This question is different from What is the optimal streamlined shape?. That question is probably about an ideal situation of flowing through space with no gravity in a fluid with zero viscosity and infinite bulk modulus. This one is about a real bullet train which is not an ideal situation like that. https://www.youtube.com/watch?v=7ccoEmlxlks shows a video of a Shinkansen 800 bullet train. It looks like a shape that will not produce much turbulence at the back end. That's based on my guess that that would be a kind of streamlined shape. My question is

Is it known for different speeds, what the force of air resistance divided by the density of air, divided by the crossectional area of the train, divided by the square of the speed of the train is when it is on rails?

I know that if atoms were infinitely small and air had the same density but with an infinite bulk modulus and zero viscosity and the train had an infinite strength and shear modulus and bulk modulus as well, that number would be constant because the laws of flow of a fluid in an ideal situation like that are preserved under a slowing down or speeding up of the process. However even if the laws allow it, we could not in general reverse the process of flowing through a fluid with turbulence because there is a forwards arrow of time because there was a beginning of time, just like if you call one heptagon in a hyperbolic heptagonal tiling the central point and start there doing a random walk, where you will be next give where you are will always be random but where you were last given where you are now and where you started will never be fully random.

12:00 into the video https://www.youtube.com/watch?v=5zI9sG3pjVU, you see the flow of air around a smooth sphere. It appears to have a jet of air a little behind the sphere nearly stationary in the frame of reference of the sphere. And the kinetic energy of the air in that jet in the frame of reference of the rest of the flowing air probably converts into kinetic energy of the eddies. So for speeds large enough to have a high enough Reynold number to be turbulent and a low enough speed to not be a high fraction of the speed of sound, for that sphere, the force of air resistance divided by the density of air, divided by the crossectional area of the sphere, divided by the square of the speed of the sphere should be very close to 1. I believe an ideal shape similar to the shape of the Shinkansen 800 in an ideal situation of going through a fluid would have zero resistance. However if it's flowing through space without gravity or real wheels like a train and no opposing gravity against a normal force like a real bullet train, there might be exponential growth in the fluid pushing it out of its original orientation.

I know almost nothing about engineering. I'm looking for a simple answer that can just satisfy me with knowledge. Then I'll leave it to other people to invest in new research based on the answer if they want to. I'm not trying to design anything myself. However, I cannot figure out an algorithm for determining in advance given what the answer is, whether it will satisfy me. When somebody is working on something, they don't propose a concrete algorithmic plan from the start. They feed themselves ideas and trust themselves to think for themselves later how to use them. Similarly, a good answer to this question would come from somebody who can function as if they are a future self of me and uses their discretion to figure out what type of answer will probably be a good answer that satisfies me.

  • $\begingroup$ I don't know if this question is better here or on Physics Stack Exchange. I didn't think but I'm not sure which is better anyway. If it's better on Physics Stack Exchange, can somebody move it there but do a after seeing if they think that community would accept it. I think the moving feature is useful. Then I don't have to use up another chance to ask a question. Plus the community of the site it was originally written on won't invest so much time on a question that does not belong there. I once wrote the question $\endgroup$
    – Timothy
    Feb 4, 2021 at 6:41
  • $\begingroup$ engineering.stackexchange.com/questions/21649/…. Much later, I become more experienced having developed the mind set as a slow discovery thing on how to contribute better. Then I realized my 3 questions on Matter Modelling Stack Exchange, mattermodeling.stackexchange.com/questions/3956/…, mattermodeling.stackexchange.com/questions/4047/…, $\endgroup$
    – Timothy
    Feb 4, 2021 at 6:43
  • $\begingroup$ and mattermodeling.stackexchange.com/questions/4057/… together are a really good substitute for that question. So I wrote in a comment under that question that that was so. $\endgroup$
    – Timothy
    Feb 4, 2021 at 6:44
  • $\begingroup$ Perhaps the engineers designing the train considered this and many other things. Have you looked to see what they published? And the last thing they want to happen is for the train to create lift which might help derail it... $\endgroup$
    – Solar Mike
    Feb 4, 2021 at 7:07
  • $\begingroup$ @SolarMike I think I tried to google search it and couldn't find it. Google normally has a good search engine. I'm not going to be able to find a way track down those engineers all on my own. It would be great if one of those engineers found this question and answered it. $\endgroup$
    – Timothy
    Feb 4, 2021 at 7:10

2 Answers 2


I'm not exactly clear what you're asking but I'll try an answer. I didn't watch the other video, but a teardrop shape is the best pure shape.

Bullet train design has diverged from the original design (shaped like a bullet), which was solely concerned with air resistance. The changes have led to elongated noses that act to help with two trains meeting or when exiting in a tunnel. Designing the shape solely for efficiency led to a shape that was louder than desired and created unwanted motion when two trains met on adjacent tracks due to the large pressure wave at the leading edge (like the bow wave on a boat/ship). Apparently designers were inspired by the kingfisher's (the bird, not the beer) beak. Because trains do ride on rails, the shape needs to be the one that works best in the environment it's designed for.

You are correct that the asymmetrical shape wouldn't be appropriate for a vehicle in some theoretical no-track environment, but the thing about engineering is that, unlike pure science, it is about solving problems as the present themselves, not solving for the theoretical .

As for your equation, as written it doesn't make sense to me. The lack of parentheses makes it hard to understand. But in general, engineering equations are a combination of science and empirical observations. I'm sure the engineers could know the answer if they wanted to.


I will add just a few points here.

Water droplets fall through the air in spherical shapes. They do not assume "teardrop" shapes until they strike and slide along the surface of an object.

Regarding drag minimization, this requires minimizing the 3 major sources of it: profile drag, which occurs because it takes work to push air out of the way at the nose end of a moving object, wetted area drag because the side walls of a moving object tend to pull air along with them as they move through it, and flow separation drag which happens when the moving air fails to follow the tail shape of a moving object, and forms a low-pressure area of suction acting on the tail end of the object.

There is another source of drag which occurs when two moving objects come close to each other; this is generally referred to as interference drag or, when the second object is a solid surface like the ground, ground effect.

Airplanes are designed to minimize the first three types of drag, which results in thin, stretched-out shapes with rounded noses and sharp tails. Ground vehicles have to take into account ground and interference effects which become significant at bullet-train speeds, but which are not very important for ordinary cars and trucks.

  • 1
    $\begingroup$ but we should note that water droplets do not have minimum drag shape, they are spherical due to surface tension $\endgroup$
    – Pete W
    Feb 5, 2021 at 2:27

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