Applied Interdisciplinary Mathematics (AIM) Seminar

Buoyancy-driven flows are central to engineering heat transport, atmosphere and ocean dynamics, climate science, geodynamics, and stellar physics. Rayleigh--B\'enard convection---the buoyancy driven flow in a fluid layer heated from below and cooled from above---is recognized as the simplest scenario in which to study such phenomena, and beyond its importance for applications this problem has served for a century as one of the primary paradigms of nonlinear physics, complex dynamics, pattern formation and turbulence. A central question about Rayleigh--B\'enard convection is how the Nusselt number $Nu$ depends on the Rayleigh number $Ra$ and the Prandtl number $Pr$---i.e., how heat flux depends on imposed temperature gradient and the ratio of the fluid's kinematic viscosity to its thermal diffusivity---as $Ra\rightarrow\infty$. Experiments and simulations have yet to rule out either `classical' $Nu \sim Ra^{1/3}$ or `ultimate' $Nu \sim Ra^{1/2}$ asymptotic scaling. Here we provide clear quantitative evidence suggesting that the ultimate regime might not exist. Our tactic is to study relatively simple time-independent states called rolls and compare heat transport by these rolls with that of turbulent convection. These steady rolls are not typically seen in large-$Ra$ simulations or experiments because they are dynamically unstable. Nonetheless, they are part of the global attractor for the infinite-dimensional dynamical system defined by Rayleigh's model, and recent results suggest that steady rolls may be one of the key coherent states comprising the `backbone' of turbulent convection. By developing novel numerical methods, we compute steady rolls between no-slip boundaries for $Ra\le 10^{14}$ with $Pr=1$ and various horizontal periods. We find that rolls of the periods that maximize $Nu$ at each $Ra$ have classical $Nu\sim Ra^{1/3}$ scaling asymptotically, and they transport more heat than turbulent experiments or simulations at similar parameters. If turbulent heat transport continues to be dominated by steady transport asymptotically, it cannot achieve ultimate scaling. Speaker(s): Baole Wen (University of Michigan)