Turbulence versus Mathematics and vice versa

It is much easier to present nice rational linear analysis than it is to wade into the morass that is our understanding of turbulence dynamics. With the analysis, professor and students feel more comfortable; even the reputation of turbulence may be improved, since the students will find it not as bad as they had expected. A discussion of turbulence dynamics would create only anxiety and a perception that the field is put together out of folklore and arm waving.” John Lumley, 1987.

From the outset I have to confess that I find myself 99% in agreement with John Lumley’s concern on “theories of turbulence”. This includes the first premise – i.e. the absence of a theory based on first principles. The second aspect concerns the importance of experiments and observations (both physical and numerical), below referred to as evidence. This lecture is intended to be, first and foremost, a critical presentation and examination of some fundamentally important issues.

* What do we really mean by ‘conventionally defined inertial range’ (CDIR)? Are its properties really independent of (the nature of) dissipation and/or large-scale forcing? Thus, is the inertial range a well defined concept or is it a mis-conception? Who is the guilty party for dissipation anomaly in turbulent flows? And what about the role of the self-amplification processes of vorticity, strain and super-helicity? Also, how well-defined and meaningful is the so-called ‘decomposition’ of energy in inertial and dissipative ranges?

* Is the ‘anomalous scaling’ an attribute of the inertial range? And of passive turbulence?

* Is the ‘4/5 law’ a purely inertial relation?

* Why should one expect that in the CDIR at very high Reynolds numbers the Navier–Stokes equations (NS) are invariant under infinitely many scaling groups (like the Euler equations), in the statistical sense of K41 labeled by an arbitrary real scaling exponent h? And more generally, should one expect to restore in some sense all the symmetries of Euler equations in the CDIR? And why necessarily Euler?

* Are weak solutions of Euler equations going to describe adequately a turbulent flow? Is the inviscid limit of NS always independent of the nature of dissipation and viscosity? Is it possible that the Reynolds dependence differs, but the limit (in distributional sense) remains the same? What does it happen to the solenoidal part of the acceleration as viscosity goes to zero? Could the ‘real’ inertial range of turbulence be adequately described by a suitable singular solution of some sort of Euler-like equations?

* About the concept of ‘non-locality’ of turbulence: is ‘cascade’ a well defined concept and is there a cascade in physical space? Is ‘cascade’ Eulerian, Lagrangian or what? These and other related questions will be briefly touched upon depending on the discussion and interest.


References
TSINOBER, A. 2009 An Informal Conceptual Introduction to Turbulence, Springer-Verlag.
TSINOBER, A. 2018 The Essence of Turbulence as a Physical Phenomenon. II edition (in press), Springer-Verlag.