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?
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.
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.)
> 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.
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.
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.
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.
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.
> but a case where multiple different outcomes can arise from identical inputs.
If they are identical but not equal and these small differences in the input lead to large deviations in the output then that's pretty much the definition of a chaotic system.
Edit: I've just browsed through the paper in question and it actually demonstrates thar an approximate (weak) solution is not unique, which means that the exact same inputs in may have multiple weak form solutions.
"Weak" here does not mean numerical approximations. These are mathematicians so any quantitative approximation would be bounded o(1) otherwise the work would be meaningless. One should think of "weak" as in constraints, for example constraints at lower spatial resolutions (but precise).
