Koopman mode representations of vortex interactions and instabilities

Coherent vortices play an important role in many systems like turbomachinery and aircraft. The dynamics of vortices affect the performance and safety of the system and neighboring ones. While stability and control of these vortices has been the subject of a great deal of work, there are still several unsolved issues. Here we propose to address these challenges using dynamical systems theory, including recent advances on Koopman mode analysis. First, a meaningful and practical definition of stability is needed. In many cases, where the overall flow is robust, linear stability analysis yields unstable eigenvalues associated with an extremely small amount of vorticity. Second, the presence of real-world imperfections may be sufficient to drive drastically different behavior from canonical predictions. Note that decompositions based on energy-containing modes (such as POD) may also fail in this case, as low-energy vortices can drive drastic changes in the dynamics. By contrast, a Koopman mode approach will be better suited to represent these flows. In this talk, we apply KMD to vortical flows, starting with vortex merger and proceeding to more complex examples. In all cases, KMD robustly recovers unstable eigenvalues.