Modal Analysis with Mori-Zwanzig Formalism: Application to Hypersonic Boundary Layer Flow

Using the recently developed data-driven Mori-Zwanzig (MZ) formulation [Lin et al, SIAM 20, 2558-2601, 2021], we present a novel methodology for performing modal analysis of large scale spatio-temporal structures from complex dynamical systems. The MZ formalism provides a mathematically exact procedure for constructing non-Markovian reduced-order models of resolved variables, where the effects due to the unresolved dynamics are captured in the memory kernel and orthogonal dynamics. We apply Mori's linear projection operator and an SVD-based compression to select resolved variables, allowing for the approximation of the modes of the measured states. We present an analysis of the modes and spectrum of MZ and Dynamic Mode Decomposition (DMD) in the context of a laminar-turbulent boundary-layer transition flow over a flared cone at Mach 6 and show the dynamical relevance of the memory kernels as well as their sensitivities to the most energetic dynamical structures. By including these memory terms, we show improvement in prediction accuracy over DMD at the same level of truncation. Furthermore, we present the spatio-temporal structures present in the MZ operators associated with the nonlinear generation of the so-called "hot" streaks on the surface of the flared cone.