Mathematicians Find Wrinkle in Famed Fluid Equations

https://www.quantamagazine.org/mathematicians-find-wrinkle-in-famed-fluid-equations-20171221/

by digital55 • 9 years ago

240 120 9 years ago

quantamagazine.org

aperrien2• 9 years ago

Does anyone have a link to this particular paper?

yorwba3• 9 years ago

The link was given in the article: https://arxiv.org/abs/1709.10033

aperrien2• 9 years ago

Thanks, I missed that somehow.

foota3• 9 years ago

Based off the abstract, there's one class of weak solutions, the Leray solutions, which are known to exist. And now they've shown for a different class of weak solutions to NS that the solutions are not unique. Is that right?

boxfire• 9 years ago

That's correct, while taking away the entropy inequality as well. The thing is that very inequality is one of these laws of thermodynamics we so love, so I kind of see this as hot air.

ianai2• 9 years ago

Aye either they have upturned a theory as august as general relativity or they’ve thrown the baby out with the bath water somewhere.

foota3• 9 years ago

Isn't that possible though if the fluid isn't a closed system? (which it probably isn't)

mar77i• 9 years ago

Who would have guessed that this story about the behavior of fluids seems to kind of leak.

qubex14• 9 years ago

Yeah, they relaxed one of the criterion’s for Leary solutions (the energy inequality) they’ve proved that these solutions are energetically unstable, and that doesn’t strike me as very profound (I need to go read the paper though, I haven’t had time yet).

Just mentioning at the bottom of the article “yeah now we are going to see if the same thing applies to proper Leary solutions, we think it does” means close to nothing, honestly.

And even if it does, the article’s author is right when he remarks that this can be seen entirely as a warning against using approximations that are too broad or coarse.

I’ll add that I find it funny that nowhere in the article (that I can see, but I am reading on mobile Safari, so maybe...) is the Navier-Stokes partial differential equation even displayed, and the relationships it defines are not explained (other than some waffle about ‘derivatives’).

BucketSort• 9 years ago

Btw this is work relating to one of the famed Millennial Problems - http://www.claymath.org/millennium-problems/navier%E2%80%93s...

FabHK2• 9 years ago

As the article pointed out.

westurner• 9 years ago

Navier-Stokes equations: https://en.wikipedia.org/wiki/Navier–Stokes_equations

CamperBob22• 9 years ago

   Using this approach, Buckmaster and Vicol 
   prove that these very weak solutions to the 
   Navier-Stokes equations are nonunique. They 
   demonstrate, for example, that if you start 
   with a completely calm fluid, like a glass 
   of water sitting still by your bedside, two 
   scenarios are possible. The first scenario 
   is the obvious one: The water starts still 
   and remains still forever. The second is 
   fantastical but mathematically permissible: 
   The water starts still, erupts in the middle
   of the night, then returns to stillness.

Also permissible, and also vanishingly unlikely, under quantum theory.

It's a stretch, but I wonder if it's reasonable to think of data points in the vector field describing fluid motion as probabilities rather than definite measurements. That might allow their behavior to be modeled with different mathematical tools.

cinquemb• 9 years ago

Yeah, I wondered about this too since the typical NS BVP, velocity vector is assumed to be confined to the domain of real numbers[0].

[0] https://warwick.ac.uk/fac/sci/statistics/staff/academic-rese...

eru5• 9 years ago

Also permissible, and also vanishingly unlikely, under quantum theory.

Navier-Stokes has nothing to do with Quantum Mechanics..

CamperBob22• 9 years ago

No one said it does. But given that statistics based on complex probabilities are effective in one domain, maybe they're worth investigating in another.

snakeboy3• 9 years ago

You missed the point of their comment.

