Dispersive to non-dispersive transition and phase velocity transient for linear waves in plane wake and channel flows

TL;DR

This paper investigates the dispersive to non-dispersive transition of phase velocity in linear waves within plane wake and channel flows, revealing how wavelength and eigenvalue spectrum influence wave behavior and transient phase velocity oscillations.

Contribution

It provides a detailed analysis of dispersive effects and phase velocity transients in shear flows, identifying a sharp transition threshold and linking transient oscillations to eigenvalue spectrum width.

Findings

Dispersive behavior occurs at low wavenumbers.

A sharp transition to non-dispersive behavior at a specific wavelength threshold.

Transient phase velocity oscillates with a frequency related to the eigenvalue spectrum width.

Abstract

In this study we analyze the phase and group velocity of three-dimensional linear traveling waves in two sheared flows, the plane channel and the wake flows. This was carried out by varying the wave number over a large interval of values at a given Reynolds number inside the ranges 20-100, 1000-8000, for the wake and channel flow, respectively. Evidence is given about the possible presence of both dispersive and non-dispersive effects which are associated with the long and short ranges of wavelength. We solved the Orr-Sommerfeld and Squire eigenvalue problem and observed the least stable mode. It is evident that, at low wave numbers, the least stable eigenmodes in the left branch of the spectrum beave in a dispersive manner. By contrast, if the wavenumber is above a specific threshold, a sharp dispersive to non-dispersive transition can be observed. Beyond this transition, the dominant mode belongs to the right branch of the spectrum. The transient behavior of the phase velocity of small three-dimensional traveling waves was also considered. Having chosen the initial conditions, we then show that the shape of the transient highly depends on the transition wavelength threshold value. We show that the phase velocty can oscillate with a frequency which is equal to the frequency width of the eigenvalue spectrum. Furthermore, evidence of intermediate self-similarity is given for the perturbation field.