Journal Article

Steady Rayleigh–Bénard convection between stress-free boundaries

Goluskin, David, LeDuc, Matthew, Doering, Charles R., Wen, Baole, Chini, Gregory P.

2020

Overview

Steady two-dimensional Rayleigh–Bénard convection between stress-free isothermal boundaries is studied via numerical computations. We explore properties of steady convective rolls with aspect ratios ${\\rm \\pi} /5\\leqslant \\varGamma \\leqslant 4{\\rm \\pi}$, where $\\varGamma$ is the width-to-height ratio for a pair of counter-rotating rolls, over eight orders of magnitude in the Rayleigh number, $10^3\\leqslant Ra\\leqslant 10^{11}$, and four orders of magnitude in the Prandtl number, $10^{-2}\\leqslant Pr\\leqslant 10^2$. At large $Ra$ where steady rolls are dynamically unstable, the computed rolls display $Ra \\rightarrow \\infty$ asymptotic scaling. In this regime, the Nusselt number $Nu$ that measures heat transport scales as $Ra^{1/3}$ uniformly in $Pr$. The prefactor of this scaling depends on $\\varGamma$ and is largest at $\\varGamma \\approx 1.9$. The Reynolds number $Re$ for large-$Ra$ rolls scales as $Pr^{-1} Ra^{2/3}$ with a prefactor that is largest at $\\varGamma \\approx 4.5$. All of these large-$Ra$ features agree quantitatively with the semi-analytical asymptotic solutions constructed by Chini & Cox (Phys. Fluids, vol. 21, 2009, 083603). Convergence of $Nu$ and $Re$ to their asymptotic scalings occurs more slowly when $Pr$ is larger and when $\\varGamma$ is smaller.

• Publisher: Cambridge University Press
• Subject: Approximation / Aspect ratio / Asymptotic methods / Balances (scales) / Boundaries / Boundary conditions / Computational fluid dynamics / Convection / Fluid flow / Fluids / Free boundaries / Heat / Heat transport / JFM Rapids / Prandtl number / Rayleigh number / Rayleigh-Benard convection / Reynolds number / Rolls / Scaling / Simulation / Strutt, John William (Lord Rayleigh) (1842-1919) / Velocity