Low-dimensional model of the large-scale circulation of turbulent Rayleigh-Benard convection in a cubic container

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. The model consists of stochastic ordinary differential equations, which were derived from approximate solutions of the Navier-Stokes equations. We test the model using Rayleigh-Benard 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 determined by the shape of the cell. The model predicts advected oscillation modes, driven by a restoring force created by the noncircular shape of the cell cross section. We observe the corresponding lowest-wave-number predicted advected oscillation mode in a cubic cell, 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-wave-number advected twisting and sloshing oscillations found in circular cylindrical cells. Using the Fokker-Planck equation to relate probability distributions of theta(0) to the potential, we find that 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. To quantitatively test the model, we report values of diffusivities and damping timescales for both the LSC orientation theta(0) and temperature amplitude for the Rayleigh number range 8 x 10(7) <= Ra <= 3 x 10(9). The new oscillation mode around corners is found above a critical Ra = 4 x 10(8). This critical Ra 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 consistent with the model if we adjust the model parameters by up to a factor of 2.9, and values are all within a factor of 3 of model predictions. 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. Since the model was developed for circular cross sections, the success of the model at predicting the potential and its relation to other flow properties for a square cross section-which has different flow modes than the circular cross section-suggests that such a modeling approach could be applied more generally to different cell geometries that support a single convection roll.