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I'm actually studying the famous paper of Kolmogorov (1941).

I am trying to derive the passages that deduces the fact that the Energy spectrum has a dependency $k^{-5/3}$. As far as I know, the dependency is demonstrated in the Obukhov 41 paper, but this paper seems disappered. I already read the book "Turbulence: The Legacy of A. N. Kolmogorov" written by Frisch, but I didn't find a clear demonstration that lead to the Energy spectrum dependency $k^{-5/3}$. Can you help me? Do you know papers or theses that show this? Or can you give me the total demonstration? Thanks in advance

Edit:

Some time ago, someone suggested me to follow the theory on this book https://it.scribd.com/document/198237823/Tatarski-Wave-Propagation-in-a-Turbulent-Medium-1961#download but it states that the dependency of the 3D Energy spectrum is $k^{-11/3}$ instead of $k^{-5/3}$. This confused me more.

Ashish Bhigah
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    Are you looking for something like this? – 2b-t Apr 01 '20 at 16:01
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    Thank you for your anser, but I was searching for a demonstration that starts from the Kolmogorov paper. I have already seen this dimensional analysis. – Ashish Bhigah Apr 01 '20 at 16:15
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    I think the dimensional analysis @2b-t linked to is the way it is derived. The K41 paper outlines the hypotheses of turbulence, and then dimensional analysis is what determines the exponent. That's consistent with how the K41 paper determines all the other relationships they lay out -- it's all based on dimensional analysis definitions. Not sure you'll find "better" demonstrations than dimensional analysis since that's where it comes from... – tpg2114 Apr 01 '20 at 18:09
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    @tpg2114 I do not know if you have read this book https://epdf.pub/turbulence-the-legacy-of-an-kolmogorov.html (the one that i cited in the first post). It explains some passages that leads to the dependency $f^{-5/3}$ (page 73), and the demonstration it is not the same. The paper of Kolmogorv does not outlines only the hypotheses but introduce the so called 2/3 law that leads to the -5/3 law (I am missing this passage) – Ashish Bhigah Apr 01 '20 at 21:00
  • Related https://physics.stackexchange.com/q/53818/226902 – Quillo Sep 29 '22 at 07:33

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If you're looking for a rigorous proof from first principles, there isn't, because that spectrum is not the right one (at least that's what experiments show, you can check the paper 'Lessons from Hydrodynamic Turbulence' by Falkovich and Sreenivasan for a nice short explanation). Basically there is a phenomenon called 'anomalous scaling', similar to the anomalous dissipation the we observe in turbulence as well. This means that even when we are 'far away' from the source that breaks the scaling symmetry (the forcing), the symmetry breaking effect persists (in the case of anomalous dissipation this means that even as the viscosity tends to 0 there is finite dissipation).

And what's the effect of this anomaly in the scaling exponents? Well, one of Kolmogorov's hypothesis is that there is an inertial range where the symmetries of the flow are recovered (because this range is far from both, the dissipation and the forcing scale), particularly the scaling symmetry, and from this you can derive the -5/3 law from dimensional analysis (in real space you derive the second order structure function exponent). The only exact exponent is that of the third-order structure function because it is derived exactly from the Karman-Howarth equation when you assume anomalous dissipation (and of course, all of this in the context of Homogeneous and Isotropic turbulence).

AdBahamonde
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