Model reduction in the back step fluid–thermal problem with variable geometry

A methodology is presented to undertake the development of reduced-order models (ROMs) in variable geometry fluid–thermal problems using the method of snapshots. First, some snapshots are calculated in computational domains that vary in both shape and number of grid points. These snapshots are projected onto a so-called virtual grid (defined in a virtual geometry) using a smooth transformation. Proper orthogonal decomposition (POD) modes are obtained from the associated virtual snapshots and projected back onto the original grids, where they are used to define expansions of the flow variables. The associated POD mode amplitudes are obtained minimizing a residual, which is calculated in terms of the reconstructed solution. POD modes are calculated using only a part of the computational domain, which will be called the projection window, and the residual is defined using only a limited number of points of the computational domain. This methodology is illustrated addressing the problem of heat transfer downstream of a backward facing step in the 2-D steady, laminar regime, with three free parameters, namely the Reynolds number, the wall temperature, and the step height.