(untitled) (Advances in Applied Mathematics and Mechanics, 2018)
The lid-driven square cavity flow is investigated by numerical experiments. 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 the critical Reynolds number for the first Hopf bifurcation localizes between Re = 7,307.75 and Re = 7,308 . Time periodical behavior begins smoothly, imperceptibly at the bottom left corner at a tiny tertiary vortex; all other vortices stay still, and then it spreads to the three relevant corners of the square cavity so that all small vortices at all levels move periodically. The primary vortex stays still. At Re = 13,393.5 the solution is time periodic; the long-term integration carried out past t_∞ = 126,562.5 and the fluctuations of the kinetic energy look periodic except slight defects. However, at Re = 13,393.75 the solution is not time periodic anymore: losing unambiguously, abruptly time periodicity, it becomes chaotic. So the critical Reynolds number for the second Hopf bifurcation localizes between Re = 13,393.5 and Re = 13,393.75 . At high Reynolds numbers Re = 20,000 until Re = 30,000 the solution becomes chaotic. The long-term integration is carried 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 another secondary vortex promptly. At Re = 30,000 another phenomenon arises—the abrupt appearance at the bottom left corner of a tiny secondary vortex, not produced by splitting of a secondary vortex.