Data-driven Mori-Zwanzig operators for boundary layer transition

Data-driven reduced-order modeling (ROM) of complex dynamical systems, such as those found in turbulent flows, is an active area of research, as it offers the potential to tackle notoriously challenging problems in engineering and the physical sciences. In this work, we apply and extend to inhomogeneous flows a recently developed data-driven learning algorithm of Mori-Zwanzig (MZ) operators, in order to extract large scale coherent structures of the Markovian and memory terms and to develop reduced order models with memory effects. MZ is based on a generalized Koopman's description of dynamical systems and provides a mathematically rigorous procedure for constructing non-Markovian reduced-order models of resolved variables from high-dimensional dynamical systems, where the effects due to the unresolved dynamics are captured in the memory kernel and orthogonal dynamics. We apply this method to the flow over a cylinder as a test case, as well as to a spatially inhomogeneous flow corresponding to laminar-turbulent boundary-layer transition. We compare this data-driven MZ approach to Dynamic Mode Decomposition (DMD) and analyze the effects of incorporating the memory terms in prediction, such as measuring the generalization errors and visualizing large-scale spatiotemporal structures.