Bounds on buoyancy driven flows with Navier-slip conditions on rough boundaries

Abstract

We consider two-dimensional Rayleigh-Bénard convection with Navier-slip and fixed temperature boundary conditions at the two horizontal rough walls described by the height function $h$. We prove rigorous upper bounds on the Nusselt number $\text{Nu}$ which capture the dependence on the curvature of the boundary $\kappa$ and the (non-constant) friction coefficient $\alpha$ explicitly. If $h\in W^{2,\infty }$ and $\kappa$ satisfies a smallness condition with respect to $\alpha$, we find $$\text{Nu}\lesssim {\text{Ra}}^{\frac{1}{2}}+\Vert \kappa {\Vert }_{\infty },$$ where $\text{Ra}$ is the Rayleigh number, which agrees with the predicted Spiegel-Kraichnan scaling when $\kappa =0$. This bound is obtained via local regularity estimates in a small strip at the boundary. When $h\in W^{3,\infty }$, the functions $\kappa$ and $\alpha$ are sufficiently small in $L^{\infty }$ and the Prandtl number $\mathrm{Pr}$ is sufficiently large, we prove upper bounds using the background field method, which interpolate between ${\text{Ra}}^{\frac{1}{2}}$ and ${\text{Ra}}^{\frac{5}{12}}$ with non-trivial dependence on $\alpha$ and $\kappa$. These bounds agree with the result in Drivas et al (2022 Phil. Trans. R. Soc. A 380 20210025) for flat boundaries and constant friction coefficient. Furthermore, in the regime $\mathrm{Pr}\ge {\text{Ra}}^{\frac{5}{7}}$, we improve the ${\text{Ra}}^{\frac{1}{2}}$-upper bound, showing $$\text{Nu}{\lesssim }_{\alpha ,\kappa }{\text{Ra}}^{\frac{3}{7}},$$ where ${\lesssim }_{\alpha ,\kappa }$ hides an additional dependency of the implicit constant on $\alpha$ and $\kappa$.

39 pages, 5 figures

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Cited by in corpus (2)

• Scaling laws for Rayleigh-Bénard convection between Navier-slip boundaries
• Large-time behavior of the 2D thermally non-diffusive Boussinesq equations with Navier-slip boundary conditions