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

TL;DR

This paper introduces a novel theoretical model that describes the laminar-turbulent transition as a topological phase change in a dynamically adaptive fractional hydrodynamics framework, linking dissipation mechanisms to turbulence onset.

Contribution

It develops a new variational approach where the fractional Laplacian order varies, providing an analytical expression for the critical Reynolds number based on spectral balance.

Findings

Derived an analytical formula for Rec.

Unified local and non-local dissipation mechanisms.

Identified topological change as a transition marker.

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.