Capturing the Edge of Chaos in a Reduced-order Model of Pipe Flow

Unstable exact coherent states (ECSs) are thought to form the core of turbulence, a state coexisting with the laminar state in some canonical shear flow configurations. We describe pipe flow from a dynamical system point of view, by focusing on the unstable and stable manifolds of these distinguished solutions that organize the geometry of the phase space. The boundary between the basins of attraction of the laminar and turbulent states is often called the edge of chaos, which is the stable manifold of an edge state in pipe flow. Accurate representation of the edge of chaos is essential for successful reduced-order modeling of this system to predict the transition to turbulence. Using simulated velocity data, we construct a low-dimensional submanifold of the edge of chaos using the recently developed theory of spectral submanifolds (SSMs). These manifolds are the unique smoothest nonlinear continuations of nonresonant spectral subspaces of the linearized system at stationary states. Motivated by the success of spectral submanifold-based reduced-order models for laminar Couette-flow, we describe the construction of a two-dimensional spectral submanifold attached to the edge state which results in a two-dimensional model of pipe flow that can predict the transition to turbulence for velocity fields near the edge of chaos.