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Chin. Phys. B, 2020, Vol. 29(3): 030503    DOI: 10.1088/1674-1056/ab6b15
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The second Hopf bifurcation in lid-driven square cavity

Tao Wang(王涛)1,2, Tiegang Liu(刘铁钢)1, Zheng Wang(王正)3
1 LIMB and School of Mathematical Science, Beihang University, Beijing 100191, China;
2 School of Mathematics and Information Science, North Minzu University, Yinchuan 750021, China;
3 Wuhan Maritime Communication Research Institute, Wuhan 430079, China
Abstract  To date, there are very few studies on the second Hopf bifurcation in a driven square cavity, although there are intensive investigations focused on the first Hopf bifurcation in literature, due to the difficulties of theoretical analyses and numerical simulations. In this paper, we study the characteristics of the second Hopf bifurcation in a driven square cavity by applying a consistent fourth-order compact finite difference scheme recently developed by us. We numerically identify the critical Reynolds number of the second Hopf bifurcation located in the interval of (11093.75,11094.3604) by bisection. In addition, we find that there are two dominant frequencies in its spectral diagram when the flow is in the status of the second Hopf bifurcation, while only one dominant frequency is identified if the flow is in the first Hopf bifurcation via the Fourier analysis. More interestingly, the flow phase portrait of velocity components is found to make transition from a regular elliptical closed form for the first Hopf bifurcation to a non-elliptical closed form with self-intersection for the second Hopf bifurcation. Such characteristics disclose flow in a quasi-periodic state when the second Hopf bifurcation occurs.
Keywords:  unsteady lid-driven cavity flows      second Hopf bifurcation      critical Reynolds number      numerical simulation  
Received:  05 October 2019      Revised:  03 December 2019      Published:  05 March 2020
PACS:  05.45.Pq (Numerical simulations of chaotic systems)  
  05.70.Jk (Critical point phenomena)  
Fund: Project supported by the National Natural Science Foundation of China (Grant Nos. 11601013 and 91530325).
Corresponding Authors:  Tiegang Liu     E-mail:  liutg@buaa.edu.cn

Cite this article: 

Tao Wang(王涛), Tiegang Liu(刘铁钢), Zheng Wang(王正) The second Hopf bifurcation in lid-driven square cavity 2020 Chin. Phys. B 29 030503

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