Natural convection in a vertical channel. Part 2. Oblique solutions and global bifurcations in a spanwise-extended domain

Vertical thermal convection is a non-equilibrium system in which both buoyancy and shear forces play a role in driving the convective flow. Beyond the onset of convection, the driven dissipative system exhibits chaotic dynamics and turbulence. In a three-dimensional domain extended in both the verti...

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Bibliographic Details
Published inarXiv.org
Main Authors Zheng, Zheng, Tuckerman, Laurette S, Schneider, Tobias M
Format Paper
LanguageEnglish
Published Ithaca Cornell University Library, arXiv.org 04.09.2024
Subjects
Bifurcation theory
Boussinesq equations
Chaos theory
Convective flow
Diamonds
Divergence
Dynamic stability
Dynamic systems theory
Dynamical systems
Free convection
Hydrodynamic configurations
Nonlinear dynamics
Shear forces
Stability analysis
Symmetry
System theory
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Summary:Vertical thermal convection is a non-equilibrium system in which both buoyancy and shear forces play a role in driving the convective flow. Beyond the onset of convection, the driven dissipative system exhibits chaotic dynamics and turbulence. In a three-dimensional domain extended in both the vertical and the transverse dimensions, Gao et al. (2018) have observed a variety of convection patterns which are not described by linear stability analysis. We investigate the fully non-linear dynamics of vertical convection using a dynamical-systems approach based on the Oberbeck-Boussinesq equations. We compute the invariant solutions of these equations and the bifurcations that are responsible for the creation and termination of various branches. We map out a sequence of local bifurcations from the laminar base state, including simultaneous bifurcations involving patterned steady states with different symmetries. This atypical phenomenon of multiple branches simultaneously bifurcating from a single parent branch is explained by the role of D4 symmetry. In addition, two global bifurcations are identified: first, a homoclinic cycle from modulated transverse rolls and second, a heteroclinic cycle linking two symmetry-related diamond-roll patterns. These are confirmed by phase space projections as well as the functional form of the divergence of the period close to the bifurcation points. The heteroclinic orbit is shown to be robust and to result from a 1:2 mode interaction. The intricacy of this bifurcation diagram highlights the essential role played by dynamical systems theory and computation in hydrodynamic configurations.
ISSN:2331-8422