In search of the 'Ultimate' state of Rayleigh-Benard convection

Center for Nonlinear Studies

Los Alamos National Laboratory

For decades, experiments (and more recently numerical simulations) have attempted to determine how the effective transport of heat (measured by the non-dimensional Nusselt number Nu) scales with the driving force (as measured by the Rayleigh number Ra) --when said driving force is asymptotically strong--in Rayleigh-Benard convection where an incompressible, Boussinesq fluid is driven by an imposed temperature gradient. To date the results are inconclusive for experiments, and finite limitations on computational resources restrict the potential usefulness of direct numerical simulation. In contrast to these approaches, we derive rigorous upper bounds on the heat transport for slippery convection in which the velocity satisfies a stress-free (or free-slip) boundary condition on the vertical boundary. For 2d convection as well as 3d convection for infinite (or even large) Prandtl number (ratio of kinematic viscosity to thermal diffusivity) this bound takes the form Nu < C Ra^{5/12}. At finite Prandtl numbers this scaling challenges some theoretical arguments regarding asymptotic high Rayleigh number heat transport by turbulent convection.