I have asked a few questions on Kolmogorov scaling before but am still struggling to understand it due to the many different notations and conventions used - as well as a lack of any rigorous resource (for which I can find). From what I can tell, there are 3 scales of interest:

  • The external or fundamental scale, $L$, is that where energy enters the system. This is the largest of the scales, and $L$ is fixed.
  • The inertial scale, $\lambda$. I am still unsure how to formally define this one, but $\lambda$ can take a range of values and is not fixed.
  • The internal scale , $\lambda_0$, is that where viscous effects take hold and dispersion starts. This scale can be fixed such that $Re=1$ at this point.

As well as these three scales we are interested in the order of magnitude of quantities such as, $\varepsilon$, the mean energy dissipated per unit time per unit mass as well as how different quantities such as $E(k)$ and velocity scale in different regions. So my question is this:

Please can you provide a list of the main results of Kolmogorov theory in each of the 3 regions and a brief explanation of how they are derived/where they come from?

In your answer please can you stick strictly to the notation $\propto$ meaning proportional to and $\sim$ meaning of the order of.

P.S. I am aware this this question it a little broad, but due to the lack of any easily accessible clear explanations of this subject (at least in my opinion) I am running the risk of it been closed since I think any good answer to will be of large benefit to others.


closed as too broad by tpg2114, Kyle Kanos, CR Drost, John Rennie, Quantum spaghettification Mar 13 '17 at 19:13

Please edit the question to limit it to a specific problem with enough detail to identify an adequate answer. Avoid asking multiple distinct questions at once. See the How to Ask page for help clarifying this question. If this question can be reworded to fit the rules in the help center, please edit the question.

  • 3
    $\begingroup$ Have you read A First Course in Turbulence? It provides the best description of things in my view. But, it takes an entire book to explain the things you are asking, so I don't know that it is answerable here. $\endgroup$ – tpg2114 Mar 13 '17 at 13:26
  • $\begingroup$ @tpg2114 (see the comments in this). All I am asking for is the main results e.g. $E(k) \propto \varepsilon^{2/3} k^{-5/3}$ laid out in consistent notation and with a concise explanation. Landau and Lifshitz do something similar in about 3 pages so I can't see how this can take an entire book. $\endgroup$ – Quantum spaghettification Mar 13 '17 at 13:46
  • $\begingroup$ Well, if Landau and Lifshitz did it in 3 pages and it was sufficient for you to understand, then you wouldn't be asking the question here right? So it needs more explanation and details that is given there, and if it took 3 pages and was too brief to understand, then it's certainly too long for the format of this site. $\endgroup$ – tpg2114 Mar 13 '17 at 14:09
  • $\begingroup$ Somebody may be able to come along and help you with it. But it's definitely too much for me to try and tackle, and I think it's going to be a really lengthy answer to explain things in any sufficient detail. $\endgroup$ – tpg2114 Mar 13 '17 at 14:11