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Decaying Turbulence as a Fractal Curve
Version 1
: Received: 12 May 2023 / Approved: 12 May 2023 / Online: 12 May 2023 (13:43:13 CEST)
Version 2 : Received: 12 May 2023 / Approved: 15 May 2023 / Online: 15 May 2023 (13:47:39 CEST)
Version 3 : Received: 15 May 2023 / Approved: 16 May 2023 / Online: 16 May 2023 (11:57:47 CEST)
Version 4 : Received: 8 June 2023 / Approved: 8 June 2023 / Online: 8 June 2023 (11:07:13 CEST)
Version 5 : Received: 5 July 2023 / Approved: 5 July 2023 / Online: 6 July 2023 (10:17:05 CEST)
Version 2 : Received: 12 May 2023 / Approved: 15 May 2023 / Online: 15 May 2023 (13:47:39 CEST)
Version 3 : Received: 15 May 2023 / Approved: 16 May 2023 / Online: 16 May 2023 (11:57:47 CEST)
Version 4 : Received: 8 June 2023 / Approved: 8 June 2023 / Online: 8 June 2023 (11:07:13 CEST)
Version 5 : Received: 5 July 2023 / Approved: 5 July 2023 / Online: 6 July 2023 (10:17:05 CEST)
How to cite: Migdal, A. Decaying Turbulence as a Fractal Curve. Preprints 2023, 2023050955. https://doi.org/10.20944/preprints202305.0955.v4 Migdal, A. Decaying Turbulence as a Fractal Curve. Preprints 2023, 2023050955. https://doi.org/10.20944/preprints202305.0955.v4
Abstract
We develop a quantitative microscopic theory of decaying Turbulence by studying the dimensional reduction of the Navier-Stokes loop equation for the velocity circulation. We have found an infinite dimensional manifold of solutions of the Navier-Stokes loop equation\cite{M93, M23PR} for the Wilson loop in decaying Turbulence in arbitrary dimension $d >2$. This family of solutions corresponds to a fractal curve in complex space $\mathbb C^d$, described by an algebraic equation between consecutive positions plus a nonlinear periodicity condition. We derive the constrained SDE for the evolution of the fractal curve at a fixed moment of physical time as a function of an auxiliary stochastic time. We expect this stochastic process to cover our fixed manifold of the solutions of the decaying Turbulence. The energy density of the fluid decays as $\mathcal E_0/t$, where $\mathcal E_0$ is an initial dissipation rate. Presumably, we have found a new phase of extreme Turbulence yet to be observed in real or numerical experiments.
Keywords
Turbulence; Fractal; Anomalous dissipation; Fixed point; Velocity circulation; Loop Equations
Subject
Computer Science and Mathematics, Analysis
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.
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Commenter: Alexander Migdal
Commenter's Conflict of Interests: Author
Added the Mathematica code and an Appendix with details of the linearized recurrent equations for this fractal curve