Bounds for convection between rough boundaries

Abstract

We consider Rayleigh-Bénard convection in a layer of fluid between rough no-slip boundaries where the top and bottom boundary heights are functions of the horizontal coordinates with square-integrable gradients. We use the background method to derive an upper bound on mean heat flux across the layer for all admissible boundary geometries. This flux, normalized by the temperature difference between the boundaries, can grow with the Rayleigh number ($Ra$) no faster than $\mathcal{O}(R{a}^{1/2})$ as $Ra\to \infty$. Our analysis yields a family of similar bounds, depending on how various estimates are tuned, but every version depends explicitly on the boundary geometry. In one version the coefficient of the $\mathcal{O}(R{a}^{1/2})$ leading term is $0.242+2.925\Vert \nabla h{\Vert }^2$, where $\Vert \nabla h{\Vert }^2$ is the mean squared magnitude of the boundary height gradients. Application to a particular geometry is illustrated for sinusoidal boundaries.

15 pages; v2: result strengthened so that the gradients of the functions defining the boundaries need only be square-integrable as opposed to uniformly bounded

References in corpus (4)

• A Comparison of Turbulent Thermal Convection Between Conditions of Constant Temperature and Constant Flux
• Tailoring boundary geometry to optimize heat transport in turbulent convection
• Internally heated convection beneath a poor conductor
• Scaling bounds on dissipation in turbulent flows

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• Thermal boundary layers in turbulent Rayleigh-Bénard convection with rough surfaces