Wave focusing and related multiple dispersion transitions in plane Poiseuille flows

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 leastdamped 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.

Global stability, sensitivity and passive control of low-Reynolds-number flows around NACA 4412 swept wings

“…As shown in figure 11( b ), whilst the frequency of the three-dimensional centrifugal modes shows a linear trend and therefore a non-dispersive behaviour, that of the von Kármán modes has a weakly parabolic behaviour, suggesting a slight dispersion in the spanwise direction (Fraternale, Nastro & Tordella 2021). Introducing a sweep angle induces a monotonic behaviour of the frequency curves with respect to the spanwise wavenumber.…”