The critical layer in pipe flow at high Reynolds number

TL;DR

This paper computes traveling wave solutions in high Reynolds number pipe flow, revealing the development of a critical layer and unexpected stability trends as viscosity decreases, with implications for shear flow universality.

Contribution

It introduces high-Reynolds-number traveling wave solutions in pipe flow, highlighting the emergence of a critical layer and stability behavior, along with new computational methods.

Findings

Solutions develop a critical layer away from the wall at high Re

Unstable eigenvalues approach zero as Re increases, indicating increased stability

Methodological advancements in GMRES-hookstep and Arnoldi iterations

Abstract

We report the computation of a family of traveling wave solutions of pipe flow up to $Re=75000$. As in all lower-branch solutions, streaks and rolls feature prominently in these solutions. For large $Re$, these solutions develop a critical layer away from the wall. Although the solutions are linearly unstable, the two unstable eigenvalues approach 0 as $Re\to\infty$ at rates given by $Re^{-0.41}$ and $Re^{-0.87}$ -- surprisingly, the solutions become more stable as the flow becomes less viscous. The formation of the critical layer and other aspects of the $Re\to\infty$ limit could be universal to lower-branch solutions of shear flows. We give implementation details of the GMRES-hookstep and Arnoldi iterations used for computing these solutions and their spectra, while pointing out the new aspects of our method.