Koopman mode representations of vortex dynamics in viscous flows

Vortex dynamics plays an important role in transitional and turbulent flows, where instabilities introduce important criteria for safety and performance of systems like turbomachinery and aerospace vehicles. In order to control these vortical flows, accurate, general and efficient models of vortex dynamics are needed. Here we explore the ability of Koopman mode decomposition (KMD) to provide such models and uncover physical mechanisms. Using pseudo-spectral simulations, we consider vortex dynamics in viscous flow for different Reynolds numbers and different initial vortex geometries. We start with co-rotating vortex pairs, for which we find that KMD modes and eigenvalues are in close agreement with linear stability predictions. We find that the onset of symmetry-breaking is weakly sensitive to Reynolds number, but highly sensitive to small details in the initial conditions. We provide a detailed comparison of the relative abilities of KMD and proper orthogonal decomposition (POD) to capture this behavior. Finally, we consider implications for modeling and controlling more general vortex flows.