The Signature of Chaos

Abstract The transition from laminar to turbulent flow is a fundamental, yet incompletely understood, phenomenon in fluid dynamics. Traditional global metrics often fail to capture the local onset of chaos. In this paper, we propose a novel methodology based on information theory to quantify fluidic complexity in three-dimensional flows. Using Smoothed-Particle Hydrodynamics (SPH) simulations of flow over airfoil profiles and spherical obstacles, we demonstrate that a local, information-centric metric—the Shannon entropy of the three-dimensional velocity field expressed in spherical coordinates—serves as a robust and unambiguous detector for the critical transition to turbulence. Our results show a distinct phase transition in which the Local Entropy Index (LEI) exhibits a sharp, non-linear increase as the Reynolds number crosses a critical threshold. This three-dimensional extension captures complex spatial structures such as vortex rings, helical instabilities, and asymmetric wake patterns characteristic of aerodynamic flows. Through high-fidelity simulations scaling from 100,000 to 600,000 particles, we demonstrate that the LEI exhibits mesh-independent convergence, validating its physical significance as a topological chaos detector. Crucially, we show that the LEI decouples mass from information, detecting turbulent complexity even in rarefied flow regimes where energy-based metrics fail. Building on these empirical findings, we derive a theoretical framework based on an Information Transport Equation, which explains turbulence as a dynamic balance between: Boundary-layer information generation Viscous information dissipation This culminates in the proposal of the Navier–Stokes–I (NSI) system—a modified set of governing equations in which local entropy acts as a physical field, exerting pressure on the fluid and creating a self-sustaining feedback loop that explains the birth and persistence of turbulent chaos. andrespirolo@gmail.com