In this talk, I will describe the general theory of a data-driven Mori-Zwanzig (MZ) formulation and present applications to boundary layer flows. This is motivated by the challenges of understanding and numerically simulating turbulent flows over the surface of an object (such as an aircraft). For example, if we want to develop new designs that can interact with the flow in real time to delay the onset of turbulence near the surface (for drag reduction), we will need reduced models as it can take months on supercomputers to solve the full Navier Stokes equations over all relevant scales for just a sliver of the airfoil. Many data-driven techniques have emerged (or matured) over the last decades to address these challenges, such as the Proper Orthogonal Decomposition, approximate Koopman learning techniques,  and recently the data-driven Mori-Zwanzig methods.

The MZ formalism provides a mathematically exact procedure for constructing non-Markovian reduced-order models of resolved variables (observables) from high-dimensional dynamical systems, where the effects due to the unresolved dynamics are captured in the memory kernel and orthogonal dynamics. The algorithm developed in this work applies Mori’s linear projection operator and an SVD based compression to the selection of the resolved variables (equivalently, a low rank approximation of the two time covariance matrices). We show that this MZ decomposition not only identifies the same spatio-temporal structures found by DMD, but it can also be used to extract spatio- temporal structures of the hysteresis effects present in the memory kernels. We perform an analysis of these structures 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. Additionally, by including these memory terms learned in our data-driven MZ approach, we show improvement in prediction accuracy over DMD at the same level of truncation and at a similar computational cost. Furthermore, an analysis of the spatio-temporal structures of the MZ operators shows identifiable structures associated with the nonlinear generation of the so-called "hot" streaks on the surface of the flared cone, which have previously been observed in experiments and direct numerical simulations.

The speaker will be presenting online.

Math, 402 and Zoom: Link https://arizona.zoom.us/j/83541348598  Password:  BB2022