Low-dimensional model of the large-scale circulation of turbulent Rayleigh-B{\'e}nard convection in a cubic container

Abstract

We test the ability of a low-dimensional turbulence model to predict how dynamics of large-scale coherent structures such as convection rolls change in different cell geometries. We performed Rayleigh-B\'enard convection experiments in a cubic container, in which there is a single convection roll known as the large-scale circulation (LSC). The model describes the motion of the orientation $\theta_0$ of the LSC as diffusion in a potential which is predicted as a function of the shape of the cell from an approximate solutions of the Navier-Stokes equations. The model predicts advected oscillation modes, driven by a restoring force created by the non-circular cell cross-section. We observe the predicted lowest-wavenumber mode in which the LSC orientation $\theta_0$ oscillates around a corner, and a slosh angle $\alpha$ rocks back and forth, which is distinct from the higher-wavenumber advected twisting and sloshing oscillations found in cylindrical cells. The potential has quadratic minima near each corner with the same curvature in both the LSC orientation $\theta_0$ and slosh angle $\alpha$, as predicted. The new oscillation mode around corners is found above a critical Ra $=4\times10^8$, which appears in the model as a crossing of an underdamped-overdamped transition. The natural frequency of the potential, oscillation period, power spectrum, and critical Ra for oscillations are all within a factor of 3 of model predictions for the Rayleigh number range $8\times10^7 \le Ra \le 3\times 10^9$. However, these uncertainties in model parameters are too large to correctly predict whether the system is in the underdamped or overdamped state at a given Ra. The success of the model at predicting the potential and flow modes for a cubic cell suggests that such a modeling approach could be applied more generally to different cell geometries that support a single convection roll.