This weekly roundup thread is intended for all culture war posts. 'Culture war' is vaguely defined, but it basically means controversial issues that fall along set tribal lines. Arguments over culture war issues generate a lot of heat and little light, and few deeply entrenched people ever change their minds. This thread is for voicing opinions and analyzing the state of the discussion while trying to optimize for light over heat.
Optimistically, we think that engaging with people you disagree with is worth your time, and so is being nice! Pessimistically, there are many dynamics that can lead discussions on Culture War topics to become unproductive. There's a human tendency to divide along tribal lines, praising your ingroup and vilifying your outgroup - and if you think you find it easy to criticize your ingroup, then it may be that your outgroup is not who you think it is. Extremists with opposing positions can feed off each other, highlighting each other's worst points to justify their own angry rhetoric, which becomes in turn a new example of bad behavior for the other side to highlight.
We would like to avoid these negative dynamics. Accordingly, we ask that you do not use this thread for waging the Culture War. Examples of waging the Culture War:
-
Shaming.
-
Attempting to 'build consensus' or enforce ideological conformity.
-
Making sweeping generalizations to vilify a group you dislike.
-
Recruiting for a cause.
-
Posting links that could be summarized as 'Boo outgroup!' Basically, if your content is 'Can you believe what Those People did this week?' then you should either refrain from posting, or do some very patient work to contextualize and/or steel-man the relevant viewpoint.
In general, you should argue to understand, not to win. This thread is not territory to be claimed by one group or another; indeed, the aim is to have many different viewpoints represented here. Thus, we also ask that you follow some guidelines:
-
Speak plainly. Avoid sarcasm and mockery. When disagreeing with someone, state your objections explicitly.
-
Be as precise and charitable as you can. Don't paraphrase unflatteringly.
-
Don't imply that someone said something they did not say, even if you think it follows from what they said.
-
Write like everyone is reading and you want them to be included in the discussion.
On an ad hoc basis, the mods will try to compile a list of the best posts/comments from the previous week, posted in Quality Contribution threads and archived at /r/TheThread. You may nominate a comment for this list by clicking on 'report' at the bottom of the post and typing 'Actually a quality contribution' as the report reason.
Jump in the discussion.
No email address required.
Notes -
Can you explain to a guy that's not deep into the math trenches what it would mean for that conjecture to be false in concrete terms irl? Or is that part of those abstractions on abstractions the mathematicians sometimes get themselves into?
In concrete terms the direct meaning of a blowup in the incompressible Navier-Stokes equations is nothing - incompressible fluids do not exist. For that matter, fluids do not exist - it's all atoms, and we pretend that atoms blend into a perfect continuum because it's a lot easier to approximate a problem using a million finite elements rather than a septillion atoms. Perhaps most fluid flow simulations are performed using incompressible Navier-Stokes anyway, though. If some of the fluid gets significantly warmer than the rest we'll partially take compressibility into account by adding a Boussinesq buoyancy term, but otherwise we generally pretend a fluid is incompressible until we're dealing with speeds near the speed of sound (in that fluid - e.g. "Mach 1" might mean 767 mph in standard air, but Mach 1 is 4 times faster in water), and we pretend flowing matter is a fluid until we're dealing with such tiny length scales or rarified gases that you have to worry about atoms slipping right past each other.
IMHO the rest of what's interesting is "abstractions on abstractions", especially for now, but ones that might lead to practical consequences in the future.
Half of what's interesting in the most practical sense here is: we want to control the error in our fluid flow simulations, and this might be another step toward figuring out how to do so more rigorously.
That "million finite elements" (in practice more likely finite volumes; unimportant distinction here) doesn't give you an exact solution to your flow problem, just an approximate solution, and that's fine because it's engineers who need the approximate solution and engineers use safety factors and the safety factors mean that if we give them a solution that's 10% off they're still fine. I once found a bug I'd written that made my code 0.0001% off, which snowballed through other equations to make a ton of engineering solutions 0.1% off. Coming from academia I was mortified, but the engineers hadn't noticed anything was wrong until I fixed the bug and thereby tripped some regression tests against past "gold" solutions. They hadn't yet bothered setting up problems where they could compare simulation solutions to exact solutions to verify the error magnitudes. Their idea of responsible testing was just validating against experiments, and the experimental measurement error was well over 0.1% so the validations said "this matches experiments well enough", and thankfully we never were simulating anything where simulation and reality diverged more than they did in experiments.
