Ultimate Rayleigh-Be<acute accent>nard turbulence

Thermally driven turbulent flows are omnipresent in nature and technology. A good understanding of the physical principles governing such flows is key for numerous problems in oceanography, climatology, geophysics, and astrophysics for problems involving energy conversion, heating and cooling of buildings and rooms, and process technology. In the physics community, the most popular system to study wall-bounded thermally driven turbulence has been Rayleigh-Be<acute accent>nard flow, i.e., the flow in a box heated from below and cooled from above. The dimensionless control parameters are the Rayleigh number Ra (the dimensionless heating strength), the Prandtl number Pr (the ratio of kinematic viscosity to thermal diffusivity), and the aspect ratio Gamma of the container. The key response parameters are the Nusselt number Nu (the dimensionless heat flux from the bottom to the top) and the Reynolds number Re (the dimensionless strength of the turbulent flow). While there is good agreement and understanding of the dependences Nu(Ra, ( Ra, Pr, Gamma) ) up to Ra similar to 1011 11 (the " classical regime"), " ), for even larger Rain the so-called ultimate regime of RayleighBe<acute accent>nard convection the experimental results and their interpretations are more diverse. The transition of the flow to this ultimate regime, which is characterized by strongly enhanced heat transfer, is due to the transition of laminar-type flow in the boundary layers to turbulent-type flow. Understanding this transition is of the utmost importance for extrapolating the heat transfer to large or strongly thermally driven systems. Here the theoretical results on this transition to the ultimate regime are reviewed and an attempt is made to reconcile the various experimental and numerical results. The transition toward the ultimate regime is interpreted as a non-normal- - nonlinear and thus subcritical transition. Experimental and numerical strategies are suggested that can help to further illuminate the transition to the ultimate regime and the ultimate regime itself, for which a modified model for the scaling laws in its various subregimes is proposed. Similar transitions in related wall-bounded turbulent flows such as turbulent convection with centrifugal buoyancy and Taylor-Couette turbulence are also discussed.