Vorticity alignment with Lyapunov vectors and rate-of-strain eigenvectors

Abstract

We derive asymptotic estimates for the projection of the vorticity onto principal directions of material stretching in 3D flows. In flows with pointwise bounded vorticity, these estimates predict vorticity alignment with Lyapunov vectors along trajectories with positive Lyapunov exponents. Specifically, we find that in inviscid flows with conservative body forces, the vorticity exactly aligns with the intersection of the planes orthogonal to the dominant forward and backward Lyapunov vectors along trajectories with positive Lyapunov exponent. Furthermore, we derive asymptotic estimates for the vorticity alignment with the intermediate eigenvector of the rate-of-strain tensor for viscous flows under general forcing. We illustrate these results on explicit solutions of Euler’s equation and on direct numerical simulations of homogeneous isotropic turbulence.