Harbingers and latecomers - The order of appearance of exact coherent structures in plane Poiseuille flow

TL;DR

This paper investigates the emergence and properties of exact coherent structures in plane Poiseuille flow, highlighting their bifurcation points, localization, and comparison with other shear flows, to better understand transition to turbulence.

Contribution

It identifies and compares various exact coherent structures in PPF, including their bifurcation Reynolds numbers and localization characteristics, revealing the lowest thresholds for such states.

Findings

Tollmien-Schlichting waves bifurcate at Re=5772 and reach down to Re=2609.

Localized solutions bifurcate at Re=2334, lower than the global ones.

Spanwise localized solutions appear at Re=316, likely the lowest in PPF.

Abstract

The transition to turbulence in plane Poiseuille flow (PPF) is connected with the presence of exact coherent structures. In contrast to other shear flows, PPF has a number of different coherent states that are relevant for the transition. We here discuss the different states, compare the critical Reynolds numbers and optimal wavelengths for their appearance, and explore the differences between flows operating at constant mass flux or at constant pressure drop. The Reynolds numbers quoted here are based on the mean flow velocity and refer to constant mass flux, the ones for constant pressure drop are always higher. The Tollmien-Schlichting waves bifurcate subcritically from the laminar profile at $Re=5772$ and reach down to $Re=2609$ (at a different optimal wave length). Their localized counter part bifurcates at the even lower value $Re=2334$. Three dimensional exact solutions appear at much lower Reynolds numbers. We describe one exact solutions that is spanwise localized and has a critical Reynolds number of $316$. Comparison to plane Couette flow suggests that this is likely the lowest Reynolds number for exact coherent structures in PPF. Streamwise localized versions of this state require higher Reynolds numbers, with the lowest bifurcation occurring near $Re=1018$.