Turbulence, and Fluid Mechanics in General

So what, you may ask, is the fabled “problem of turbulence”? In essence, this: what on Earth do our statistics and our equation have to do with each other? A solution to the problem of turbulence would be, more or less, a valid derivation from the Navier-Stokes equation (and statements about the appropriate conditions) of our measured statistics. Physicists are very far from this at present. Our current closest approach stems from the work of Kolmogorov, who, by means of some statistical hypotheses about small-scale motion, was able to account for the empirical laws I mentioned. Unfortunately, no one has managed to coax the hypotheses from the Navier-Stokes equation (sound familiar?) and the hypotheses hold exactly only in the limit of infinite Reynolds number, i.e. they are not true of any actual fluid.

http://vserver1.cscs.lsa.umich.edu/~crshalizi/notebooks/turbulence.html