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. Preprints2023, 2023121357. https://doi.org/10.20944/preprints202312.1357.v1
Migdal, A. Dual Theory of Decaying Turbulence. 2. Numerical Simulation and Continuum Limit. Preprints 2023, 2023121357. https://doi.org/10.20944/preprints202312.1357.v1
Migdal, A. Dual Theory of Decaying Turbulence. 2. Numerical Simulation and Continuum Limit. Preprints2023, 2023121357. https://doi.org/10.20944/preprints202312.1357.v1
APA Style
Migdal, A. (2023). Dual Theory of Decaying Turbulence. 2. Numerical Simulation and Continuum Limit. Preprints. https://doi.org/10.20944/preprints202312.1357.v1
Chicago/Turabian Style
Migdal, A. 2023 "Dual Theory of Decaying Turbulence. 2. Numerical Simulation and Continuum Limit" Preprints. https://doi.org/10.20944/preprints202312.1357.v1
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.
Copyright:
This is an open access article distributed under the Creative Commons Attribution License which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
