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Title: Steady Rayleigh–Bénard convection between stress-free boundaries
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.  more » « less
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Journal of Fluid Mechanics
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National Science Foundation
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