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Chin. Phys. B, 2021, Vol. 30(9): 090201    DOI: 10.1088/1674-1056/abf7a6
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Multiple solutions and hysteresis in the flows driven by surface with antisymmetric velocity profile

Xiao-Feng Shi(石晓峰)1, Dong-Jun Ma(马东军)1,†, Zong-Qiang Ma(马宗强)1, De-Jun Sun(孙德军)3, and Pei Wang(王裴)1,2,†
1 Institute of Applied Physical and Computational Mathematics, Beijing 100094, China;
2 Center for Applied Physics and Technology, Peking University, Beijing 100871, China;
3 Department of Modern Mechanics, University of Science and Technology of China, Hefei 230027, China
Abstract  Multiple steady solutions and hysteresis phenomenon in the square cavity flows driven by the surface with antisymmetric velocity profile are investigated by numerical simulation and bifurcation analysis. A high order spectral element method with the matrix-free pseudo-arclength technique is used for the steady-state solution and numerical continuation. The complex flow patterns beyond the symmetry-breaking at Re≈320 are presented by a bifurcation diagram for Re<2500. The results of stable symmetric and asymmetric solutions are consistent with those reported in literature, and a new unstable asymmetric branch is obtained besides the stable branches. A novel hysteresis phenomenon is observed in the range of 2208 < Re < 2262, where two pairs of stable and two pairs of unstable asymmetric steady solutions beyond the stable symmetric state coexist. The vortices near the sidewall appear when the Reynolds number increases, which correspond to the bifurcation of topology structure, but not the bifurcation of Navier-Stokes equations. The hysteresis is proposed to be the result of the combined mechanisms of the competition and coalescence of secondary vortices.
Keywords:  bifurcation and continuation      symmetry-breaking      hysteresis      surface driven cavity flow  
Received:  09 December 2020      Revised:  02 February 2021      Accepted manuscript online:  14 April 2021
PACS:  02.30.Oz (Bifurcation theory)  
  47.20.Ky (Nonlinearity, bifurcation, and symmetry breaking)  
  47.15.Fe (Stability of laminar flows)  
Fund: Project supported by the National Natural Science Foundation of China (Grant Nos. 11902043 and 11772065), and the Science Challenge Project (Grant No. TZ2016001).
Corresponding Authors:  Dong-Jun Ma, Pei Wang     E-mail:;

Cite this article: 

Xiao-Feng Shi(石晓峰), Dong-Jun Ma(马东军), Zong-Qiang Ma(马宗强), De-Jun Sun(孙德军), and Pei Wang(王裴) Multiple solutions and hysteresis in the flows driven by surface with antisymmetric velocity profile 2021 Chin. Phys. B 30 090201

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