Hacker News Board Hacker News, browsed like a classic message board
Live data from Hacker News

Mathematicians Find Wrinkle in Famed Fluid Equations

quantamagazine.org

Post reply on HN
Print view View original

91–100 of 124 posts

Re: Mathematicians Find Wrinkle in Famed Fluid Equations

#91

Post by neutronicus »

rusk wrote: post #54
alephnil wrote: post #49

Earlier quoted context omitted.

The Navier-Stokes equations assume that the medium is continuous even at infinitely small scales, which obviously not the case for natural fluids, that are made of discrete atoms. Thus the equations are only correct at sufficiently large scales. They work fine for describing the airflow around an aeroplane, but not the airflow around the head of a hard drive, which is small enough that the finite size of atoms must b…

How are Navier Stokes the other way, on the macro scale? e.g. in the context of meteorology. Asking because I had a discussion with somebody recently where they claimed the fundamental flaw to the science underlying Climate Change science is over-reliance on NS at macro scale as a way of predicting climate behaviour, or something. I took it to be Baloney, but I'm wondering if there is some strands of truth to it ...

Fluid equations assume that there's a single characteristic velocity for the atoms at each point in space. So, for physical systems where local velocity distributions are wide or even multi-modal, fluid equations won't capture the physics.

In plasma physics, Laser Wakefield dynamics is an example of a system that can't be modeled as a fluid.

I sort of doubt these considerations apply to the atmosphere, but this is one of the main heuristics for when you can't use a fluid equation.

Re: Mathematicians Find Wrinkle in Famed Fluid Equations

#92

Post by qubex »

throwwit wrote: post #90
qubex wrote: post #88

Earlier quoted context omitted.

I don't understand what the first sentence of your reply is supposed to mean, but the second sentence is a very mature response and belies your wisdom: yes, science and rational thinking is all about putting ideas out there and rejoicing when somebody helps you etch away at those that are not compatible with reality, so that only plausible ones remain.

Well what I meant was shrinking the n body system to a point then extending that to a field. But it’s beyond me how the math works to invert those same force vectors to an impulse.

Fluid dynamics and gravitation are not about the things that are interacting (of which there can be many in both cases) but in the forces that arise between these things, and the forces are profoundly different in both cases.

Re: Mathematicians Find Wrinkle in Famed Fluid Equations

#93

Post by sdfadfa »

pge wrote: post #74

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 (w…

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.

Re: Mathematicians Find Wrinkle in Famed Fluid Equations

#94

Post by throwwit »

throwwit wrote: post #90
qubex wrote: post #88

Earlier quoted context omitted.

I don't understand what the first sentence of your reply is supposed to mean, but the second sentence is a very mature response and belies your wisdom: yes, science and rational thinking is all about putting ideas out there and rejoicing when somebody helps you etch away at those that are not compatible with reality, so that only plausible ones remain.

Well what I meant was shrinking the n body system to a point then extending that to a field. But it’s beyond me how the math works to invert those same force vectors to an impulse.

Apologies if I’m cargo-culting math... just positing.

Re: Mathematicians Find Wrinkle in Famed Fluid Equations

#95

Post by btrettel »

pge wrote: post #74

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 (w…

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.)

Re: Mathematicians Find Wrinkle in Famed Fluid Equations

#97

Post by qubex »

davidw wrote: post #96

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

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).

Re: Mathematicians Find Wrinkle in Famed Fluid Equations

#98

Post by FabHK »

mannykannot wrote: post #80

"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.

"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

Re: Mathematicians Find Wrinkle in Famed Fluid Equations

#99

Post by btrettel »

sdfadfa wrote: post #93
pge wrote: post #74

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 (w…

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 univers…

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.

Re: Mathematicians Find Wrinkle in Famed Fluid Equations

#100

Post by qubex »

throwwit wrote: post #94
throwwit wrote: post #90

Earlier quoted context omitted.

Well what I meant was shrinking the n body system to a point then extending that to a field. But it’s beyond me how the math works to invert those same force vectors to an impulse.

Apologies if I’m cargo-culting math... just positing.

https://xkcd.com/451/

;)

Post reply on HN