preprints.org > computer science and mathematics > analysis > doi: 10.20944/preprints202305.0955.v3

Preprint Article Version 3 Preserved in Portico This version is not peer-reviewed

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)

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. The probability measure is explicitly constructed in terms of products of conventional measures for orthogonal group $SO(d)$ and a sphere $\mathbb S^{d-3}$. In three dimensions $d=3$, we compute a fractal dimension $d_f = 1.39$ for this fractal curve and the step size PDF with fat tail $x^{-2.} $. We also compute the enstrophy PDF with fat tail $x^{-5/3}$, corresponding to an infinite mean value (anomalous dissipation). 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 not yet observed in real or numerical experiments.

Turbulence; Fractal; Anomalous dissipation; Fixed point; Velocity circulation; Loop Equations

Computer Science and Mathematics, Analysis

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