Variational determination of minimal absorbing zones in incompressible shear flows

TL;DR

This paper introduces a variational method to identify minimal absorbing zones in incompressible shear flows, providing new insights into flow dynamics and potential ways to estimate turbulent mean states without extensive simulations.

Contribution

It develops a novel variational approach to determine minimal absorbing zones, advancing understanding of shear flow attractors and turbulence characteristics.

Findings

Absorbing zones act as attractors for all trajectories in shear flows.

The minimal absorbing zone's centroid approximates turbulent mean flow.

Method offers a new foundation for stability analysis and turbulence modeling.

Abstract

The dynamical analysis of shear flows remains challenging, as turbulence generation and evolution are not fully understood. Here, a lesser-explored feature of incompressible shear flows-the absorbing zone-is investigated. This region in the infinite dimensional state space is shown to act as an attractor for all trajectories: any solution initialised outside eventually enters and remains inside. Consequently, the absorbing zone must contain all possible attractors, both chaotic and non-chaotic. Existence is established through the Reynolds-Orr identity, which indicates that nonlinear terms do not directly influence the temporal evolution of kinetic energy. The zone is constructed around a so-called shift flow, and multiple such regions may exist, even when the laminar state is linearly unstable. Gradient based optimisation is employed to identify the absorbing zone of minimal radius. Assuming a chaotic trajectory explores state space randomly, it is hypothesised that the centroid of this minimal zone approximates the turbulent mean flow. This central state is computed for plane Poiseuille and Couette flows and compared with established turbulent mean profiles. Although the proposed hypothesis is found to be an oversimplification-yielding profiles that qualitatively resemble but do not quantitatively reproduce the turbulent means-the methodology provides novel insights into shear flow dynamics. Furthermore, it offers a promising foundation for refining estimates of global stability thresholds. With continued development, this framework may facilitate the direct computation of turbulent mean states without reliance on empirical turbulence models or time-dependent numerical simulations.