B{\'e}nard convection in a slowly rotating penny shaped cylinder subject to constant heat flux boundary conditions

TL;DR

This paper investigates axisymmetric Boussinesq convection in a shallow, rotating cylinder with constant heat flux, comparing asymptotic analysis with numerical simulations, and developing hybrid methods to better understand flow features at various Prandtl numbers.

Contribution

It introduces hybrid analytical-numerical methods to analyze flow in a rotating cylinder, improving understanding of azimuthal flow behavior at low Prandtl numbers.

Findings

Asymptotics match DNS at high Prandtl numbers

Azimuthal flow is poorly predicted by asymptotics at low Prandtl numbers

Hybrid methods improve flow feature predictions at large Rayleigh numbers

Abstract

We consider axisymmetric Boussinesq convection in a shallow cylinder radius, L, and depth, H (<< L), which rotates with angular velocity $\Omega$ about its axis of symmetry aligned to the vertical. Constant heat flux boundary conditions, top and bottom, are adopted, for which the onset of instability occurs on a long horizontal length scale provided that $\Omega$ is sufficiently small. We investigate the nonlinear development by well-established two-scale asymptotic expansion methods. Comparisons of the results with the direct numerical simulations (DNS) of the primitive governing equations are good at sufficiently large Prandtl number, $\sigma$. As $\sigma$ is reduced, the finite amplitude range of applicability of the asymptotics reduces in concert. Though the large meridional convective cell, predicted by the DNS, is approximated adequately by the asymptotics, the azimuthal flow fails almost catastrophically, because of significant angular momentum transport at small $\sigma$, exacerbated by the cylindrical geometry. To appraise the situation, we propose hybrid methods that build on the meridional streamfunction $\psi$ derived from the asymptotics. With $\psi$ given, we solve the now linear azimuthal equation of motion for the azimuthal velocity v by DNS. Our ''hybrid'' methods enable us to explain features of the flow at large Rayleigh number, found previously by Oruba, Davidson \& Dormy (J. Fluid Mech.,vol. 812, 2017, pp. 890-904).