Effect of non-uniform heating on cylindrical Rayleigh–Bénard instability

In this study, we analyze the cylindrical Rayleigh–Bénard convection using linear stability analysis, while introducing radially non-uniform heating at the bottom wall prescribed as a Gaussian function of the radial coordinate. We vary the standard deviation σ of this radial profile and the Rayleigh number Ra based on the radially averaged temperature, which characterizes the relative importance of thermal diffusion and buoyancy-driven convection time scales, to observe their impact on the steady flow and the stability properties. For Ra=1000, we identified five distinct axisymmetric steady-state solution branches that satisfy the same governing equations and boundary conditions resulting from two bifurcations occurring at σ−1=0.223 and σ−1=0.270. Interestingly, the first bifurcation point coincides with a loss of stability with respect to non-axisymmetric azimuthal perturbations, in this case m=1, pointing to the irremediable loss of axisymmetry of the flow. We observe that the critical Rayleigh number Rac, i.e., for the onset of instability, decreases with σ−1 before reaching a saturation around Rac=600 and σ−1>0.480. Modifying the heating distribution by increasing the parameter σ−1 results in a notable reduction of the critical Rayleigh number. This alteration facilitates the initiation of instability at lower Rayleigh numbers than those observed in the context of uniformly heated Rayleigh–Bénard convection. This investigation holds relevance for a range of practical applications such as atmospheric and oceanic circulation, magma transport, and convection in diverse engineering systems.