The responders' point was that Quantum Mechanics is a different framework for modeling physical phenomena which takes a probabilistic framework, and so if fluid were to be modeled in a similar framework, you could work more naturally with these "vanishingly unlikely" events.

btrettel6• 9 years ago

You are on the right track. You have reinvented the basic idea behind the "Reynolds-averaged Navier–Stokes" (RANS) equations, dating back to 1895. The goal is to predict an average velocity field, and this requires modeling unclosed terms (the "turbulence problem" in one respect). RANS and statistical methods in general are the most popular turbulence modeling approaches and much of turbulence theory is based around these ideas. My opinion is that RANS answers are typically what you want, but the models don't work that well. LES (large eddy simulation; applying a low pass filter to NS) is a newer approach that is gaining popularity, and makes physical sense, but is a lot more computationally expensive. And direct simulation (DNS) is an option too, but the computational costs are prohibitive outside of some relatively simple academic problems.

wiz21c• 9 years ago

Also FTA :

>> Nonunique Leray solutions would mean that, according to the rules of Navier-Stokes, the exact same fluid from the exact same starting conditions could end up in two distinct physical states, which makes no physical sense and implies that the equations aren’t really describing what they’re supposed to describe.

This basically say that from one given starting conditions one only expect one (and only one) outcome. Doesn't this conflate a model with the actual physical reality ?

emileokada2• 9 years ago

I'm not sure what you're getting at. Classically we expect that given the same initial conditions, the outcome of a fluids experiment should always be the same. If an equation meant to describe fluid flows doesn't have this property, it is probably a bad model.

rocqua• 9 years ago

If you are going to describe the fluid motion as probabilities, are you going to assume those probabilities to be independent or not?

An independence assumption really helps with computation. However, considering the `eddies' in turbulence, I'd suppose there is large and complex coupling between the probabilities of flow even for positions that are separated.

You might deal with the coupling, but at a first guess that feels like it requires tracking many branching paths, which would have exponential memory requirements.

contravariant2• 9 years ago

No need for quantum mechanics, it's permissible under statistical mechanics. It is incredibly unlikely to happen though (and any way of making it happen necessarily takes a lot of effort).

xaedes• 9 years ago

   The second is 
   fantastical but mathematically permissible: 
   The water starts still, erupts in the middle
   of the night, then returns to stillness.

Big bang in the middle of the night.

Some quirling around until all matter and energy is equally distributed. What a journey!

Then returns to stillness.

ericbowman• 9 years ago

That is fascinating. Is there any sort of immediate real-world impact (like to weather forecasting)?

labster2• 9 years ago

Immediate? Certainly not. Weather models do have incomplete information, but the equations are approximated by Taylor series (1st to 3rd order, depending on the model, last time I checked).

It's possible that this can explain some of the differences between models or ensemble runs... but you have to realize that most of the error comes from incomplete data in the initial and boundary conditions. Looking for weather model effects is like looking for relativistic effects in automobiles.

m_mueller2• 9 years ago

Almost. Most models use some variations of Runge-Kutta approximation, to third order usually.

Edit: typos

throwwit7• 9 years ago

Financial forecasting and risk-elimination in a system like a blockchain would be my guess — and in that respect it’s probably the most important problem out there IMO.

tzahola• 9 years ago

What??

btrettel6• 9 years ago

Fluid dynamicist here. At the risk of looking like a heretic, I am willing to say that I don't think the NS Millennium Prize problem and related things will end up being very useful practically.

I can't see how a definitive answer to the question will result in better turbulence modeling, which is what matters from a practical point of view. If it turns out that the solutions are not unique then we could probably find an additional condition to add (e.g., the entropy condition) to make the solutions unique. If the solutions are unique, bounded, etc. then that's great and it would have no impact practically speaking aside from perhaps helping the reputation NS has for accuracy. Some people seem to think that solving the NS Millennium Prize problem would likely lead to a solution for the turbulence problem, but as I said, I can't see how. I'd be interested if anyone could explain this belief better.

There may be other benefits. I've found papers that find bounds on different fluid dynamics quantities to be interesting, and the motivation for these studies are the NS problem from what I understand. Unfortunately the results from these papers tend to be less useful than bounds I can derive specifically for applications myself.

