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Mathematicians prove universal law of turbulence
- esch89 7y agoMy not very mathematically-inclined mind: the universe is patterns.
- Ameo 7y agoThat may be true, but it might just be a side effect of the fact that most of our perception is based off of layered pattern recognition.
- HenryKissinger 7y agoTo any fluid physicist or mathematician here: Does this move us closer to a general solution for Navier-Stokes?
- kkylin 7y agoIf what you mean by "general solution for Navier-Stokes" is something like the Millennium Prize problem (proving general existence & uniqueness of solutions for NS), then no, at least not as far as I can see. This work is about a different problem than NS: they are mainly interested in the advection of passive tracers, e.g., a blob of ink, in a chaotically-evolving fluid field. To do this, they assume a number of things: their fluid equations are subjected to stochastic forcing, which makes available a bigger bag of mathematical tools than deterministic chaotic systems (like high Reynolds number NS without stochastic forcing). They also assume periodic boundary conditions, which enables explicit calculations using Fourier series representations. (Physical boundaries play an important role in many kinds of turbulent flows but make the mathematical analysis harder.) And, in 3d, a hyperviscosity term (4th derivatives in space instead of the usual 2nd derivatives associated with viscosity), which makes solutions more smooth. That being said, proving new theorems in this area is hard. I think it's very nice work.
- conformist 7y agoThat's a nice summary! As far as I remember, a periodic domain is fine for one version of the Millennium problem.
- kkylin 7y agoThanks! (I had the benefit of sitting down with one of the authors and having some of this explained to me.) And thanks for the clarification about the Millennium Prize.
- atrettel 7y agoThis result applies to turbulent flows. Not all flows are turbulent, and the Navier-Stokes equations should describe laminar, transitional, and turbulent flows (3 major regimes of fluid flow). Transitional flows are not yet fully turbulent and can behave differently than a fully turbulent flow behaves. My background is in fluid dynamics, so I can't speak directly about the mathematical difficulties in proving the existence and uniqueness of solutions of the Navier-Stokes equations. However, I can say that any general solution to the Navier-Stokes equations must apply to laminar, transitional, and turbulent flows, so finding a particular result for turbulent flows alone is insufficient. Any such result would have to transcend its regime in some way. Surely I think more proofs like this couldn't hurt, but I do not see any direct connection between this result and the Navier-Stokes existence/uniqueness problem (coming from a fluid dynamics background). Hacker News has already discussed this result previously in an earlier thread: https://news.ycombinator.com/item?id=21771684 https://news.ycombinator.com/item?id=21771684
- mar77i 7y agoDamn it, you beat me to bringing up laminar flow!
- lgmargolin 7y agoI think it is relevant to note this proof is based on the incompressible Navier--Stokes equations. Real fluids are compressible. From the point of view of kinetic theory, even the compressible Navier--Stokes equations are only a first-order approximation to the Boltzmann equation. Further, many authors (including myself) have questioned whether the Chapman--Enskog approximation is justifiable for high Reynolds number flows. Certainly it is known that NS does not correctly describe shock flows, which are also high Reynolds number flows. I would say these papers represent an interesting mathematical exercise whose relevance to the physics of turbulence remains to be assessed.
- kfmu 7y agoWonder what the implications are for magnetic confinement fusion.
- petschge 7y agoThis is not a bad question (turbulent transport is very important for magnetic confinement fusion), but the answer is unfortunately "very little". This work considered only hydrodynamics, which does not describe the inclusion of magnetic fields. Turbulence in magnetohydrodynamics is much more complicated and it is not clear how the present result would be generalized to cover that case as well. Also we do have results for neo-classical transport in MHD turbulence, even we don't have mathematically rigorous proofs.
- pge 7y agoAlso, there are almost no implications to real-world applications from this proof because the result has been assumed to be true for a long time. I don't mean to diminish the value of the proof, which is an impressive bit of work; it's just that the value here is more academic and in the theory realm than in the practical or experimental where the theorem has been treated as true without a formal proof.
- willis936 7y agoOh hey I’m sitting in an office down the hall from a neoclassically optimized stellarator. :) I’m not a scientist, but from my layman view there seems to be a lot of empirical work being done to characterize turbulence beyond what MHD models tell us. Afaict, hydrodynamics is a set of empirically derived equations that describe fluid macro scale behavior. Neoclassical MHD models apply Maxwell’s equations to these, but that isn’t sufficient for most plasma regimes. Gyrokinetic approximations of particle simulations are the lead that most people are following, but they aren’t able to agree with real world measurements very well yet. I have the impression that particle level simulation works but we are several orders of magnitude away from that in terms of computational capacity.
