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.