Data-driven time-dependent bases for turbulent airfoil wake-extreme gust interactions

We analyze interactions between turbulent airfoil wakes and extreme gusts using a data-driven framework with time-dependent bases. The current approach represents each snapshot with time-varying bases consisting of two-dimensional in-plane modes and one-dimensional spanwise modes, together with a reduced covariance matrix. By deriving closed-form evolution equations, we advance these components using a small rolling window, eliminating the need for full-history storage. Applied to extreme vortex gust-airfoil interactions at Re=5000, we reveal that the first in-plane mode dominates before impingement, while the second mode gains energy post-impingement, amplified by gust intensity. During impingement, leading modes exhibit multiscale features such as shear-layer roll-up and fine-scale turbulence. The temporal evolution of the mode-energy spectrum and its rank gaps further quantifies these interactions. A large leading-mode gap indicates energy concentrated in a few coherent structures, leading to faster recovery, whereas a smaller gap indicates energy distributed across many modes, consistent with richer multiscale activity and delayed re-stabilization. These trends also follow the transient lift dynamics, with higher amplitude and more oscillations indicated by a rise in the leading singular values. This work may provide a foundation for time-varying data-driven modal analysis of extreme gust encounters.