Reduced order modeling of turbulent reacting flows on low-rank matrix manifolds

A new low-rank approximation, referred to as time-dependent principal component analysis (t-PCA), is developed for reduced-order modeling (ROM) of scalar transport in turbulent reactive flows. In t-PCA, the evolution of the composition matrix is constrained to a low-rank matrix manifold, similar to that in standard PCA. Specifically, the t-PCA approximates the composition matrix through the multiplication of two thin, time-dependent matrices that represent spatial and composition subspaces. The evolution equations for these subspaces are derived by projecting the composition transport equation onto the tangent space of the low-rank matrix manifold. While the evolution equations for the spatial subspace in both PCA and t-PCA are similar, there are differences in how the composition subspace is obtained: (i) In t-PCA, the composition subspace is time-dependent, whereas in PCA, it is static. (ii) The t-PCA does not require any prior data, and an evolution equation for the composition subspace is derived. In PCA, the composition subspace is obtained from data. The t-PCA can be regarded as an on-the-fly low-rank approximation that can adapt to changes in the flow instantaneously. It is shown that the low-rank t-PCA approximations achieve residual levels lower than those obtained via PCA. For demonstrations and a comparative assessment of the ROMs, simulations are conducted of a non-premixed CO/H₂ flame in a temporally evolving jet. Two cases are considered, based on the mechanisms previously suggested for combustion kinetics of this flame: (i) the GRI-Mech 3.0 model involving 53 species for a two-dimensional flame, (ii) the skeletal syngas model involving 11 species for a three-dimensional turbulent flame. The results are appraised via a posteriori comparisons against data generated via full-rank direct numerical simulation (DNS) of the same flame, and also with the PCA-reduced data from the DNS. It is shown that t-PCA yields excellent predictions of various features of the thermo-chemistry field.