Steady Rayleigh--B\'enard convection between stress-free boundaries

TL;DR

This study numerically investigates steady Rayleigh--Bénard convection between stress-free boundaries across a wide range of parameters, revealing asymptotic scaling laws for heat transport and flow velocity at high Rayleigh numbers.

Contribution

It provides comprehensive numerical analysis of steady convective rolls over broad parameter ranges, confirming asymptotic scalings and their dependence on aspect ratio and Prandtl number.

Findings

Nu scales as Ra^{1/3} at large Ra

Re scales as Pr^{-1} Ra^{2/3} at large Ra

Scaling prefactors depend on aspect ratio, with specific maxima

Abstract

Steady two-dimensional Rayleigh--B\'enard convection between stress-free isothermal boundaries is studied via numerical computations. We explore properties of steady convective rolls with aspect ratios $\pi/5\le\Gamma\le4\pi$, where $\Gamma$ is the width-to-height ratio for a pair of counter-rotating rolls, over eight orders of magnitude in the Rayleigh number, $10^3\le Ra\le10^{11}$, and four orders of magnitude in the Prandtl number, $10^{-2}\le Pr\le10^2$. 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 $\Gamma$ and is largest at $\Gamma \approx 1.9$. The Reynolds number $Re$ for large-$Ra$ rolls scales as $Pr^{-1} Ra^{2/3}$ with a prefactor that is largest at $\Gamma \approx 4.5$. All of these large-$Ra$ features agree quantitatively with the semi-analytical asymptotic solutions constructed by Chini \& Cox (2009). Convergence of $Nu$ and $Re$ to their asymptotic scalings occurs more slowly when $Pr$ is larger and when $\Gamma$ is smaller.