Abstract
The long-standing puzzle of diverging heat transport measurements at very high Rayleigh numbers (Ra) is addressed by a simple model based on well-known properties of classical boundary layers. The transition to the ‘ultimate state’ of convection in Rayleigh–Bénard cells is modeled as sub-critical transition controlled by the instability of large-scale boundary-layer eddies. These eddies are restricted in size either by the lateral wall or by the horizontal plates depending on the cell aspect ratio (in cylindrical cells, the cross-over occurs for a diameter-to-height ratio around 2 or 3). The large-scale wind known to settle across convection cells is assumed to have antagonist effects on the transition depending on its strength, leading to wind-immune, wind-hindered or wind-assisted routes to the ultimate regime. In particular winds of intermediate strength are assumed to hinder the transition by disrupting heat transfer, contrary to what is assumed in standard models. This phenomenological model is able to reconcile observations from more than a dozen of convection cells from Grenoble, Eugene, Trieste, Göttingen and Brno. In particular, it accounts for unexplained observations at high Ra, such as Prandtl number and aspect ratio dependences, great receptivity to details of the sidewall and differences in heat transfer efficiency between experiments.
Highlights
We explore if published data support this second conjecture, in particular: (a) An instability of bl-eddies occurring at the observed critical Rayleigh number. (b) A transition controlled by the characteristic length scale L defined above
The motivation of this study was to sketch out a minimalist model able to reconcile all very high Rayleigh numbers experiments
A simple phenomenological model meeting this objective has been elaborated based on 3 conjectures, all directly elaborated from present knowledge of transitions in classical boundary layers:
Summary
The Boussinesq approximation consists in assuming fluid incompressibility (except for a linear temperature dependence of density yielding the buoyant term), constant fluid properties, and decoupling of heat and mechanical energies [1, 29]. In this approximation, the flow is determined by only two control parameters which are traditionally chosen as the Rayleigh and Prandtl numbers. The Rayleigh number can be seen as the control parameter associated with the thermal forcing of the flow while the Prandtl number is the only relevant property of the fluid. In cylindrical cells of diameter φ and height h, it results in an extra dimensionless control parameter: the cell aspect ratio Γ defined as
Sign-in/Register to view highlights & summary 
Published Version (
Free)
Talk to us
Join us for a 30 min session where you can share your feedback and ask us any queries you have
Schedule a call