Streamwise localization of traveling wave solutions in channel flow

TL;DR

This paper investigates localized traveling wave solutions in channel flow, demonstrating how their upstream and downstream behavior can be analytically predicted, thus providing a theoretical understanding of exponential localization observed in shear flows.

Contribution

It introduces an analytical method to compute the asymptotics of localized solutions, linking numerical observations with theoretical predictions in shear flow dynamics.

Findings

Analytical asymptotics match numerical exponential localization.

Localized solutions approach laminar flow far upstream and downstream.

The method applies to spanwise uniform states in channel flow.

Abstract

Channel flow of an incompressible fluid at Reynolds numbers above 2400 possesses a number of different spatially localized solutions that approach laminar flow far upstream and downstream. We use one such relative time-periodic solution, which corresponds to a spatially localized version of a Tollmien-Schlichting wave, to illustrate how the upstream and downstream asymptotics can be computed analytically. In particular, we show that for these spanwise uniform states the asymptotics predict the exponential localization that has been observed for numerically computed solutions of several canonical shear flows but never properly understood theoretically.