Natural convection in a vertical channel. Part 3. Bifurcations of many (additional) unstable periodic orbits and their dynamical relevance

Abstract

Vertical thermal convection system exhibits weak turbulence and spatio-temporally chaotic behaviour. In this system, we report seven equilibria and 26 periodic orbits, all new and linearly unstable. These orbits, together with four previously studied in Zheng et al. (2024b) bring the number of periodic orbit branches computed so far to 30, all solutions to the fully non-linear three-dimensional Navier--Stokes equations. These new invariant solutions capture intricate spatio-temporal flow patterns including straight, oblique, wavy, skewed and distorted convection rolls, as well as bursts and defects in rolls. These interesting and important fluid mechanical processes in a small flow unit are shown to appear locally and instantaneously in a chaotic simulation in a large domain. Most of the solution branches show rich spatial and/or spatio-temporal symmetries. The bifurcation-theoretic organisation of these solutions is discussed; the bifurcation scenarios include Hopf, pitchfork, saddle--node, period-doubling, period-halving, global homoclinic and heteroclinic bifurcations, as well as isolas. These orbits are shown to be able to reconstruct statistically the core part of the attractor, and these results may pave the way to quantitatively describing transitional fluid turbulence using periodic orbit theory.

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