Low-dimensional models of transitional free convective flow and heat transfer

In this study, low-order representations of transitional, free convective flow and heat transfer in a vertical channel with discrete heaters are 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-dependent numerical solutions of the full model equations at Gr = 22500 and Pr=0.71. Using the computed eigenfunctions in a truncated series expansion, we are able to reconstruct the original flow and temperature fields in an optimal way. It is found that almost all the flow and temperature 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 eigen-modes for each field result in stable, self-sustained oscillations with correct amplitude.