It was said in book "A first course in turbulence - Tennekes H., Lumley J.L", 1970:

Randomness and nonlinearity combine to make the equa- tions of turbulence nearly intractable; turbulence theory suffers from the absence of sufficiently powerful mathematical methods. This lack of tools makes all theoretical approaches to problems in turbulence trial-and-error affairs. Nonlinear concepts and mathematical tools have to be developed along the way; one cannot rely on the equations alone to obtain answers to problems. This situation makes turbulence research both frustrating and challenging: it is one of the principal unsolved problems in physics today.

So my questions are:

1. Is this saying still true?

2. What are the main progresses in turbulence research during last 40 years?

Thanks!

I think the excellent book by Pope, Turbulent Flows, should answer your questions. Tennekes and Lumley is still an excellent book on turbulence, however, there has been progress since the '70s, and Pope presents an excellent overview of turbulence in beginning of the 21st century.