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Abstract

Thermal convection governed by the Navier–Stokes–Boussinesq equations provides a canonical framework for analyzing instability, bifurcation, and transitions to complex dynamics. This thesis studies convection in a two-dimensional annulus with vertical heating. The fluid is confined between an inner and an outer circular boundary, and the flow is periodic in the azimuthal direction. The velocity satisfies no-slip conditions at both boundaries. At the inner boundary the temperature is thermally insulated, while at the outer boundary it is prescribed to vary linearly with height. This annular configuration, first analyzed by Huang and Moore, is closely related to classical Rayleigh–Bénard convection while retaining azimuthal periodicity. Chapter 2 presents the governing equations specialized to this geometry and reviews a low-order truncation expressed in terms of angular momentum and temperature center-of-mass coordinates. Chapter 3 develops a Fourier–Chebyshev pseudo-spectral solver with semi-implicit time integration and verifies its accuracy by convergence tests. Chapter 4 revisits the conductive reference state, showing that under the present boundary conditions a strictly motionless equilibrium is not admitted, and an analytically corrected weak-flow baseline is constructed. Chapter 5 reports direct numerical simulations for two representative Prandtl numbers. For the larger Prandtl number, the large-scale dynamics remain broadly Lorenz-like, with circulation and reversals, while also exhibiting shear-layer instabilities and Rayleigh–Taylor-type plume detachment. For the smaller Prandtl number, the loss of steadiness is dominated by shear-related mechanisms. Bifurcation diagrams compiled from both simulations and the truncated model—using time-averaged angular momentum, Nusselt number, and Reynolds number—show that the reduced system captures the principal state transitions, although quantitative thresholds and finer-scale features may differ. An appendix provides the explicit mapping between the truncated annular model and the Lorenz system, clarifying the scope of reduced-order interpretations in this geometry.