Optimally time-dependent mode analysis of vortex gust-airfoil wake interactions*

Understanding transient dynamics in flows with time-varying base states is critical for evaluating the performance of wings in gusty environments. In this study, optimally time-dependent (OTD) modes, a set of orthonormal modes that traces the most amplified directions of the dynamics with respect to time-varying base flow, are considered to reveal the primary transient flow features. Instead of identifying the asymptotic behaviors, the non-linear trajectory of highly unsteady flows is tracked in a real-time manner.

We apply the OTD mode analysis to gust vortex-airfoil wake interactions, which exhibit strong transient amplification as the gust vortex impacts the airfoil. The OTD modes capture the dominant amplification dynamics stemming from the shear layer and the wake. For both positive and negative vortex-airfoil interactions, positive instantaneous growth rates are captured from the reduced-order linear operators in the OTD subspace.  Moreover, we uncover that the most amplified direction associated with the largest growth rate is closely related to the largest magnitude of lift force variation experienced by the airfoil.