Low-order dynamical modeling of natural convective air-cooling system in a vertical channel

Low-order dynamical models of transitional natural convective air-cooling system in a vertical channel with periodically repeated discrete heat sources are developed. Proper orthogonal decomposition (POD) methodology has been applied to supercritical oscillatory solutions, obtained by solving the flow governing partial differential equations (PDEs) with a spectral element method at Grashof number, Gr=25000. POD is used to extract the empirical eigenfunctions, to compress the data and to identify the organized spatio-temporal structures. Low-order models (LOMs), consisting of reduced number of nonlinear ordinary differential equations (ODEs), are derived using the computed empirical eigenfunctions as basis functions and applying Galerkin projection (GP). The ability of the reduced models to describe the dynamics of the flow and temperature fields at design conditions is studied. In this study, at least four modes for both velocity and temperature are required to predict self-sustained oscillations in time. The LOM predictions based on four modes are in excellent agreement with the full model results, capturing the short- and long-time nonlinear dynamical behavior of the thermo-fluid system. The developed LOMs may be used to make feasible parametric studies with less computational effort and storage requirements, to investigate stability behavior of the forced/natural convective air-cooling systems, and to explore possible flow control strategies.