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Here I found that:
Sixty years later, Russian mathematician Andrey Kolmogorov furthered our mathematical understanding of turbulence when he proposed that energy in a turbulent fluid at length $R$ varies in proportion to the five-thirds power of $R$.
Was he the first one to describe the turbulence rigorously? I don't define the term "rigorously" on purpose, because I'd like to hear different views.
Systematic study of turbulence originates with a series of experiments conducted by Osborne Reynolds starting in 1870s. His mathematical theory was developed in On the dynamical theory of incompressible viscous fluids and the determination of the criterion (1895). However, already in 1883 he described a classical experiment with a jet of dyed water at the center of a flow in a glass pipe that demonstrated the transition to turbulence. He also introduced what is now called the Reynolds number as the parameter that governs the transition from laminar to turbulent flows. Between the values of 2000 and 13000, depending on entry conditions (if extreme care was taken the threshold could even be increased to 40000), the dyed layer which remained distinct at smaller values, broke up and spread out across the entire cross-section of the pipe.
Here are some excerpts from the 1895 paper describing the relation of Reynolds's work to Stokes's before him. The "assumption" referred to below is the assumption shown by St. Venant and Stokes in 1845 to underlie the Navier-Stokes equations: that the stresses, other than pressure, which is uniform in all directions, are linear functions of the rates of distortion, with a coefficient depending on the fluid:
By obtaining a singular solution of these equations as applied to the cases of pendulums in steady periodic motion, Sir G. Stokes was able to compare the theoretical results with the numerous experiments that had been recorded... these results, both theoretical and practical, were directly at variance with common experience as to the resistance encountered by larger bodies moving with higher velocities through water, or by water moving with greater velocities through larger tubes. This discrepancy Sir G. Stokes considered as probably resulting from eddies which rendered the actual motion other than that to which the singular solution referred and not as disproving the assumption...
[...]These experimental results [Reynolds's from 1883] completely removed the discrepancy previously noticed, showing that, whatever may be the cause, in those cases in which the experimental results do not accord with those obtained by the singular solution of the equations, the actual motions of the water are different. But in this there is only a partial explanation, for there remains the mechanical or physical significance of the existence of the criterion to be explained.
My object in this paper is to show that the theoretical existence of an inferior limit to the criterion follows from the equations of motion as a consequence: --
(1) Of a more rigorous examination and definition of the geometrical basis on which the analytical method of distinguishing between molar-motions and heat motions in the kinetic theory of matter is founded ; and
(2) Of the application of the same method of analysis, thus definitely founded, to distinguish between mean-molar-motions and relative-molar-motions..."
