What would the implications be if someone were to find and release a solution to Navier-Stokes?
Imagine such a solution works for compressible fluids and all viscosity profiles. Also presume that the general solution isn't some fuckhuge summation but is rather a plug and chug formula for finding fluid behavior at (x, y, z, t). What all fields would be affected, and how?
We'd be able to make cars that can be driven to the moon!
"General solutions" to PDEs like this one aren't really a thing, some sort of boundary conditions or symmetries have to be specified, since PDEs depend on arbitrary functions.
In any case, finding a practical, useful solution to the equations is not the Navier Stokes problem. Proving the existence, uniqueness, and continuity/differentiability of solutions is, which is a great deal trickier.
Of course, you make up boundary conditions to start with and work from there. With that, you can find very close and consistent approximations for the turbulent effects that happen in real fluids.
I'm talking Navier-Stokes as it applies to physics and how much of science would be uprooted if a computer could find and model true fluid behavior just by plugging in the numbers. I imagine a lot of otherwise insoluble problems in physics could be solved if fluids didn't have to be dealt with using weeks of calculations on clusters of supercomputers.
>I'm talking Navier-Stokes as it applies to physics and how much of science would be uprooted if a computer could find and model true fluid behavior just by plugging in the numbers. I imagine a lot of otherwise insoluble problems in physics could be solved if fluids didn't have to be dealt with using weeks of calculations on clusters of supercomputers.
Literally nothing would happen you mongoloid, have you heard of numerical approximations? We can calculate them very efficiently to a arbitrarily high level of precision. Getting an analytical solution (which is impossible btw) wouldn't speed that up significantly.
>oh geez guys how would the world change if we could calculate this pde analytically instead of numerically
>Literally nothing would happen you mongoloid
Do you realize how much effort and skill is required to arrive at reasonably accurate results for turbulent flow around complex geometry?
An analytical solution to the Navier-Stokes equation would be a godsend. We could have accurate solutions to every problem instead of relying on correlations so much.
>In any case, finding a practical, useful solution to the equations is not the Navier Stokes problem. Proving the existence, uniqueness, and continuity/differentiability of solutions is, which is a great deal trickier.
But if you find a solution then you trivially proved the existence of a solution.
>>Literally nothing would happen you mongoloid
>Do you realize how much effort and skill is required to arrive at reasonably accurate results for turbulent flow around complex geometry?
Yes, but that's not numerical/analytical divide. Numerical solutions are relatively easy, just throw computer power on it. What, you think that they have a basement full of niggers with slide rules? Even if analytical solutions were magically available right now, they wouldn't decrease computer power requirements that much.
>An analytical solution to the Navier-Stokes equation would be a godsend. We could have accurate solutions to every problem instead of relying on correlations so much.
What correlations? Nigga, you can solve them numerically right now with arbitrary precision. Only limitation is your computer power.
The Navier-Stokes equations are not the be-all and end-all of fluid mechanics, it's actually limited in validity to the continuum regime and to conditions where the state variables are "smooth".
Having an analytic solution will not change much. As said, CFD can get you results right now. And nothing profound is being held back due to the computational cost of CFD. It's not what's keeping us from going to Mars or having fusion energy. Having an analytic solution would improve transportation, energy and possibly medicine, but it won't be a revolutionary change like the transistor.
>ITT: weather men downplaying the problem so they can deny that they'll one day lose their jobs to a computer
>if fluids didn't have to be dealt with using weeks of calculations on clusters of supercomputers
He literally states the method we use for our numerical approximations right there.
Before going for a STEM degree, I worked in a hospital, and fluid simulations took weeks to check and recheck every possible case to ensure safety to the patient for, say, a replacement organ, ESPECIALLY if it's a part of a lung or the heart. Over half the time in getting a new heart valve is spent running those simulations, and in that time, your patient could die without or even with expensive life support systems. Really, dealing with cardiovascular or lymphatic delivery on anything that has to be evaluated on a case-by-case basis is a nightmare plaguing medicine that a lot of the educated higher-ups are cognizant of, and a good chunk of the research meant to improve those systems is done by improving methods of fluid analysis, specifically in the context that Navier-Stokes describes.
Whoever earns that Millennium Prize will be worth well over the million dollars he's offered.
Let's say that they are solved. How simple do you think the solutions will be, really? How less computationally intensive will the symbol manipulation vs numeric approximation be? It's a hard problem dude, if they can even be solved it will be an improvement, but not a huge one. Hell, I'd bet that 10 years of Moore's law will be a bigger change than that.
>implying it's possible to accurately capture the boundary conditions for an entire fucking planet
This. An analytical solution is not an urgent need.
However, proving that smooth solutions always exist is an important first step for understanding turbulence.
>implying it isn't
The computational need will be at least one exponential order less intensive than what we use now, which is a pretty big improvement. Right now, we iterate for all points and then iterate again for all points for the next instant in time, et cetera. With an analytical solution, we can remove the need to iterate over instances in time, and every data point won't be necessarily dependent on every single data point that comes before it. Computationally, that's pretty significant, especially since large numbers of time instances are needed to approximate just ten seconds of continuous flow.
More like thousands of approximate boundary conditions taken from satellite images and Doppler radar that can be used to reasonably approximate actual boundary conditions.
Why not?
Nah, it's a periodic boundary condition over a sphere. Super simple.
I mean it's not ridiculous, I worked in a weather research lab where they're working on some pretty cool improvements to WRF which incorporate an immersed boundary method for 50 m near surface grid size simulations, resolves terrain pretty accurately without the negative aspects of terrain following coordinates. With a good geological survey, and enough computing power, this could be pretty accurately scaled up to a much larger size than the 10k x 10k grid I was working with.
But did you account for things like the latent energy of surface/ subsurface moisture resevoirs? :^)
There's just way too many factors to build an accurate transient state CFD model of weather patterns, since tiny differences in the boundary conditions will amplify over time to produce results that are waaaayyyy different from reality. A much more robust way to do weather forecasting is by an empirical machine learning approach, in that implicitly learns the tendencies of the boundary conditions in a given area by observing actual weather patterns.
>Numerical solutions are relatively easy, just throw computer power on it.
That's not how it works.
That is largely how it works. He just doesn't grasp the cost of using those operations and why they can't be employed by your average physicist at a scale necessary for answering most of our substantive questions about fluids.
It's not a question oft heard on Veeky Forums. The problem isn't that we can't do it. The problem is that our current methods require finance, time, and the availability of some powerful capital. Despite Hollywood-esque desensitization to the true rarity of the highest-end computing systems, supercomputers are terribly expensive and specialized investments that no one has just laying around, and no sane researcher is gonna use his limited research budget to pay for something that's only planned to be used to resolve a few of his conjectures and then for nothing else.
If we had all the time and money in the world, it wouldn't be a problem, but real life can be a real bitch.
>That is largely how it works
No it isn't. Turbulence models are still in their infancy.
For example, we still don't have a good model for the transition region. The only way to accurately model that is DNS.
But the problem with DNS is that even if you utilized the computing power of every single supercomputer in the world, you still wouldn't be able to get an answer for a real world industrial application.
Well that's settled, just solve an unsolvable PDE! Meanwhile somebody also should make prime number factorisation O(n).
Thank fuck we have wishful thinking.
>implying it is