Preprint Article Version 2 Preserved in Portico This version is not peer-reviewed

Dual Theory of Decaying Turbulence. 2. Numerical Simulation and Continuum Limit

Version 1 : Received: 16 December 2023 / Approved: 18 December 2023 / Online: 19 December 2023 (09:40:22 CET)
Version 2 : Received: 19 December 2023 / Approved: 20 December 2023 / Online: 20 December 2023 (16:43:56 CET)
Version 3 : Received: 23 December 2023 / Approved: 25 December 2023 / Online: 26 December 2023 (10:00:43 CET)
Version 4 : Received: 26 December 2023 / Approved: 26 December 2023 / Online: 27 December 2023 (09:38:03 CET)
Version 5 : Received: 27 December 2023 / Approved: 27 December 2023 / Online: 27 December 2023 (12:38:44 CET)

How to cite: Migdal, A. Dual Theory of Decaying Turbulence. 2. Numerical Simulation and Continuum Limit. Preprints 2023, 2023121357. https://doi.org/10.20944/preprints202312.1357.v2 Migdal, A. Dual Theory of Decaying Turbulence. 2. Numerical Simulation and Continuum Limit. Preprints 2023, 2023121357. https://doi.org/10.20944/preprints202312.1357.v2

Abstract

This is the second paper in a cycle investigating the exact solution of loop equations in decaying turbulence. We perform numerical simulations of the Euler ensemble, suggested in the previous work, as a solution to the loop equations. We designed novel algorithms for simulation, which take a small amount of computer RAM so that only the CPU time grows linearly with the size of the system and its statistics. This algorithm allows us to simulate systems with billions of discontinuity points on our fractal curve, dual to the decaying turbulence. For the vorticity correlation function, we obtain quantum scaling laws with regime changes between effective indexes drifting with the logarithm of scale. The traditional description of turbulence with fractal or multifractal scaling laws does not apply here. In particular, the effective fractal dimension of our curve is a function of the scale. The measured conditional probabilities of fluctuating variables are smooth functions of the logarithm of scale, with statistical errors being negligible at our large number of random samples $T = 167,772,160$ with $N = 200,000,000$ points on a fractal curve. The quantum jumps arise from the analytical distribution of scaling variable $X = \frac{\cot^2(\pi p/q)}{q^2}$ where $\frac{p}{q}$ is a random fraction. This distribution is related to the totient summatory function and is a discontinuous function of $X$. In particular, the energy spectrum has quantum levels on top of a continuous background.

Keywords

turbulence; fractal; anomalous dissipation; fixed point; velocity circulation; numerical simulations; GPU; loop equations

Subject

Physical Sciences, Fluids and Plasmas Physics

Comments (1)

Comment 1
Received: 20 December 2023
Commenter: Alexander Migdal
Commenter's Conflict of Interests: Author
Comment: Modified several formulast u=in the last section. The rest of the results didn't change
+ Respond to this comment

We encourage comments and feedback from a broad range of readers. See criteria for comments and our Diversity statement.

Leave a public comment
Send a private comment to the author(s)
* All users must log in before leaving a comment
Views 0
Downloads 0
Comments 1


×
Alerts
Notify me about updates to this article or when a peer-reviewed version is published.
We use cookies on our website to ensure you get the best experience.
Read more about our cookies here.