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?
Answer
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.
Why it is an unsolved problem from a physical point of view, read Ruelle and Takens here.
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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