- petschge 7y ago
- btilly 7y agoThe result is fascinating. Any form of self-similarity across scales means that it is a fractal with a specific dimension. The dimension describes how thoroughly it is mixed.
- core-questions 7y agoThis should be somewhat self-evident to anyone familiar with fractals who has spent time watching turbulence in clouds, fluids, etc. (particularly if under the influence of LSD). It's fractal down to the Kolmogorov limit, was my understanding - beyond a certain size it's not meaningfully a "fluid" anymore.
- agf 7y agoThere are lots of things that appear to be true and aren't, and there is a long way between observing something that appears to be true and proving it mathematically. Your comment is misleading because it implies we should all have been assuming this based on the evidence you state -- and we certainly should not have.
- core-questions 7y agoEverywhere else in nature that we have seen things that appear fractal, it's been borne out by analysis, as far as I've ever seen.
- agf 7y agoThat's not even remotely true. Most things in nature people talk about being fractal aren't, in the mathematical sense, at all. They show statistical variation between areas and only have a few levels of self-similarity. A few minutes of Googling with confirm that.
- core-questions 7y agoYou're missing the forest for the trees - perhaps literally. Trees are a perfect example. Fractal in form, down to a certain level; but also an example of something that builds over time like a cellular automaton and is being influenced by the environment. Compromises are made, the perfect form is impossible to achieve, but the gestalt is still there. The fact that the most realistic simulated / computer-generated trees we can render are made primarily of simple fractals is a great indicator. We see the same thing in terms of self-similarity in mountain ranges, lightning, rivers, lungs, and now clouds and water. At this point if you want to deny it, all I can surmise is that you either never go outside, or simply don't know what to look for.
- fernly 7y agoFor anyone looking for a simple writeup, the Wikipedia topic title is "Batchelor vortexes", and it has not been updated to reflect the papers mentioned here. https://en.wikipedia.org/wiki/Batchelor_vortex https://en.wikipedia.org/wiki/Batchelor_vortex
- semi-extrinsic 7y agoNo, that's a different thing, what you link to is an analytical vortex that can approximate a boundary layer. The result posted here is that the power density of the turbulent energy spectrum scales with the inverse of wavenumber.
- steve76 7y agoMixed grey: low wavenumber, high power density. Marbled black and white: High wavenumber, low power density. So I can zoom into that mixed grey, and eventually get marbled black and white unless I hit the molecular bonds. Did I get that right?
- semi-extrinsic 7y agoPrevious discussion of this same result from 2 months ago, 141 points and 16 comments: https://news.ycombinator.com/item?id=21771684 https://news.ycombinator.com/item?id=21771684
- whatitdobooboo 7y agoNot too well on mathematical proofs, but to me this sounds like the "proved" something by not being able to disprove Batchelor's law? Also, the part about randomness makes sense in theory but the jump to an actual proof seems a little wide to me. Are many mathematical laws proved in this way?
- flafla2 7y agoFrom the article: > In their first paper, the mathematicians focused on what happens during the mixing process to two points of black paint that begin the process right next to each other. They proved that the points follow chaotic paths and go off in their own directions. In other words, the nearby points can’t ever get stuck in a vortex that will keep them close forever. > “The particles move together initially,” Blumenthal said, “but eventually they split apart and go in completely different directions.” > In the second and third papers, they took a broader look at the mixing process. They proved that in a chaotic fluid, generally speaking, the black and white paint mixes as quickly as possible. This further established that the turbulent fluid doesn’t form the kinds of local imperfections (vortices) that would prevent the elegant global picture described by Batchelor’s law from being true. > In these first three papers, the authors did the hard mathematics required to prove that the paint mixes in a thorough, chaotic fashion. In the fourth, they showed that in a fluid with those mixing properties, Batchelor’s law follows as a consequence. So no, they are not "proving something by not being able to disprove it." A better way of phrasing their strategy is, "proving something by proving that disproving it is impossible." In Computer Science, there is a similar concept for proving asymptotic bounds of algorithms called an "adversarial proof." The idea is, given some query that your algorithm performs (e.g. in a graph algorithm, a query could be "are two vertices connected") come up with a worst-case adversary that answers queries in the absolute worst way possible, that would necessitate even more queries to complete the problem. In this way, you can prove a universal lower bound for the cost of solving some problem. See [1]. In this case, the adversary is trying to come up with the worst-case initial conditions for this particular brand of turbulence. Basically they are saying, no matter what, you couldn't come up with an initial condition that challenges Batchelor's law more. [1] https://www.cs.cmu.edu/afs/cs/academic/class/15451-s20/www/lectures/lec2.pdf https://www.cs.cmu.edu/afs/cs/academic/class/15451-s20/www/l... Section 3.2