Low-dimensional representation of transitional buoyancy-driven flow in a vertical channel with discrete heaters

A low-dimensional representation of transitional, buoyancy-induced flow in a vertical channel with discrete heaters is developed. The governing equations are solved using a spectral element method. Proper Orthogonal Decomposition (POD) is applied to extract the most energetic eigenfunctions (coherent structures) from time-independent numerical solutions of the full model equations. Using the computed eigenfunctions we are able to reconstruct the original flow and temperature fields in an optimal way. It is found that almost all the flow energy is captured by the first 6 modes. A low-dimensional set of nonlinear ordinary differential equations that describes the dynamics of the flow and temperature fields is also derived. It is found that low-order models based on retaining at least 4 eigenmodes for each field result in stable, self-sustained oscillations with correct amplitude.