Why is it that turbulence is considered to be an unsolved problem of classical mechanics? What is meant by "unsolved"? Don't the Navier-Stokes equations apply to turbulent flows? It's difficult to grasp these concepts because there doesn't seem to be a mathematically precise definition of turbulence.
Let's take a provocative definition of turbulence:
Of course, this is a moving definition as time varies. But it ensures that turbulence will remain an unsolved problem, forever.
Post scriptum. Never say in a talk "I dont't think we shall understand turbulence within my lifetime". Perhaps someone in the audience will stand up, show a knife and say "I am not so patient to wait that long".
Indeed, from a mathematical point of view, turbulence is what happens to a solution of Navier-Stokes when it develops a singularity. And we do not know that the solution exists to begin with. So turbulence is the regime where something which we are not sure to exist ceases to exist. Quite shady, isn't it :)
More seriously, there have been some attempts to model turbulence. I think one basic problem is that it is not even clear what exactly one should model. Does it make sense to describe the fluid by a field $u(t,x)$, surely a very singular one, governed by some PDE? A classical discussion of this problem is in Chapter III of the sixth volume of Landau-Lifshitz' treatise, and there are some more recent books entirely devoted to this question.