We test the ability of a general low-dimensional model for turbulence to predict geometry-dependent dynamics of large-scale coherent structures, such as convection rolls. The model consists of stochastic ordinary differential equations, which are derived as a function of boundary geometry from the Navier-Stokes equations. The model describes the motion of the LSC in terms of diffusion in a potential. We test the model using 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 predicts an oscillation mode in which the LSC oscillates around a corner. To characterize the dynamics of the large-scale circulation in a cube, we report values of diffusivities and damping time scales for both the LSC orientation $\theta_0$ and temperature amplitude, and of the mean temperature amplitude, for $8\times10^7 \le Ra \le 3\times 10^9$. We show that the potential for the orientation $\theta_0$ and slosh angle $\alpha$ has quadratic minima near each corner, with curvature about a factor of 2 smaller than predicted. The potential is nearly symmetric in $\theta_0$ and $\alpha$ near the corners as predicted, indicating the 2 parameters are uncoupled. The prediction of the natural frequency $\omega_r$ of the potential was too large by factor of 2.5, which is typical of this model. We observed a new oscillation mode around corners of the cell above a critical Ra $=5.\times10^8$. This critical Ra, appears in the model as a crossing of an underdamped-overdamped transition. The oscillation period, power spectrum, and critical Ra for oscillations, are consistent with the model if we adjust the model parameters by about a factor of 2. The uncertainty of a factor of 2 in model parameters is too large to correctly predict whether resonance exists. The structure of the oscillation mode consisted of oscillations in $\theta_0$ and $\alpha$ each around the same corner. The structure of the oscillation of $\hat \theta_0$ is consistent with the predicted $n = 1$ advected oscillation mode, the oscillation in $\hat \alpha$ is not quite that of the prediction. The observed oscillation in $\alpha_b$ is consistent with the prediction of $n = 1$, oscillation in $\alpha_t$ is having the consistent phase shift but the opposite correlation as the prediction, and there is oscillation in $\alpha_m$ while the prediction is 0. The observed oscillation is more complex. Since the model was developed in a cylindrical geometry, the continued success of the model at predicting the potential and its relation to other flow properties in a cubic geometry – which has very different flow modes – still suggests great promise for the potential to predict the dynamics of large-scale coherent structures in complicated geometries. However, the failure to predict the complexity of the oscillation structure indicates the need for a more complex model.
Thesis Advisor: Eric Brown
eric.brown@yale.edu