preprints.org > physical sciences > theoretical physics > doi: 10.20944/preprints202308.2042.v1

Preprint Article Version 1 Preserved in Portico This version is not peer-reviewed

Version 1 : Received: 29 August 2023 / Approved: 29 August 2023 / Online: 30 August 2023 (09:01:51 CEST)
Version 2 : Received: 31 August 2023 / Approved: 4 September 2023 / Online: 4 September 2023 (12:00:54 CEST)
Version 3 : Received: 6 September 2023 / Approved: 7 September 2023 / Online: 7 September 2023 (09:28:55 CEST)
Version 4 : Received: 15 September 2023 / Approved: 18 September 2023 / Online: 18 September 2023 (13:56:35 CEST)

We have found an infinite dimensional manifold of exact solutions of the Navier-Stokes loop equation for the Wilson loop in decaying Turbulence in arbitrary dimension d>2. This solution family is equivalent to a fractal curve in complex space Cd with random steps parametrized by N Ising variables σi=±1, in addition to a rational number pq and an integer winding number r, related by ∑σi=qr. This equivalence provides a dual theory describing a strong turbulent phase of the Navier-Stokes flow in Rd space as a random geometry in a different space, like ADS/CFT correspondence in gauge theory. This is a quantum statistical system with integer-valued parameters, satisfying some number theory constraints. Its long-range interaction leads to critical phenomena when its size N→∞ or its chemical potential μ→0. The system with fixed N has different asymptotics at odd and even N→∞, but the limit μ→0 is well-defined. The energy dissipation rate is analytically calculated as a function of μ using methods of number theory. It grows as ν/μ2 in the continuum limit μ→0, leading to anomalous dissipation at μ∝ν→0. The same method applies to other observables.

Turbulence, Fractal, Anomalous dissipation, Fixed point, Velocity circulation, Loop Equations, Euler Phi, Prime numbers

Physical Sciences, Theoretical Physics

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