Spatiotemporal structures of buoyancy-induced flow in a vertical channel

The present investigation deals with the development of low-order representations of transitional free convection in a vertical channel with discrete heaters. The governing equations are solved using a spectral element method. Proper orthogonal decomposition (POD) is applied to extract the most energetic eigenfunctions (and the related spatiotemporal structures) from time-dependent numerical solutions of the full model equations at a Grashof number higher than the critical value. Using the computed eigenfunctions in a truncated series expansion, reconstruction of the original flow and temperature fields is achieved in an optimal way. It is found that almost all the flow and temperature fluctuation energy is captured by the first six eigenmodes. 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 four eigenmodes for each field predict stable, self-sustained oscillations with correct amplitude and frequency.