Multicelluar thermal convection in tall slot cavities: Influences of Prandtl number and abstract ratio on structure and stability

Vertical thermal convection in tall slot geometries gives rise to the onset and persistence of multicellular vortical structures, where multiple vertically stacked rolls exist upon each other. In this study, we investigate the mechanisms of the onset of multicellular vertical structures and roll-states in natural convection in a bounded geometry, focusing on the influences of the Prandtl and cavity aspect ratio. Through direct numerical simulations, we examine the flow, thermal transport phenomena, and number of cellular roll structures over a wide range of Prandtl and Rayleigh numbers and aspect ratios. 

Conducting modal analysis of the flow fields, we show the existence of transitional Rayleigh numbers, where the number of steady and quasi-steady state rolls in the cavities increases with Ra. Furthering this, stability analysis finds that near the transitional Rayleigh number, the long-term behavior of the flow field and roll state is dependent on the initial condition. When approaching the transitional Rayleigh number, the core is unstable when starting from a lower Rayleigh number, whereas starting from a Rayleigh number larger than the transitional Rayleigh number, the higher roll state will persist.

Stability analysis is employed to investigate the influence of the cavity aspect ratio on the formation of vertical roll states. We find that a lower bound cavity aspect ratio exists for the formation of coherent multicellular vertical structures, where stable multicellular vortical structures cannot form beyond this aspect ratio. Through the lens of the vortical evolution equation, we find that at a given Ra and Pr, the aspect ratio plays a significant role in the balance and magnitudes of vortical convection, diffusion, and buoyancy contributions of the flow state, in which narrowing of the cavity changes the flow dynamics and leads to multicellular roll formation. 

Examining the role of the Prandtl number in the formation and structure of multicellular vortical structures, we find that there are distinct dominant roll states dependent on the Prandtl number. For example, in the range of 0.005 ≤ Pr < 0.02, the three vertical roll roll-state is the dominant flow mode in the cavity. Upon reaching Pr = 0.02, there is a sharp transition from the triple VRS (vertically roll state) to double VRS, which persists from 0.02 < Pr < 0.1.