Steady Rayleigh--Bénard convection between stress-free boundaries

Abstract

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 $\pi /5\le \Gamma \le 4\pi$, where $\Gamma$ is the width-to-height ratio for a pair of counter-rotating rolls, over eight orders of magnitude in the Rayleigh number, $10^3\le Ra\le 10^{11}$, and four orders of magnitude in the Prandtl number, $10^{-2}\le Pr\le 10^2$. At large $Ra$ where steady rolls are dynamically unstable, the computed rolls display $Ra\to \infty$ asymptotic scaling. In this regime, the Nusselt number $Nu$ that measures heat transport scales as $R{a}^{1/3}$ 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 $P{r}^{-1}R{a}^{2/3}$ 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.

References in corpus (3)

• Scaling of large-scale quantities in Rayleigh-Bénard convection
• Heat Transport by Coherent Rayleigh-Bénard Convection
• Dissipation Layers in Rayleigh-Bénard Convection: A Unifying View

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• Rigorous scaling laws for internally heated convection at infinite Prandtl number
• Deterministic and Stochastic Euler-Boussinesq Convection
• Transient and steady convection in two dimensions
• Enhancing wall-to-wall heat transport with unsteady flow perturbations