Streamlines Near a Closed Curve and Chaotic Streamlines in Steady Cavity Flows

Numerical simulations are carried out for the three-dimensional steady flow in a lid-driven rectangular cavity. We study the incompressible flow in a cavity with spanwise aspect ratio 1.0 and aspect ratios 0.4, 0.6, 1.0 and 1.4. Streamlines are obtained from the steady velocity field and Poincaré sections are plotted from the streamlines for Reynolds numbers ranging from 100 and 400. There are two types of streamlines: localized streamlines near a closed curve and chaotic streamlines. In the Poincaré sections we can find various structures of ovals of invariant tori and resonant islands by localized streamlines in the regions near the end-wall and irregularly distributed points by chaotic streamlines. The structures in the Poincaré sections are similar to those in the phase portraits of one-dimensional non-autonomous Hamiltonian system. The Reynolds number ranges where the 3:1 and 2:1 resonances occur are presented for various aspect ratios.