Can a theory of everything solve the oldest problem of turbulence? Can a theory of everything solve the oldest problem of turbulence?
Can we have unified theory of universe without solving the problem of turbulence?
 A: 
Can we have unified theory of universe without solving the problem of turbulence?

Yes. Because the "unified" in the name refers to the basic interactions, from which in principle everything could be explained.
In practice, a hierarchy of models, with effective theories emerging from lower-level ones (think fluids laws from molecules' interactions, or quasiparticles in condensed matter) is likely to remain the most feasible and useful description.
A: I’d like to answer by expanding the analogy made by @Charlie. 
A theory of everything would be like knowing the rules of chess. We could understand all the rules, the pieces, and their moves and interactions. But there would remain many deep mathematical problems: e.g., what’s the perfect strategy in chess? It seems unlikely that this will be solved in our lifetimes. What’s even a good strategy? Knowing the rules does not mean that playing or understanding the game is easy. 
So knowing a theory of everything would be like knowing the rules. Even with the rules, fully understanding the Universe would still be extremely challenging. There would still be very hard mathematical problems, like turbulence.
However,  there’s no reason to think that the additional rules provided by string theory or another theory of everything would help at all explain turbulence. We already know the relevant rules for turbulence, we just can’t solve them. 
