Hopf bifurcations to quasi-periodic solutions for the two-dimensional plane Poiseuille flow

TL;DR

This study investigates Hopf bifurcations in two-dimensional plane Poiseuille flow, revealing the stability and nature of quasi-periodic solutions and introducing a numerical scheme for analyzing complex bifurcations in fluid dynamics.

Contribution

The paper improves the precision of bifurcation analysis in plane Poiseuille flow and develops a novel numerical scheme for studying quasi-periodic solutions in fluid flows.

Findings

Bifurcating flows are unstable and move forward with respect to Reynolds number.

Stable quasi-periodic solutions and 3-tori are identified at certain bifurcations.

The proposed numerical method efficiently analyzes complex bifurcations in Navier-Stokes equations.

Abstract

This paper studies various Hopf bifurcations in the two-dimensional plane Poiseuille problem. For several values of the wavenumber $\alpha$, we obtain the branch of periodic flows which are born at the Hopf bifurcation of the laminar flow. It is known that, taking $\alpha\approx1$, the branch of periodic solutions has several Hopf bifurcations to quasi-periodic orbits. For the first bifurcation, previous calculations seem to indicate that the bifurcating quasi-periodic flows are stable and go backwards with respect to the Reynolds number, $Re$. By improving the precision of previous works we find that the bifurcating flows are unstable and go forward with respect to $Re$. We have also analysed the second Hopf bifurcation of periodic orbits for several $\alpha$, to find again quasi-periodic solutions with increasing $Re$. In this case the bifurcated solutions are stable to superharmonic disturbances for $Re$ up to another new Hopf bifurcation to a family of stable 3-tori. The proposed numerical scheme is based on a full numerical integration of the Navier-Stokes equations, together with a division by 3 of their total dimension, and the use of a pseudo-Newton method on suitable Poincar\'e sections. The most intensive part of the computations has been performed in parallel. We believe that this methodology can also be applied to similar problems.