Wave focusing and related multiple dispersion transitions in plane Poiseuille flows

TL;DR

This paper investigates wave dispersion and focusing phenomena in plane Poiseuille flows, revealing complex dispersive transitions and linear focusing effects through detailed analysis of wave behavior at different parameters.

Contribution

It uncovers new dispersive transition regions and characterizes wave packet morphology, linking linear dispersion features to nonlinear coupling in shear flows.

Findings

Identification of regions with significant dispersion variation

Observation of linear dispersive focusing in wave envelopes

Correlation between dispersive focusing and nonlinear coupling

Abstract

Motivated by the recent discovery of a dispersive-to-nondispersive transition for linear waves in shear flows, we accurately explored the wavenumber-Reynolds number parameter map of the plane Poiseuille flow, in the limit of least-damped waves. We have discovered the existence of regions of the map where the dispersion and propagation features vary significantly from their surroundings. These regions are nested in the dispersive, low-wavenumber part of the map. This complex dispersion scenario demonstrates the existence of linear dispersive focusing in wave envelopes evolving out of an initial, spatially localized, three-dimensional perturbation. An asymptotic wave packet's representation, based on the saddle-point method, allows to enlighten the nature of the packet's morphology, in particular the arrow-shaped structure and spatial spreading rates. A correlation is also highlighted between the regions of largest dispersive focusing and the regions which are most subject to strong nonlinear coupling in observations.