Steady Rayleigh--B\'enard convection between no-slip boundaries

TL;DR

This paper investigates steady convection roll solutions in Rayleigh--Bénard convection at very high Rayleigh numbers, finding they follow classical scaling and transport more heat than turbulent flows, challenging the ultimate scaling hypothesis.

Contribution

The study computes high-Rayleigh-number steady solutions and demonstrates they follow classical scaling, providing new insights into heat transport mechanisms in turbulent convection.

Findings

Steady solutions follow classical $Nu \,\sim\, Ra^{1/3}$ scaling.

Steady convection transports more heat than turbulent flows at high $Ra$.

Implication that turbulent heat transport may not reach the ultimate scaling.

Abstract

The central open question about Rayleigh--B\'enard convection -- buoyancy-driven flow in a fluid layer heated from below and cooled from above -- is how vertical heat flux depends on the imposed temperature gradient in the strongly nonlinear regime where the flows are typically turbulent. The quantitative challenge is to determine how the Nusselt number $Nu$ depends on the Rayleigh number $Ra$ in the $Ra\to\infty$ limit for fluids of fixed finite Prandtl number $Pr$ in fixed spatial domains. Laboratory experiments, numerical simulations, and analysis of Rayleigh's mathematical model have yet to rule out either of the proposed `classical' $Nu \sim Ra^{1/3}$ or `ultimate' $Nu \sim Ra^{1/2}$ asymptotic scaling theories. Among the many solutions of the equations of motion at high $Ra$ are steady convection rolls that are dynamically unstable but share features of the turbulent attractor. We have computed these steady solutions for $Ra$ up to $10^{14}$ with $Pr=1$ and various horizontal periods. By choosing the horizontal period of these rolls at each $Ra$ to maximize $Nu$, we find that steady convection rolls achieve classical asymptotic scaling. Moreover, they transport more heat than turbulent convection in experiments or simulations at comparable parameters. If heat transport in turbulent convection continues to be dominated by heat transport in steady rolls as $Ra\to\infty$, it cannot achieve the ultimate scaling.