@Article{AAMM-1-4,
author = {Salvador, Garcia},
title = {Hopf Bifurcations, Drops in the Lid-Driven Square Cavity Flow},
journal = {Advances in Applied Mathematics and Mechanics},
year = {2009},
volume = {1},
number = {4},
pages = {546--572},
abstract = {<p style="text-align: justify;">The lid-driven square cavity flow is investigated by numerical
experiments. It is found that from $ \mathrm{Re} $$=$ $5,000 $ to $
\mathrm{Re}
$$=$$ 7,307.75 $ the solution is stationary, but at $
\mathrm{Re}$$=$$7,308 $ the solution is time periodic. So the
critical Reynolds number for the first Hopf bifurcation localizes
between $ \mathrm{Re} $$=$$ 7,307.75 $ and $ \mathrm{Re} $$=$$ 7,308
$. Time periodical behavior begins smoothly, imperceptibly at the
bottom left corner at a tiny tertiary vortex; all other vortices
stay still, and then it spreads to the three relevant corners of the
square cavity so that all small vortices at all levels move
periodically. The primary vortex stays still. At $ \mathrm{Re}
$$=$$ 13,393.5 $ the solution is time periodic; the long-term
integration carried out past $ t_{\infty} $$=$$ 126,562.5 $ and the
fluctuations of the kinetic energy look periodic except slight
defects. However, at $ \mathrm{Re} $$=$$ 13,393.75 $ the solution is
not time periodic anymore: losing unambiguously, abruptly time
periodicity, it becomes chaotic. So the critical Reynolds number for
the second Hopf bifurcation localizes between $ \mathrm{Re} $$=$$
13,393.5 $ and $ \mathrm{Re} $$=$$ 13,393.75 $. At high Reynolds
numbers $ \mathrm{Re} $$=$$ 20,000 $ until $ \mathrm{Re} $$=$$
30,000 $ the solution becomes chaotic. The long-term integration is
carried out past the long time $ t_{\infty} $$=$$ 150,000 $,
expecting the time asymptotic regime of the flow has been reached.
The distinctive feature of the flow is then the appearance of drops:
tiny portions of fluid produced by splitting of a secondary vortex,
becoming loose and then fading away or being absorbed by another
secondary vortex promptly. At $ \mathrm{Re}
$$=$$ 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.</p>},
issn = {2075-1354},
doi = {https://doi.org/10.4208/aamm.09-m0924},
url = {https://global-sci.com/article/73702/hopf-bifurcations-drops-in-the-lid-driven-square-cavity-flow}
}