Lagrangian chaos, Eulerian chaos, and mixing enhancement in converging–diverging channel flows

Abstract

A study of Lagrangian chaos, Eulerian chaos, and mixing enhancement in converging–diverging channel flows, using spectral element direct numerical simulations, is presented. The time‐dependent, incompressible Navier–Stokes and continuity equations are solved for laminar, transitional, and chaotic flow regimes for 100≤Re≤850. Classical fluid dynamics representations and dynamical system techniques characterize Eulerian flows, whereas Lagrangian trajectories and finite‐time Lagrangian Lyapunov exponents identify Lagrangian chaotic flow regimes and quantify mixing enhancement. Classical representations demonstrate that the flow evolution to an aperiodic chaotic regime occurs through a sequence of instabilities, leading to three successive supercritical Hopf bifurcations. Poincaré sections and Eulerian Lyapunov exponent evaluations verify the first Hopf bifurcation at 125<Re<150 and the onset of Eulerian chaos at Re≊550. Lagrangian trajectories and finite‐time Lagrangian Lyapunov exponents reveal the onset of Lagrangian chaos, its relation with the appearance of the first Hopf bifurcation, the interplay between Lagrangian and Eulerian chaos, and the coexistence of Lagrangian chaotic flows with Eulerian nonchaotic velocity fields. Last, Lagrangian and Eulerian Lyapunov exponents are used to demonstrate that the onset of Eulerian chaos coincides with the spreading of a strong Lagrangian chaotic regime from the vortex region to the whole fluid domain.