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?
[Physics] the thestery of turbulence
fluid dynamicsnavier-stokes;turbulence
Related Solutions
I belive you have it pretty much settled already. If I was to change anything, I would shrink instead of adding more items:
Identify the relevant quantities of your system: Energy, Momentum, entropy, electric charge, mass ...
Which may or may not be conserved. If you have boundary conditions, most probably you don't have energy and/or momentum conservation on the system alone. Either way, you need to write continuity equations for the quantities of your system, which may have source terms that are due to this non-conservations.
So, if think on a non-relativistic, free of boundaries fluid, with no additional conserved charge.
The relevant variables would be the (in principle well defined) velocity $\vec v$, the mass density $\rho$, the stress tensor $\sigma$, which gives the interaction inside the fluid, and also the external force density that acts on the fluid, $\vec f$.
Write general equations for your system based on this general laws. Navier Stokes would come from the conservation of momentum, with a particular choice for the stress tensor.
So, an example of equations in this case is to put an non-relativistic fluid without boundaries, so the general equations would (mostly) respect Galilei symmetries and you would end up with:
$\frac{\partial \rho}{\partial t}+\nabla \cdot \left(\rho \vec v\right)=0$
$\frac{\partial}{\partial t}\left(\rho \vec v\right)+\nabla \cdot \left(\rho \vec v\otimes \vec v + \sigma\right)=\vec f$
So you end up with 2 continuity equations, the first without source, which describes the conservation of mass of the fluid. The second, which describes the conservation of momentum of the fluid, in princible have a source term, that is the external force density (think on gravity).
You would still need an equation for the conservation of energy and entropy.
Here is also where you would need to include boundary conditions and the sort. You can try inserting this the $\vec f$ (at least numerically, like in particle methods), or writing constraining equations, which can be, for example, zero relative velocity near a wall with a viscous fluid, or finite energy and momentum system with no free boundaries.
Write down the Constitutive Equations. I believe this is the trickiest part, since you can't deduce the general form of them from symmetry principles alone. This is where you need to include the Equation of State (EoS) for the system, and also where you need to include memory effects if applicable.
Here, if you are studying newtonian fluids, you have the following form of the stress tensor:
$\sigma_{ij} = p\delta_{ij}- \eta\left(\frac{\partial v_i}{\partial x_j} + \frac{\partial v_j}{\partial x_i}\right) - \delta_{ij} (\zeta-\frac{2}{3}\eta) \nabla \cdot \mathbf{v}$
Where you now need an equation of state for $\eta$ and $\zeta$ and $p$, but in principle, they are functions of $\rho$ and $T$(temperature), but that's it. Once you arrive here, you should have a 'complete' hydrodynamical description of your system.
For the 'working physicist/engineer', most probably you will need to, at least on some point, work out a computational solution to your problems, since most of the time hydrodynamics, even in it's simplest forms, are very hard to attack only with pencil and paper. Thus, I would add another item:
Formulate your problem numerically: Discretize your problem, write down what would be the approximate equations, either using mesh-full (Finite Differences, Finite Elements...) or mesh-free (Smoothed Particle Hydrodynamics...) methods, and implement on the computer using your favorite programming language.
Much of the advances on the practical point of view are done this way, instead of trying to solve 'by hand'. This is even more important with non-newtonian fluids.
Still it's important not to forget alternative approaches to specific cases. The first example that comes to my mind is the Statistical Theory of Turbulence (K41 theory and the ones that followed it). Even though you don't get all the information that is there to get, you can attack the problem phenomenologically, which in many cases is more than you get if you attack the problem directly.
Lastly:
I haven't included anything in there about boundary layers or thermal boundary layers or turbulent/laminar flow, etc. What would be the best spot to put these considerations?
I believe they would fit on section 2, since, in the case of turbulence, you usually treat different from laminar flow, thus requiring a different set of equations. Consequently you would also need to adapt 1 and 3, but, in my view, the crux is at 2.
There are groups that try to solve turbulent flows directly, but it's necessarily via numerical methods, and it's incredibly costly to do.
There are known solutions to the Navier-Stokes equations. A simple example would be laminar shear-driven flow between two moving plates. Just as in the case of Einstein's equations, the known solutions regard simple situations with particular boundary conditions; a general solution that covers all possible cases is not known in either case. One should not expect such a thing ever to be found, since these systems can exhibit chaos; the difficulty of finding a general solution is the same as for the n-body problem in Newtonian gravity.
However, the millennium problem for the Navier-Stokes equations is something different. It doesn't ask for a single solution, it asks whether a solution always exists, for any initial and boundary conditions. Then, if solutions do always exist, it asks whether they will always have certain technical properties. But it doesn't ask us to find the solution, because we know that in general this will not be possible.
I'm no expert, but I believe the reason this is harder for the Navier-Stokes equations than for Einstein's equations is that the Navier-Stokes equations for an incompressible fluid are inherently non-local; the incompressibility constraint means that something happening in one part of the fluid can instantly affect every other part of the system. (This is, of course, an approximation to the physical reality.) In contrast, general relativity is inherently local.
Best 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.