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

TL;DR

This work reframes the laminar–turbulent transition as a topological change in the fluid’s dissipative operator by promoting the fractional Laplacian order $s$ to a dynamic field $s(\boldsymbol{x},t)$. An adaptive fractional Navier–Stokes (AFNS) model couples a variationally derived free-energy functional to a local Reynolds number $Re_\ell$ to produce a Fermi–Dirac–type transition in $s$, bridging local diffusion ($s\approx 1$) and non-local inertial-range dissipation ($s\approx 1/3$). A central result is an analytical expression for the critical Reynolds number $Re_c$ in terms of a geometric spectral constant $\mathcal{K}_\Omega$ and the ratio of operator weights, $Re_c \sim \mathcal{K}_\Omega [\mathcal{W}(s_{lam})/\mathcal{W}(s_{min})]^{1/\Delta s}$, yielding values of order $10^3$ that align with pipe, channel, and Couette transitions as lower bounds to metastability. The framework naturally accounts for 3D–2D dimensionality effects, reproduces Kolmogorov scaling with $s_{min}=1/3$ as the dissipative fixed point, and predicts a fractal dissipation geometry with $D \approx 3 - s$, consistent with observed vorticity structures. By embedding dissipation in an adaptive, topology-dependent operator, the AFNS model offers a parameter-free, first-principles pathway to unify laminar dissipation, turbulence onset, and fractal scaling within a single theoretical structure, with potential numerical validation via DNS.

Abstract

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.