(In a nutshell the turbulence problem is that NS has far too high a computational cost/complexity to be used in practical simulations. So cheaper approximations to NS are used, which you can cladsify as "turbulence models". How steep the drop-off in accuracy is as you reduce complexity is an open question. My opinion is that fluids probably require high computational cost for accuracy a-priori. Things like correlations from experiments can get around this as you are using pre-computed results, and that may be what we should go for in my philosophy.)

jerf• 9 years ago

In addition to the other fine replies, weather forecasting's inaccuracies are dominated by the lack of information, then by lack of processing power. Lack of closed-form solutions to NS or better solutions rates quite a ways down the list of issues it has, or put another way, even if we had a magic box that completely accurately solved NS for weather forecasting, it would not get that much accurate. (My suspicion is that it would literally be measured in "minutes" more accurate, rather than the "days" you'd like, but I concede I can't prove that... but bear in mind that it may well be the cases that it would be milliseconds more accurate or something, not just that I could be wrong about it being "days", as the errors in the initial data compound over the course of the simulation no matter what math you throw at the problem.)

btrettel6• 9 years ago

It seems popular to believe that for fluids in general forecast accuracy is dominated by errors in the initial conditions (ICs), but my own look at the problem suggests that's not so clear. I recall skimming a book on forecasting by a weather forecaster and he addressed this misconception. It appears that there are multiple sources of error, from errors in the ICs to numerical integration to the fact that the models they use are approximate (i.e., they don't solve NS; they solve a filtered version of NS with a turbulence model and additional models for other physics like chemistry), etc. My impression is that the dominant two are model inadequacy (the models are approximate) and compounded errors due to IC errors and non-linearity, but which is larger likely depends on the problem, and I am not particularly confident about this in general as I don't have hard data. (Certain types of turbulence models get more accurate as the resolution/computational cost increases, but I can't speak for other models. This fits with what you said about lack of computational power.)

The right way to do this is through uncertainty quantification techniques, and I don't know a lot about those at the moment. Until then, all I can say is that there are multiple sources of error.

uptownfunk• 9 years ago

So will they get the million?

emileokada2• 9 years ago

No. They've shown non-uniqueness of solutions of a strictly weaker problem. They have yet to say anything about the actual Navier-Stokes problem.

floatingatoll2• 9 years ago

It's disappointed to see that their reaction to non-unique results is "must be broken", rather than "we've rediscovered the quantum physics uncertainty principle in fluid mechanics".

snakeboy3• 9 years ago

The Uncertainty Principle is a very well established result which is quite generalized mathematically beyond Heisenberg's result. This has nothing to do with that principle.

floatingatoll2• 9 years ago

If you weaken the vector density until a point that you can generate multiple possible outputs from a given weakened input, then what level of weakening guarantees no unique outputs from the equations?

These equations were designed for a system in which every 'atom' (vector) has a perfectly knowable 'spin', and they begin to produce unexpected results as the uncertainty is dialed up through weakening.

It's just a shame that the considerations stop at "therefore we broke the equations" as opposed to "gee, that looks familiar". What's the Navier-Stokes equivalent of the diffraction experiment? What does the interference pattern of two vector fields even look like? Why aren't they trying to study the interference patterns of the one input, two outputs scenario?

I get that this is all "obviously pointless" to others, but no one I've asked can actually explain why these comparisons are unacceptable. You completely dismiss it without any explanation other than "science is well-established", as if somehow that's meaningful.

So, yeah, disappointment.

bmcooley• 9 years ago

Could you possibly dive into this? The electron slit experiment shows that superposition exists in a fundamental way in our universe. You can even measure quantum effects with objects as large as buckyballs. Isn't it sort of obvious that a precise mathematical description of a macroscale, emergent system based on objects with quantum and probabilistic effects is always going to be an inaccurate abstraction?

throwwit7• 9 years ago

Isn’t Navier-Stokes smoothness just a reframing of the three-body problem? Karl Sundman solved that one in 1909, and n-body was generalized in 1990 by Qiudong Wang.

Edit- to answer my own question: http://www.scholarpedia.org/article/N-body_simulations_(grav...

qubex14• 9 years ago

I'm sorry if it seems I am persecuting you and putting all your ideas down, but I really don't see how this can be true either.

The 3- or N-body problem is about point-particles interacting gravitationally at nonzero distance according to an inverse square law. Navier-Stokes is, at root and in the limit, about elastic collisions about infinitesimal corpuscles that transfer momentum between each other.

