Hopf Bifurcations, Drops in the Lid-Driven Square Cavity Flow (Advances in Applied Mathematics and Mechanics, 2018)
The lid-driven square cavity flow is investigated by numericalexperiments. It is found that from $ Re $$=$ $5,000 $ to $Re$$=$$ 7,307.75 $ the solution is stationary, but at $Re$$=$$7,308 $ the solution is time periodic. So thecritical Reynolds number for the first Hopf bifurcation localizesbetween $ Re $$=$$ 7,307.75 $ and $ Re $$=$$ 7,308$. Time periodical behavior begins smoothly, imperceptibly at thebottom left corner at a tiny tertiary vortex; all other vorticesstay still, and then it spreads to the three relevant corners of thesquare cavity so that all small vortices at all levels moveperiodically. The primary vortex stays still. At $ Re$$=$$ 13,393.5 $ the solution is time periodic; the long-termintegration carried out past $ t_{∞} $$=$$ 126,562.5 $ and thefluctuations of the kinetic energy look periodic except slightdefects. However, at $ Re $$=$$ 13,393.75 $ the solution isnot time periodic anymore: losing unambiguously, abruptly timeperiodicity, it becomes chaotic. So the critical Reynolds number forthe second Hopf bifurcation localizes between $ Re $$=$$13,393.5 $ and $ Re $$=$$ 13,393.75 $. At high Reynoldsnumbers $ Re $$=$$ 20,000 $ until $ Re $$=$$30,000 $ the solution becomes chaotic. The long-term integration iscarried out past the long time $ t_{∞} $$=$$ 150,000 $,expecting the time asymptotic regime of the flow has been reached.The distinctive feature of the flow is then the appearance of drops:tiny portions of fluid produced by splitting of a secondary vortex,becoming loose and then fading away or being absorbed by anothersecondary vortex promptly. At $ Re$$=$$ 30,000 $ another phenomenon arises—the abrupt appearance atthe bottom left corner of a tiny secondary vortex, not produced bysplitting of a secondary vortex.