Spectral analysis of mixing in 2D high-Reynolds flows

Abstract: We use spectral analysis of Eulerian and Lagrangian dynamics to study the advective mixing in an incompressible 2D bounded cavity flow. A significant property of such a rotational flow at high Reynolds numbers is that mixing in its core is slower than wall-adjacent areas and corner eddies. We explain this property by considering the resonance between frequencies of unsteady motion -- captured by the Koopman spectral analysis of the velocity field -- and the circulation frequency of Lagrangian tracers in the mean flow. In high-Reynolds rotational 2D flows, the vorticity in the rotational core is uniformly distributed, which leads to uniform distribution of circulation periods in the mean flow, i.e., the kinematics in the core of mean flow is like rigid-body rotation. When this ``rigid" core is exposed to velocity fluctuations arising from bifurcations at high Reynolds, it shows more resilience toward resonance in Lagrangian motion and hence mixes more slowly compared to other areas. We also show how our qualitative resonance argument extends to chaotic flows where the classical tools of dynamical systems are not applicable.