The role of long waves in the stability of the plane wake

TL;DR

This paper investigates the stability of long three-dimensional perturbation waves in a plane wake using multiscale analysis and numerical methods, revealing persistent asymptotic instability regardless of initial conditions.

Contribution

It introduces a multiscale analytical framework for analyzing long wave perturbations in wakes, providing new insights into their transient behavior and stability characteristics.

Findings

Amplification factor of transversal perturbations does not show typical growth-damping trend.

Asymptotic instability is consistently observed in long waves.

Numerical results align with first-order analysis, validating the approach.

Abstract

This work is directed towards investigating the fate of three-dimensional long perturbation waves in a plane incompressible wake. The analysis is posed as an initial-value problem in space. More specifically, input is made at an initial location in the downstream direction and then tracing the resulting behavior further downstream subject to the restriction of finite kinetic energy. This presentation follows the outline given by Criminale and Drazin [Stud. in Applied Math. \textbf{83}, 123 (1990)] that describes the system in terms of perturbation vorticity and velocity. The analysis is based on large scale waves and expansions using multi scales and multi times for the partial differential equations. The multiscaling is based on an approach where the small parameter is linked to the perturbation property independently from the flow control parameter. Solutions of the perturbative equations are determined numerically after the introduction of a regular perturbation scheme analytically deduced up to the second order. Numerically, the complete linear system is also integrated. Since the results relevant to the complete problem are in very good agreement with the results of the first order analysis, the numerical solution at the second order was deemed not necessary. The use for an arbitrary initial-value problem will be shown to contain a wealth of information for the different transient behaviors associated to the symmetry, angle of obliquity and spatial decay of the long waves. The amplification factor of transversal perturbations never presents the trend - a growth followed by a long damping - usually seen in waves with wavenumber of order one or less. Asymptotical instability is always observed.