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One of the great unsolved problems in physics is turbulence but I'm not too clear what the mystery is. Does it mean that the Navier-Stokes equations don't have any turbulent phenomena even if we solve it computationally? Or does it mean we simply don't have a closed-form solution to turbulent phenomena?

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See also e.g. this Wikipedia page. –  Qmechanic Mar 11 '13 at 1:54
I've read and that's why I'm asking hear. It just says it's hard to model but so are other systems that aren't really a mystery like the n-body problem. Does it really come down to it being hard to model? –  cspirou Mar 11 '13 at 2:03
this question here is relevant physics.stackexchange.com/q/15738 –  anna v Mar 12 '13 at 6:29

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Turbulence is indeed an unsolved problem both in physics and mathematics. Whether it is the "greatest" might be argued but for lack of good metrics probably for a long time.

Why it is an unsolved problem from a mathematical point of view read Terry Tao (Fields medal) here : http://terrytao.wordpress.com/2007/03/18/why-global-regularity-for-navier-stokes-is-hard/

Why it is an unsolved problem from a physical point of view, read Ruelle and Takens here : http://www.ihes.fr/~ruelle/PUBLICATIONS/%5B29%5D.pdf

The difficulty is in the fact that if you take a dissipative fluid system and begin to perturb it for example by injecting energy, its states will qualitatively change. Over some critical value the behaviour will begin to be more and more irregular and unpredictable. What is called turbulence are precisely those states where the flow is irregular. However as this transition to turbulence depends on the constituents and parameters of the system and leads to very different states, there exists sofar no general physical theory of turbulence. Ruelle et Takens attempt to establish a general theory but their proposal is not accepted by everybody.

So in answer on exactly your questions :

yes, solving numerically Navier Stokes leads to irregular solutions that look like turbulence

no, it is not possible to solve numerically Navier Stokes by DNS on a large enough scale with a high enough resolution to be sure that the computed numbers converge to a solution of N-S. A well known example of this inability is weather forecast - the scale is too large, the resolution is too low and the accuracy of the computed solution decays extremely fast.

This doesn't prevent establishing empirical formulas valid for certain fluids in a certain range of parameters at low space scales (e.g meters) - typically air or water at very high Reynolds numbers. These formulas allow f.ex to design water pumping systems but are far from explaining anything about Navier Stokes and chaotic regimes in general.

While it is known that numerical solutions of turbulence will always become inaccurate beyond a certain time, it is unknown whether the future states of a turbulent system obey a computable probability distribution. This is certainly a mystery.

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What do you mean by not being "possible to solve numerically Navier Stokes by DNS on a large enough scale with a high enough resolution to be sure that the computed numbers converge to a solution of N-S?" Do you mean that computer power is not up to par so that you can include all temporal and spatial scales in order to get a truly physical turbulent solution? Or do you mean that not even with a full DNS simulation can you capture the true nature of turbulence? –  Isopycnal Oscillation Jun 7 '13 at 17:03
I mean that DNS can never be extended to a large enough size and a small enough time increment to get a converged solution for a fully turbulent flow. Example is atmosphere, weather, oceanic currents etc. DNS can only give usable results for scales of the order of meters. The reason is of course computing power which in the case of atmosphere would have to be infinite for all practical purposes. –  Stan Won Jun 19 '13 at 9:09

Turbulence is not one of the great unsolved problems in physics. Physics tells us exactly how turbulence emerges as a direct consequence of local mass and momentum conservation. We can create multiparticle computer models such as lattice gas automata that generate turbulence at large length and time scales. We can write down the equations that govern turbulence. These are the Navier-Stokes equations.

From a fundamental physics perspective, turbulence is a solved problem that has entered the engineering realm a long time ago.

So what is the unsolved problem associated with turbulence? In short, turbulence is an unsolved problem not in physics but in mathematics. The point is that mathematicians struggle to answer the question if the Navier-Stokes equation always allows for solutions that at fine enough length and time scales are well behaved. In fact, if you manage to conclusively answer this question ("yes" or "no"), you will win a math prize that comes with a handsome cheque of $ 1,000,000.

In case you want to give it a try, the precise question is:

Prove or give a counter-example of the following statement: In three space dimensions and time, given an initial velocity field, there exists a vector velocity and a scalar pressure field, which are both smooth and globally defined, that solve the Navier–Stokes equations.

The math difficulties have to do with the fact that turbulence emerges when the highest derivative term in the Navier-Stokes equations (the viscosity term) becomes small compared to the other terms. You can take almost any non-linear partial differential equation, and get mathematicians to cringe simply by multiplying the highest derivative term with a factor $\epsilon$ and ask about the limiting behavior of the equation when $\epsilon$ approaches zero.

Fundamental physicists shrug and continue studying real mysteries such as quantum gravity.

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I feel your answer misses the question in a way. What you say is true and a good answer to "what is an unsolved problem relating to turbulence" and you might interpret the question like that. However, when usually people complain about their unsolved problems with turbulence, what they mean is that they feel there is no proper practical treatment of turbulence with reasonable computational effort. No engineer and maybe only theoretical physicists care about the solution to the associated mathematical problem. Unless it's constructive. Polemically: The math problem isn't all that interesting. –  NikolajK May 9 '13 at 18:17
Well, OP uses words like "mystery". The fact that it takes progressively more computing power to resolve turbulence at higher Reynolds number is a practical CFD issue, certainly not a mystery. –  Johannes May 9 '13 at 19:40
@Johannes you mean that the physical mechanism that gives rise to a turbulent flow is completely known? –  Zorich May 9 '13 at 19:54
@Monopole - Absolutely. You can accurately simulate turbulence using nothing more than particles hopping on a grid (google "Lattice Gas Automata"). All that is needed is a minimal amount of symmetry and the fact that inter particle collisions conserve momentum. –  Johannes May 9 '13 at 20:06
Is this the same as the "grey area" of a moody diagram around Reynolds values of 2000-3000, where the flow changes from laminar to turbulent? –  fibonatic Jun 3 '13 at 10:57

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