Again, I can see the analogy “lots of things interacting”, but they have quite little in common beyond that.

throwwit7• 9 years ago

Just the inverse n-vectors applied (regardless of space and field strength or possibly just instantaneous field strength). Always good to be called out tho.

epx• 9 years ago

Luce Irigaray knew it already!

phkahler• 9 years ago

This makes senses to me. If you run a simulation at too low a resolution you run into ambiguity. By refining the grid you can resolve that ambiguity. Turbulent flow is chaotic, so this makes perfect sense to me. What this may lead to is a method of determining criteria for adaptive grid refinement. But I thought that already existed, so maybe just an improvement over what's out there.

reikonomusha• 9 years ago

This doesn’t have to do with numerical error introduced by discretization. This has to do with uniqueness of solutions.

Glauc• 9 years ago

I wonder what the interplay of the material's EOS has with constraining these solutions. Note I only had time to skim the article.

pge• 9 years ago

I'm struggling to understand the significance of this, at least as the N-S equations are used in the real world. Many years ago, I interned with the Navy writing fortran code for fluid dynamics simulations on submarine hulls, and IIRC there were flow dynamics we observed consistently in the real world (e.g. oscillating vortices) that were fundamentally inconsistent with the results coming from our N-S calculations (which would say there could not be an oscillation because it was a steady state flow). There was always a hand-waving of "N-S is actually right; our computer models are just not fine-grained enough." But at the same time, given the computational limits of our grids(particularly at the time -25 years ago), it was understood and accepted that N-S would yield only an approximation. That's only one data point, but it certainly seemed to me that no one was relying on N-S as an accurate predictor of motion (as you would a newtonian model of a ball rolling or something like that), but rather just as a first order approximation. If that impression is accurate, a result that says N-S isn't always accurate is kind of a statement of the obvious. What am I missing?

eadmund• 9 years ago

What I got from the article is that there's a possibility of a chaotic result: it may not be possible to get computed results which are arbitrarily close to the real-world results. I was reminded a bit of the famous Lorenz system (originally for weather prediction, IIRC), which turns out to be chaotic under some conditions.

incidentnormal• 9 years ago

I think it's not a chaotic result as such (as the Lorenz weather prediction models you're referencing are) - which would mean large differences in outcome from very small differences in input, but a case where multiple different outcomes can arise from identical inputs.

dkarl• 9 years ago

The beginning of the article does a horrible job of explaining the big question. The Millennium Prize problem[1] is a math problem: whether the Navier-Stokes equations always have a solution with certain properties. Some possible answers to the problem would mean there are some physical situations where they don't produce any prediction at all about what might happen next, or they produce multiple predictions, or they produce physically implausible predictions. (If you have a strong math background, the official problem description might be interesting to you; it's a bit beyond me.[2])

The article does get around to explaining it better if you keep going.

it certainly seemed to me that no one was relying on N-S as an accurate predictor of motion (as you would a newtonian model of a ball rolling or something like that), but rather just as a first order approximation

Numerical methods for solving the Navier-Stokes equations are approximate and therefore diverge from the correct solution. The same is true for a ball rolling down an incline, but the inaccuracies are smaller than you would ever care about in the real world. What your colleagues were saying about the Navier-Stokes equations is that the numerical error was often large enough that the calculated solutions were known to diverge from mathematical reality in significant ways, and therefore seeing them diverge from physical reality was consistent with physical reality and mathematical reality being the same.

[1] http://www.claymath.org/millennium-problems

[2] http://www.claymath.org/sites/default/files/navierstokes.pdf

ekelsen• 9 years ago

I think it's likely you were solving the Reynold's Averaged NS - https://en.wikipedia.org/wiki/Reynolds-averaged_Navier%E2%80... and dropping the turbulence term to get what you call "steady state".

sdfadfa• 9 years ago

N-S aren't "right" though. The derivations make a lot of good assumptions that breaks down in certain materials/situations.

