidly. If the internal pressure of the flow matches the surrounding (ambient) pressure, say at sea level, then we call it "perfectly expanded" because it is at exactly the right pressure to be ejected in a perfectly straight line out of the back of the engine. However, if the engine bell is too large/small, things get wonky. If the engine bell is too large, the internal gas pressure drops below the surrounding ambient pressure (IE, the gas has "expanded" too much), which causes the surrounding atmosphere pushes inwards towards the flow and causes the classic "shock diamond" effect" seen in an overexpanded flow. Conversely, if the engine bell is too small, the internal pressure of the gas will be considerably higher than the surrounding atmosphere, which will cause the exhaust to push "outwards" and create that classic expansion plume I attached earlier. This is exactly why plumes go from over-> perfect-> under when you see a rocket rise through the atmosphere; as the pressure drops, the relationship between the internal pressure (Pe) and the ambient pressure (Pa) goes from Pe<Pa to Pe=Pa to Pe>Pa. And yes, since the engine cannot ever be infinitely long, it is nigh impossible to ever get Pe = 0 in space with a conventional bell, so it will always have a bit of a parabolic plume. That being said, this scenario where Pe>Pa is called "under expansion" because it is not expanded *enough* to eject its fuel optimally. As you can probably tell its a very weird concept, but TLDR an "Under-Expanded Nozzle" has a higher exit pressure than its surroundings, causing that outward expansion you saw in my post. And while yes, that effect will be dramatized further in space, that doesnt change the fact that it will still be under expanded in a vacuum. (Not to be rude, just trying to be helpful) So, I was correct in calling it under expanded at sea level in my previous post, as the engine they had shown was clearly underexpanded. Just as a heads up, I take my profession very seriously, and while im sure you meant no offense, starting a comment with "As an aerospace engineer, you should know..." could comes across as quite demeaning. I am sure you didn't mean it in that way though, so please if you have any other questions about basically anything else rocketry related please let me know!! I love talking about this kind of thing, and would be happy to answer any questions you may have!

Really enjoying the game so far, but as an aerospace engineer the vacuum engine plume spread has really been confusing me, and I was wondering if this is an aesthetic decision. IRL, rocket engines that are under expanded have a parabolic-esque flow geometry like: (This was a test of a partial nozzle, the reason it is underexpanded ASL, and actually bends back before forming a shock diamond) But in KSP2, it seems to expand more outwards, almost hyperbolically: I saw that, in a previous post someone mentioned that this hyperbolic-esq plume may have been inspired by a post made here, but I honestly dont know if that is correct: After looking into a few papers, I believe the hyperbolic plume may be wrong, atleast partially. From what I understand about rocket engines, it should really only expand in that "hyperbolic" way when the visible flow is in direct contact with something, IE a surface or stage. For a more in-depth explanation of the physics, ive been explaining everything I personally know here: EDIT: Discussing this in these posts actually brought another possibility to light: All of this being said, there is a way for both the parabola AND the hyperbola to exist simultaneously: Here, the engine plume DOES still expand in a bell shape, but due to the low pressure & temp it is barely visible. Further inwards, the high pressure region in the middle of the plume (which sometimes does have a hyperbolic shape) is dense and hot enough to be seen easily, and so is that what's being modelled instead? If so, is there any plan to expand on the current engine plume model to include this low pressure low light emission region? END OF EDIT I know you have been doing alot of great research into this so I will trust your response here, as I am definitely no expert on rocket plume visualization. If I am wrong about the engine having a bell though, I do have a follow up question: (If the bell is correct, please ignore everything below) In one of your previous posts, you mentioned that there is a difference between transitional and vacuum flow: Assuming the hyperbolic assumption is correct, would the flow not look more like this, with an initial expansion first: For the transitional plume to have the bell shape that it does, it must have a higher pressure than its surroundings, which would hold even more true when it reaches vacuum. Therefore, if the hyperbolic plume is correct, would it not *also* have a rapid expansion at the exit? P.S. loving the game and all of the amazing work so far, you and your teams passion has really shone through, and the fundamental core of the physics engine is one of the most impressive ive ever seen. Keep up the good work!!