By Tsinober, A. (1937-)
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Additional info for An informal introduction to turbulence
From the mathematical point of view the transitions from one flow regime to another with increasing Reynolds number – as we observe them in physical space – are believed to be a manifestation of generic structural changes of the mathematical objects called phase flow and attractors in the phase space through bifurcations in a given flow geometry (Hopf, 1948). However, partly turbulent flows (a special feature of these flows is the coexistence of regions with laminar and turbulent states of flow) are not easily ‘fit’ in this picture.
An individual description of the fields of velocity, pressure, temperature, and other characteristics of turbulent flow is in principle impossible. Moreover, such description would not be useful even if possible, since the extremely complicated and irregular nature of all the fields eliminates the possibility of using exact values of them in any practical problems... In the present-day statistical fluid mechanics, it is always implied that the fluid mechanical fields of a turbulent flow are random fields in the sense used in probability theory.
In other words, nothing less than a thorough understanding of the [global behaviour of the] system of all their [NSE] solutions would seem to be adequate to elucidate the phenomenon of turbulence (Neumann, 1949). That is in order to understand the dynamics, or the main characteristics of the dynamics, it is necessary to understand a significant portion of the phase flow (see glossary of terms), especially the unstable solutions14. However, at present (if ever) it is impossible due to very high dimension and complicated structure of the underlying attractors (assumed to exist): one may never be able to realistically determine the fine-scale structure and dynamical details of attractors of even moderate dimension.