What is right is its starting point on the conservation of momentum and energy. Then it makes certain assumptions about the stress-tensor which are not necessarily true. Meaning, you can derive the N-S from consv. of mass and E and a certain stress tensor (ST), but its not derived from a universal ST.

btrettel6• 9 years ago

NS are a pretty good model for the underwater scenarios a Navy would be interested in, so I think they can be called "right" here. The fluids are regarded as "Newtonian" so the stress tensor model is good, and the density of the fluid is high enough that the continuum approximation is good. The largest source of error is likely the approximations made to model the turbulence, or in other words, reduce the computational complexity while also reducing accuracy.

btrettel6• 9 years ago

As ekelsen mentioned, you probably were not doing a direct numerical simulation (DNS; using NS specifically) and instead were using an approximation to NS which has much lower computational cost/complexity but also reduced accuracy. Good LES would converge to the DNS result if the grid were fine enough. The term for this sort of error is "model inadequacy error", that is, error from the model being wrong.

My impression is that so far DNS matches experimental results well given that the experiment actually represents the situation of interest. For example, I am aware that at least some "Kelvin-Helmholtz" experiments don't match DNS well at all, and the DNS is considered more credible than the experiments because in the DNS case you know all of the inputs, whereas in the experiments the initial conditions might be close to the desired case, but apparently not close enough. (The Kelvin-Helmholtz instability is of fundamental importance but is not easy to obtain in isolation experimentally.) "Sensitivity dependence on initial conditions"/chaos means that close may not be enough.

There also is the issue of numerical error from the fact that you are using discrete equations, but usually simulators take steps to check this is negligible. (Which may not be enough.)

geezerjay2• 9 years ago

That's only one data point, but it certainly seemed to me that no one was relying on N-S as an accurate predictor of motion

You're confusing how models are used by (some) engineers (in some applications) with how models are used by physicists and mathematicians. Engineers have to deal with all kinds of uncertainties, from matetial parameters to use cases to limit states to wear and fatigue and geometrical deviations etc etc etc. Therefore, engineers develop robust designs to comply with all design requirements under any plausible and probable scenario given a design life. To accomplish this, engineers use models to provide approximate but accurate results that are on the safe side of any limit state. Yet, eventhough designs need to be robust, simulations do need to be accurate.

These findings suggest that low-resolutoon Navier-Stokes simulations that were believed to be on the safe side may actually not be on the safe side. These finding are important, as they will illicit significant changes on how Navier-Stokes simulations are used in cases where accuracy matters.

justaman• 9 years ago

I'm not going to pretend I understand anything about fluid dynamics.

It brings to mind how people say "Bumble bee's defy the laws of physics to fly".

mannykannot• 9 years ago

"When I meet God, I am going to ask him two questions: Why relativity ? And why turbulence ? I really believe he will have an answer for the first." - Werner Heisenberg.

FabHK2• 9 years ago

"I am an old man now, and when I die and go to heaven there are two matters on which I hope for enlightenment. One is quantum electrodynamics, and the other is the turbulent motion of fluids. And about the former I am rather optimistic." - Horace Lamb

tulasinandan• 9 years ago

The author explaining the paper: https://www.youtube.com/watch?v=F71SRP3MZcw

davidw2• 9 years ago

Totally unfamiliar with the math and physics, but I recall the name from one of the earlier parts of Cryptonimicon.

qubex14• 9 years ago

Navier & Stokes is the name of the scientists who indecently formulated the equation that bears their name that describes the motion of most fluids. In Cryptonomicon the main character is given a simple question about a boat on a river during his admission to the army, deploys heavy-duty fluid dynamics to give a non-obvious answer, and consequentially gets classified as a moron and relegated to menial duties (which suit him just fine, as I recall).

davidw2• 9 years ago

Yeah it was a pretty funny scene - it's supposed to be this simple math question, but the ... "Asperger"... he might be called these days - character, Lawrence Waterhouse, really digs into the problem. He ends up discovering something that he submits to a math journal for publication, but the army folks think he's an idiot.

bitL• 9 years ago

Is there any online course I can take that would teach me Navier-Stokes?

derjames• 9 years ago

I would recommend this text book: Transport Phenomena. Bird, Steward and Lightfoot. The first part is related to the motion of fluids/transfer of momentum ( The one you are interested ). The second part is the transfer of energy and the third part is related to the transfer of mass.

Mathematicians Find Wrinkle in Famed Fluid Equations

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