Streamwise decay of localized states in channel flow

TL;DR

This paper investigates the streamwise decay properties of localized coherent structures in channel flow, deriving analytical expressions that match observed exponential decay patterns and extending previous stationary state analyses to advected structures.

Contribution

It extends the analysis of localized states in channel flow to advected structures, providing analytical decay expressions based on eigenvalues and eigenfunctions.

Findings

Decay constants differ on upstream and downstream sides.

Analytical decay expressions agree with observed structures.

Extension of stationary state analysis to advected states.

Abstract

Channel flow, the pressure driven flow between parallel plates, has exact coherent structures that show various degrees of localization. For states which are localized in streamwise direction but extended in spanwise direction, we show that they are exponentially localized, with decay constants that are different on the upstream and downstream side. We extend the analysis of Brand and Gibson [J. Fluid Mech. 750, R1 (2014)] for stationary states to the case of advected structures that is needed here, and derive expressions for the decay in terms of eigenvalues and eigenfunctions of certain second order differential equations. The results are in very good agreement with observations on exact coherent structures of different transversal wave lengths.