The trouble is that, even with an approximate solution, we always like to know how approximate it is, and if we always had an experiment to validate against then we wouldn't bother running a simulation. Grossly oversimplifying: with the simplest partial differential equations, we can prove results that look like "if the boundary conditions and 'forcing function' are of magnitude ‖f‖, then the solution u must be of magnitude ‖u‖≤c‖f‖, for a constant c that depends on the boundary". We can't actually find the solution u exactly, but we can approximate it, and we can turn those results about the exact solution into results about the error e=u-uₕ on an approximate solution uₕ, results that look something like ‖e‖≤mc‖f‖, for a constant m that depends on what type of approximation we use and how expensive we make it. If we want to guarantee a small error, for a given c and ‖f‖ we can pick an approximation with m small enough to get it.
We couldn't do this with Navier-Stokes in the most general cases, and now it's clear why: this does not work if c is infinity. Techniques that use the fact that the solution can't grow too badly to prove that our error can't grow too badly do not work if the solution can blow up. Often we can still prove that we're dealing with a "nice" case where the fluid viscosity can "overpower" a weak forcing enough to prevent a blowup, but now we know that "often" isn't "always"; we weren't just missing some clever trick to get there. This counterexample might stop people from wandering down a ton of blind alleys looking for one, or by showing us what we need to avoid it might point the way to expanding our criteria for "nice".
And the other half of what might become interesting down the road here: We don't understand turbulence nearly well enough, and yet most of the liquid flow and essentially all of the gas flow problems we want to solve are turbulent.
If we have a turbulent flow, that grid of "a million finite elements" becomes laughably small for a direct numerical simulation of it, because smaller and smaller turbulent eddies quickly become so small that we can't represent them accurately, while still being powerful enough to greatly affect the flow on the larger "macroscale" we care about. Instead we need to use, say, "35 trillion grid points", if we want a solution to just the equations of fluid flow that resembles the actual fluid flow. We can't afford to make every engineering problem millions of times more expensive to solve. So we've come up with "turbulence models", additional equations that try to approximate the statistically-averaged-out effects of turbulence on the macroscale flow, rather than precisely tracking every microscale vortex from its creation until viscosity damps it out. Now our equations are only 10% more expensive to solve (with the cheapest models) or 50% or 200% or whatever as we switch to better and better models, and we hope the results are good enough. It really is a mix of "hope" and "this matches experiments well enough" (for a much looser definition of "well enough" this time), because we don't have any really good turbulence models, not in the same "so close we never notice a difference" or "this is an inevitable consequence of the laws of physics with a few simplifications" senses in which incompressible-Navier-Stokes is a good model for viscous macroscopic fluid flow. Heisenberg supposedly said "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." - apocryphal quote from nearly a century ago, but the turbulence experts I've talked to during the 21st century didn't disagree. But now we've got some fairly unexpected Navier-Stokes results, specifically dealing with the ways in which viscosity can or can't damp down a microscale vortex? This might go nowhere, or it might end up leading someone to a much better turbulence model down the road. I haven't looked at this stuff in a decade, but at a brief glance right now it looks like the cutting-edge in turbulence modeling is split between "tweak some 1990-era models to make them a bit better" and "if we enforce a few laws of physics on top of a machine-learning black-box it sort of works". Breaking out of those ruts wouldn't have a flashy Millennium Prize in the headlines but it could be a huge breakthrough.
That was super-informative! I just gained more useful insight from 2 minutes of reading your comment then I did from an hour of reading other explainers of the Navier-Stokes result. Many thanks!
More options
Context Copy link
More options
Context Copy link
More options
Context Copy link