Critical Reynolds Number as a Topological Phase Transition in Adaptive Fractional Hydrodynamics

We present a theoretical framework that models the laminar-turbulent transition as a topological change in the dissipative operator. The order s of the fractional Laplacian is promoted from a fixed parameter to a dynamic field, governed by a variational principle that minimizes a regularized free-energy functional. This adaptive formulation continuously interpolates between the local, viscous dissipation of the Navier-Stokes equations and the non-local, anomalous dissipation characteristic of the inertial range in Kolmogorov turbulence. From this framework, we derive an analytical expression for the critical Reynolds number, Rec, by establishing a spectral balance condition where the effective dissipative capacity of the laminar operator is saturated.

💡 Research Summary

The paper proposes a novel theoretical framework that treats the laminar‑to‑turbulent transition as a topological phase transition of the dissipative operator in the Navier‑Stokes equations. Instead of the classical Laplacian (−Δ) which represents local viscous diffusion, the authors introduce a fractional Laplacian (−Δ)^s, where the order s∈(0,1] interpolates between the local viscous regime (s→1) and the non‑local, scale‑invariant dissipation observed in the inertial range of Kolmogorov turbulence (s→1/3). Crucially, s is not fixed; it is promoted to a dynamic field s(x,t) that evolves according to a variational principle derived from a regularized free‑energy functional

 F(s)=α(s−s_min)−β ln Re.

Minimizing this functional yields a Fermi‑Dirac‑type transition law

 s(Re)=s_min+(s_max−s_min)/