Summer of Math Exposition

The simulation wasn’t loading for me at first, but the second time it did, it did not render anything. Otherwise, great article. Very succinct and simple, yet memorable; great job breaking it down to three simple memorable points!

Yes! They had a simulation to showcase the CFD

I really like the idea of presenting the topic with increasing levels of complexity. I feel like the jump from level 1 to level 2 was maybe a bit extreme: Really the only math in level 1 was 100 - 5 = 95 and then the first things we see in level 2 are limits and partial derivatives and that feels like a pretty big gap. That said, the concept was really nicely explained in level 1 giving good motivation for wanting to learn more.

I’d have liked more visuals in levels 2 and 3. Both to help picture what all the equations are actually doing and also to break up the presentation a bit and give the reader a bit of a break from parsing all the math. Even though I’m familiar with calculus at some point it just becomes too many symbols to juggle and starts to get confusing.

This was really good, I loved how simple it was to follow the ideas, and I learnt a lot from this post. I hope you write more articles like this!

I found the explanations on three levels really valuable, though I would suggest making the level descriptions independent of the US school system (I have no idea what physics AP, C or BC means), or adding a small explanation on the knowledge required. Level 1: Possibly some further information on how non-fluid objects, i.e. the wing itself which disturbs the air, are modeled, and on the feasibility of doing these kinds of tests in practice, making clearer the immense benefit of numerical modeling. Level 2: The mention of viscosity comes a bit out of nowhere and is then not mentioned for a while. The part about partial derivative might benefit from an image, showing this “cross-section of the graph. “mass x acceleration” might be confused with cross product. Figure 3 would benefit from more description in the caption of the displayed variables. Personally, I don’t know/remember Newton’s method from school and was a bit confused during that part but would have immediately gotten it if its relation to the Taylor series was mentioned. Level 3: The meaning of the colon in the equations was not clear.

The article explains very nicely how CFD works, and that proves helpful for the engineering students. However, it contains little to no mathematical content, which renders it irrelevant to the competition.

This was a fun peek into how some of these numerical solvers work.

I really like the “3 Levels” presentation, and I especially like the highlighted “What to remember” boxes. I also liked your use of analogies and colorful, friendly language.

The crash course in multivariable calculus presented in Level 2 came rather quickly, but that’s fair since that wasn’t the main goal.

I think some of the equations you presented could have benefited from labeling via “underbraces” to help aid the reader in knowing what the different parts of equations are about, and to organize everything.

I also think appropriately chosen illustrative examples could have been nice, like a graphic with as few as two regions and vectors drawn or scalars denoted, followed by the implications of that for the next timestep. Perhaps something at the level of “(100C hot | 0C cold) ---> heat should flow to the right across the interface” to summarize the different effects.

The interactive simulation didn’t load after several minutes, thought it eventually did after toggling the “Run Interactive Simulation” collapsible heading. Downloading the code worked fine. The “Reset” button seemed to delete the velocity slider. The program ran once though, and it made a cool image. I’d love to see some examples of drawing “actionable take-aways” from such images.

I really loved how you broke down the article into 3 sections of different difficulties. Reading through them in order gave a nice gradual introduction into each concept. Your writing was solid throughout.

I also liked how you put discretizing fluids as ‘breaking up a swimming pool into a ball pit.’ That’s probably the best analogy I’ve heard for it.

Level 1: good introduction to CFD, very clear explanation and analogies/metaphors. I appreciate the way the topic is introduced very gently, without immediately diving deep into complex mathematical formulas.

Level 2: A bit more challenging, but still, the explanation seems quite in-depth, giving the reader enough information to grasp everything without needing external research. I appreciate calling the conservation of fluid momentum formula the “F = ma” of fluids, that immediately made it click for me. Although I’m not entirely sure what the general transport equation is supposed to be. Where does it come from? What is its purpose? What are “transported quantity”, “diffusive coefficient”, and “source term” supposed to mean? It seems this formula is dropped in out of nowhere.

Level 3: Now things are starting to get a lot more complicated for me. I had to look a lot of things up to be able to understand this part, although, to be fair, I’m not a physics graduate, so maybe I’m not part of the target audience anyway. So yeah, not sure if I can be an accurate judge of this part of the submission.

Conclusively, I think this was a really good submission! Also, I appreciated the sense of humor in this article. Although, the demo at the bottom of the page didn’t work for me, when I press “Solve” it says “starting…” but it never actually starts.

I really like the “what to remember” boxes! Maybe a few too many overall, but super helpful!

I like that you split it into 3 levels. I feel that the content could be broken down into slightly more digestible chunks. It wasn’t really clear to me what the interactive simulation was doing - maybe integrating it into the article itself would have been nicer.

Thank you for this nice introduction on CFD. I really liked the fact that there are multiple level of difficulties to adapt to the level of everyone. However, I missed a lot about images or simulations : you provided some nice schemes, but never any fluid simulation output about what we build (or even some videos you could add) ! And I think that this is one of a few topic in physics where you are able to easily show nice visuals / videos / simulations to the reader, so I was a bit disappointed that there were none.

I like the idea of breaking up a complex topic into parts or acts, each act with finer detail than the act before it. I also like that there are banners that summarizes a few paragraph every now and then. I would say that more details/description could have been added to the equations.