Proper orthogonal decomposition of direct numerical simulation data: Data reduction and observer construction

In this paper, direct numerical simulation (DNS) data of an opposed-jet hydrogen/air diffusion flame are, in a postprocessing step, analyzed using the proper orthogonal decomposition (POD) technique. The aim of this work is twofold. The first goal is to compute a small number of space-dependent empirical eigenfunctions, so that a low-dimensional representation of the data generated by the large model of the discretized partial differential equations can be obtained using a weighted sum of these few eigenfunctions (POD modes). It is found that only six modes are needed for an accurate representation of the data in an extended range of inflow velocities. This large data reduction takes into account not only chemical kinetics but also transport phenomena in a full two-dimensional context and constitutes the first step toward the construction of low-dimensional dynamic models for the opposed-jet system. It is also found that the PODs have very good interpolatory properties. The second goal is to use part of the available data (i.e., partial measurements), together with the computed modes, to estimate, or, in the terminology of process control, to observe, the "unmeasured" quantities. It is found that only a small number of measurements are needed to obtain accurate estimates of the rest of the data.