Turbulent Rayleigh-B\'enard convection in spherical shells

TL;DR

This study numerically investigates turbulent Rayleigh-Bénard convection in spherical shells, analyzing boundary layer asymmetries, plume spacing, and scaling laws, and finds consistency with classical laminar boundary layer profiles and existing theoretical models.

Contribution

First comprehensive numerical analysis of turbulent convection in spherical shells across various geometries and gravity profiles, validating boundary layer and scaling theories.

Findings

Boundary layer asymmetry depends on radius ratio and gravity profile.

Average plume spacing is similar at inner and outer boundaries.

Scaling laws for Nusselt and Reynolds numbers agree with classical theories.

Abstract

We simulate numerically Boussinesq convection in non-rotating spherical shells for a fluid with a unity Prandtl number and Rayleigh numbers up to $10^9$. In this geometry, curvature and radial variations of the gravitationnal acceleration yield asymmetric boundary layers. A systematic parameter study for various radius ratios (from $\eta=r_i/r_o=0.2$ to $\eta=0.95$) and gravity profiles allows us to explore the dependence of the asymmetry on these parameters. We find that the average plume spacing is comparable between the spherical inner and outer bounding surfaces. An estimate of the average plume separation allows us to accurately predict the boundary layer asymmetry for the various spherical shell configurations explored here. The mean temperature and horizontal velocity profiles are in good agreement with classical Prandtl-Blasius laminar boundary layer profiles, provided the boundary layers are analysed in a dynamical frame, that fluctuates with the local and instantaneous boundary layer thicknesses. The scaling properties of the Nusselt and Reynolds numbers are investigated by separating the bulk and boundary layer contributions to the thermal and viscous dissipation rates using numerical models with $\eta=0.6$ and a gravity proportional to $1/r^2$. We show that our spherical models are consistent with the predictions of Grossmann \& Lohse's (2000) theory and that $Nu(Ra)$ and $Re(Ra)$ scalings are in good agreement with plane layer results.