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Invariant states in inclined layer convection. Part 1. Temporal transitions along dynamical connections between invariant states

Published online by Cambridge University Press:  08 July 2020

Florian Reetz  and
Tobias M. Schneider Open the ORCID record for Tobias M. Schneider [Opens in a new window]
Florian Reetz
Affiliation:
Emergent Complexity in Physical Systems Laboratory (ECPS), Ecole Polytechnique Fédérale de Lausanne, CH-1015, Switzerland
Tobias M. Schneider*
Affiliation:
Emergent Complexity in Physical Systems Laboratory (ECPS), Ecole Polytechnique Fédérale de Lausanne, CH-1015, Switzerland
*
†Email address for correspondence: [email protected]
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Abstract

Thermal convection in an inclined layer between two parallel walls kept at different fixed temperatures is studied for fixed Prandtl number $\mathit{Pr}=1.07$. Depending on the angle of inclination and the imposed temperature difference, the flow exhibits a large variety of self-organized spatio-temporal convection patterns. Close to onset, these patterns have been explained in terms of linear stability analysis of primary and secondary flow states. At a larger temperature difference, far beyond onset, experiments and simulations show complex, dynamically evolving patterns that are not described by stability analysis and remain to be explained. Here we employ a dynamical systems approach. We construct stable and unstable exact invariant states, including equilibria and periodic orbits of the fully nonlinear three-dimensional Oberbeck–Boussinesq equations. These invariant states underlie the observed convection patterns beyond their onset. We identify state space trajectories that, starting from the unstable laminar flow, follow a sequence of dynamical connections between unstable invariant states until the dynamics approaches a stable attractor. Together, the network of dynamically connected invariant states mediates temporal transitions between coexisting invariant states and thereby supports the observed complex time-dependent dynamics in inclined layer convection.

JFM classification

Nonlinear Dynamical Systems: Pattern formation Instability: Nonlinear instability
Type
JFM Papers
Information
Journal of Fluid Mechanics , Volume 898 , 10 September 2020 , A22
Copyright
© The Author(s), 2020. Published by Cambridge University Press

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