Low-order dynamical models of thermal convection in high-aspect ratio enclosures

Low-dimensional dynamical models for transitional buoyancy-driven flow in differentially heated enclosures are presented. The full governing partial differential equations with the associated boundary conditions are solved by a spectral element method. Proper orthogonal decomposition is applied to the oscillatory solutions obtained from the full model in order to construct empirical eigenfunctions. Using the most energetic empirical eigenfunctions for the velocity and temperature fields as basis functions and applying Galerkin's method, low-order models consisting of few non-linear ordinary differential equations are obtained. For all cases, close to the "design" conditions (Pr_o, Gr_o), the low-order model (LOM) predictions are in excellent agreement with the predictions of the full model. In particular, the critical Grashof number at the onset of the first temporal flow instability (Hopf bifurcation) as well as the frequency and amplitude of oscillations at slightly supercritical conditions are in excellent agreement with the predictions of the full model. Far from design conditions, the LOMs capture some important characteristic properties of the full model solutions. For example, the low-order model derived for a cavity of A = 20 and Gr_o = 3.2 × 10⁴, Pr_o = 0.71, captures the multiplicity of solutions for large values of Grashof number, while it predicts a unique steady solution at small values of Grashof number. In addition, the model predicts that a stationary instability precedes the onset of oscillatory convection. On the other hand, low-order models derived for low-aspect ratio cavities predict that the solution is unique and stable for sufficiently small values of Grashof number and that the primary instability leads to oscillatory time-dependent flow in agreement with experimental and numerical studies based on the full model.