中国物理B ›› 2020, Vol. 29 ›› Issue (3): 30503-030503.doi: 10.1088/1674-1056/ab6b15

• SPECIAL TOPIC—Recent advances in thermoelectric materials and devices • 上一篇    下一篇

The second Hopf bifurcation in lid-driven square cavity

Tao Wang(王涛), Tiegang Liu(刘铁钢), Zheng Wang(王正)   

  1. 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
  • 收稿日期:2019-10-05 修回日期:2019-12-03 出版日期:2020-03-05 发布日期:2020-03-05
  • 通讯作者: Tiegang Liu E-mail:liutg@buaa.edu.cn
  • 基金资助:
    Project supported by the National Natural Science Foundation of China (Grant Nos. 11601013 and 91530325).

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. 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
  • Received:2019-10-05 Revised:2019-12-03 Online:2020-03-05 Published:2020-03-05
  • Contact: Tiegang Liu E-mail:liutg@buaa.edu.cn
  • Supported by:
    Project supported by the National Natural Science Foundation of China (Grant Nos. 11601013 and 91530325).

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摘要： 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.

关键词: unsteady lid-driven cavity flows, second Hopf bifurcation, critical Reynolds number, numerical simulation

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.

Key words: unsteady lid-driven cavity flows, second Hopf bifurcation, critical Reynolds number, numerical simulation

中图分类号:  (Numerical simulations of chaotic systems)

  • 05.45.Pq
05.70.Jk (Critical point phenomena)