Steady Rayleigh–Bénard convection between stress-free boundaries

• DOI: 10.1017/jfm.2020.812
• Publication Date: 2020-11-04T11:46:46Z
• Authors: Baole Wen; David Goluskin; Matthew LeDuc; Gregory P. Chini; Charles R. Doering
• Keywords: Rayleigh–Bénard convection

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 $\Gamma$ the width-to-height ratio for a pair counter-rotating rolls, over eight orders magnitude in Rayleigh number, $10^3\le Ra\le10^{11}$, and four Prandtl $10^{-2}\le Pr\le10^2$. At large $Ra$ are dynamically unstable, computed display $Ra \rightarrow \infty$ asymptotic scaling. In this regime, Nusselt number $Nu$ that measures heat transport scales as $Ra^{1/3}$ uniformly $Pr$. The prefactor scaling depends on largest at $\Gamma \approx 1.9$. Reynolds $Re$ large-$Ra$ $Pr^{-1} Ra^{2/3}$ 4.5$. All these features agree quantitatively semi-analytical solutions constructed by Chini \& Cox (2009). Convergence to their scalings occurs more slowly when $Pr$ larger smaller.