Volume 1, Issue 4
Hopf Bifurcations, Drops in the Lid-Driven Square Cavity Flow

Adv. Appl. Math. Mech., 1 (2009), pp. 546-572.

Published online: 2009-01

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• Abstract

The lid-driven square cavity flow is investigated by numerical experiments. It is found that from $\mathrm{Re} $$= 5,000 to \mathrm{Re}$$=$$7,307.75 the solution is stationary, but at \mathrm{Re}$$=$$7,308 the solution is time periodic. So the critical Reynolds number for the first Hopf bifurcation localizes between \mathrm{Re}$$=$$7,307.75 and \mathrm{Re}$$=$$7,308 . Time periodical behavior begins smoothly, imperceptibly at the bottom left corner at a tiny tertiary vortex; all other vortices stay still, and then it spreads to the three relevant corners of the square cavity so that all small vortices at all levels move periodically. The primary vortex stays still. At \mathrm{Re}$$=$$13,393.5 the solution is time periodic; the long-term integration carried out past t_{\infty}$$=$$126,562.5 and the fluctuations of the kinetic energy look periodic except slight defects. However at \mathrm{Re}$$=$$13,393.75 the solution is not time periodic anymore: losing unambiguously, abruptly time periodicity, it becomes chaotic. So the critical Reynolds number for the second Hopf bifurcation localizes between \mathrm{Re}$$=$$13,393.5 and \mathrm{Re}$$=$$13,393.75 . At high Reynolds numbers \mathrm{Re}$$=$$20,000 until \mathrm{Re}$$=$$30,000 the solution becomes chaotic. The long-term integration is carried out past the long time t_{\infty}$$=$$150,000 , expecting the time asymptotic regime of the flow has been reached. The distinctive feature of the flow is then the appearance of drops: tiny portions of fluid produced by splitting of a secondary vortex, becoming loose and then fading away or being absorbed by another secondary vortex promptly. At \mathrm{Re}$$=$$30,000 another phenomenon arises---the abrupt appearance at the bottom left corner of a tiny secondary vortex, not produced by splitting of a secondary vortex. • Keywords Navier-Stokes equations Hopf bifurcations chaos • AMS Subject Headings 76D05 76F06 • Copyright COPYRIGHT: © Global Science Press • Email address • BibTex • RIS • TXT @Article{AAMM-1-546, author = {Salvador Garcia}, title = {Hopf Bifurcations, Drops in the Lid-Driven Square Cavity Flow}, journal = {Advances in Applied Mathematics and Mechanics}, year = {2009}, volume = {1}, number = {4}, pages = {546--572}, abstract = { The lid-driven square cavity flow is investigated by numerical experiments. It is found that from \mathrm{Re}$$=$ $5,000$ 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.

}, issn = {2075-1354}, doi = {https://doi.org/10.4208/aamm.09-m0924}, url = {http://global-sci.org/intro/article_detail/aamm/8385.html} }
TY - JOUR T1 - Hopf Bifurcations, Drops in the Lid-Driven Square Cavity Flow AU - Salvador Garcia JO - Advances in Applied Mathematics and Mechanics VL - 4 SP - 546 EP - 572 PY - 2009 DA - 2009/01 SN - 1 DO - http://doi.org/10.4208/aamm.09-m0924 UR - https://global-sci.org/intro/article_detail/aamm/8385.html KW - Navier-Stokes equations KW - Hopf bifurcations KW - chaos AB -

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.

Salvador Garcia. (1970). Hopf Bifurcations, Drops in the Lid-Driven Square Cavity Flow. Advances in Applied Mathematics and Mechanics. 1 (4). 546-572. doi:10.4208/aamm.09-m0924
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