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What is isotropic turbulence and why is it important?

David McComb

Abstract

This article begins with an overview, then gives the precise definition of isotropic turbulence, and follows that with the basic conservation equations, in both real space and wavenumber space. These provide the foundations of all theoretical approaches, both fundamental and phenomenological. After that, my intention is to try to highlight the main unresolved issues and give some indication of what progress there has been over decades (in all cases), and what still needs to be done. I should emphasise that I am not trying to provide either a conventional review or even a pedagogical treatment. Instead I am giving concise summaries, supplemented (where I can) by my own observations, which make substantial points that I believe are original, and which have not been made in the literature. To take just one example, it is known by some people that Kolmogorov's 1962 theory is not correctly described as a 'refinement' of his 1941 theory. This was pointed out by Kraichnan in 1974. However, what does not appear to have been recognized is that the 1962 theory is physically invalid, and also that a plausible implementation of it destroys the Kolmogorov (1941) scaling of energy spectra which has been widely observed over many years. This is discussed in Section 4 below. Lastly, I have tried to give an informal treatment in order to make everything easily accessible, to reach the widest possible audience. In particular, the section on renormalization methods is written without giving the equations of the various theories, merely stating in words what has been done, what are the different methods and also what still needs to be done.

What is isotropic turbulence and why is it important?

Abstract

This article begins with an overview, then gives the precise definition of isotropic turbulence, and follows that with the basic conservation equations, in both real space and wavenumber space. These provide the foundations of all theoretical approaches, both fundamental and phenomenological. After that, my intention is to try to highlight the main unresolved issues and give some indication of what progress there has been over decades (in all cases), and what still needs to be done. I should emphasise that I am not trying to provide either a conventional review or even a pedagogical treatment. Instead I am giving concise summaries, supplemented (where I can) by my own observations, which make substantial points that I believe are original, and which have not been made in the literature. To take just one example, it is known by some people that Kolmogorov's 1962 theory is not correctly described as a 'refinement' of his 1941 theory. This was pointed out by Kraichnan in 1974. However, what does not appear to have been recognized is that the 1962 theory is physically invalid, and also that a plausible implementation of it destroys the Kolmogorov (1941) scaling of energy spectra which has been widely observed over many years. This is discussed in Section 4 below. Lastly, I have tried to give an informal treatment in order to make everything easily accessible, to reach the widest possible audience. In particular, the section on renormalization methods is written without giving the equations of the various theories, merely stating in words what has been done, what are the different methods and also what still needs to be done.
Paper Structure (41 sections, 69 equations, 3 figures)

This paper contains 41 sections, 69 equations, 3 figures.

Table of Contents

  1. Introduction
  2. Is turbulence research still in its infancy?
  3. A pragmatic taxonomy for turbulence
  4. Definition of isotropic turbulence
  5. The implications of isotropy for the statistical formulation
  6. The energy balance equations in $x$-space and $k$-space
  7. The Karman-Howarth equation
  8. The Lin equation
  9. Inertial transfer and Onsager's observation
  10. The scale-invariance paradox
  11. The Richardson-Kolmogorov cascade
  12. Is Kolmogorov (1962) physically valid?
  13. The viscous dissipation rate
  14. The asymptotic nature of the dimensionless dissipation rate of turbulence
  15. Stationary turbulence
  16. ...and 26 more sections

Figures (3)

  • Figure 1: This figure shows the dimensionless dissipation rate plotted against Taylor-Reynolds number and is reproduced from the article McComb15a The continuous line labelled as 'Present DNS' is a plot of equation (\ref{['ourdiss']}) with constants given by equation (\ref{['ourconstants']}). The citations to other investigations may be obtained by reference to the article McComb15a
  • Figure 2: Schematic view of the energy spectrum for turbulence at a Reynolds number large enough to exhibit an inertial range.
  • Figure 3: Schematic view of the energy spectrum showing the application of renormalization group to the Navier-Stokes equation for stirred fluid motion and for turbulence.