Skip to main content
SpringerLink
Log in

Secondary steady-state and time-periodic flows from a basic flow with square array of odd number of vortices

  • Published:
Applied Mathematics and Mechanics Aims and scope Submit manuscript

Abstract

In a magnetohydrodynamic (MHD) driven fluid cell, a plane non-parallel flow in a square domain satisfying a free-slip boundary condition is examined. The energy dissipation of the flow is controlled by the viscosity and linear friction. The latter arises from the influence of the Hartmann bottom boundary layer in a three-dimensional (3D) MHD experiment in a square bottomed cell. The basic flow in this fluid system is a square eddy flow exhibiting a network of N2 vortices rotating alternately in clockwise and anticlockwise directions. When N is odd, the instability of the flow gives rise to secondary steady-state flows and secondary time-periodic flows, exhibiting similar characteristics to those observed when N = 3. For this reason, this study focuses on the instability of the square eddy flow of nine vortices. It is shown that there exist eight bi-critical values corresponding to the existence of eight neutral eigenfunction spaces. Especially, there exist non-real neutral eigenfunctions, which produce secondary time-periodic flows exhibiting vortices merging in an oscillatory manner. This Hopf bifurcation phenomenon has not been observed in earlier investigations.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. FITZGERALD, J. G. and FARRELL, B. F. Mechanisms of mean flow formation and suppression in two-dimensional Rayleigh-Bénard convection. Physics of Fluids, 26, 054104 (2014)

    Article  Google Scholar 

  2. BAILOOR, S., SEO, J. H., and MITTAL, R. Vortex shedding from a circular cylinder in shear-thinning Carreau fluids. Physics of Fluids, 31, 011703 (2019)

    Article  Google Scholar 

  3. KIM, S., WILSON, P. A., and CHEN, Z. M. Large-eddy simulation of the turbulent near wake behind a circular cylinder: Reynolds number effect. Applied Ocean Research, 49, 1–8 (2015)

    Article  Google Scholar 

  4. NARENDRAN, K. and JAIMAN, R. K. Effect of near-wake jet on the lock-in of a freely vibrating square cylinder. Physics of Fluids, 31, 053603 (2019)

    Article  Google Scholar 

  5. ASHCROFT, G. and ZHANG, X. Vortical structures over rectangular cavities at low speed. Physics of Fluids, 17, 015104 (2005)

    Article  MATH  Google Scholar 

  6. BIRINGEN, S. and DANABASOGLU, G. Oscillatory flow with heat transfer in a square cavity. Physics of Fluids A: Fluid Dynamics, 1, 1796 (1989)

    Article  Google Scholar 

  7. SENGUPTA, T. K., SHARMA, N., and SENGUPTA, A. Non-linear instability analysis of the two-dimensional Navier-Stokes equation: the Taylor-Green vortex problem. Physics of Fluids, 30, 054105 (2018)

    Article  Google Scholar 

  8. VALENTINE, D. T. Decay of confined, two-dimensional, spatially periodic arrays of vortices: a numerical investigation. International Journal for Numerical Methods in Fluids, 2, 155–180 (1995)

    Article  MATH  Google Scholar 

  9. BONDARENKO, N. F., GAK, M. Z., and DOLZHANSKII, F. V. Laboratory and theoretical models of a plane periodic flow. Izvestiya, Atmospheric and Oceanic Physics, 15, 711–716 (1979)

    Google Scholar 

  10. CHEN, Z. M., and PRICE, W. G. Secondary fluid flows driven electromagnetically in a two-dimensional extended duct. Proceedings of the Royal Society A, 461, 1659–1683 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  11. PARET, J. and TABELING, P. Intermittency in the two-dimensional inverse cascade of energy: experimental observations. Physics of Fluids, 10, 3126 (1998)

    Article  Google Scholar 

  12. SOMMERIA, J. Experimental study of the two-dimensional inverse energy cascade in a square box. Journal of Fluid Mechanics, 170, 139–168 (1986)

    Article  Google Scholar 

  13. SOMMERIA, J. and VERRON, J. An investigation of nonlinear interactions in a two-dimensional recirculating flow. Physics of Fluids, 27, 1918 (1984)

    Article  Google Scholar 

  14. XIA, H. and FRANCOIS, N. Two-dimensional turbulence in three-dimensional flows. Physics of Fluids, 29, 111107 (2017)

    Article  Google Scholar 

  15. BONDARENKO, N. F., GAK, E. Z., and GAK, M. Z. Application of MHD effects in electrolytes for modeling vortex processes in natural phenomena and in solving engineering-physical problems. Journal of Engineering Physics and Thermophysics, 75, 1234–1247 (2002)

    Article  Google Scholar 

  16. THESS, A. Instabilities in two-dimensional spatially periodic flows, part II: square eddy lattice. Physics of Fluids A: Fluid Dynamics, 4, 1396 (1992)

    Article  MathSciNet  MATH  Google Scholar 

  17. CHEN, Z. M. Instability of two-dimensional square eddy flows. Physics of Fluids, 31, 044107 (2019)

    Article  Google Scholar 

  18. CHEN, Z. M. Steady-state bifurcation of a non-parallel flow involving energy dissipation over a Hartmann boundary layer. Journal of Nonlinear Science, 31, 91 (2021)

    Article  MathSciNet  MATH  Google Scholar 

  19. MEUNIER, P., LE DIZÈS, S., and LEWEKE, T. Physics of vortex merging. Comptes Rendus Physique, 6, 431–450 (2005)

    Article  Google Scholar 

  20. CERRETELLI, C. and WILLIAMSON, C. H. The physical mechanism for vortex merging. Journal of Fluid Mechanics, 475, 41–77 (2003)

    Article  MathSciNet  MATH  Google Scholar 

  21. CHEN, Z. M. Secondary flows from a linear array of vortices perturbed principally by a Fourier mode. Journal of Nonlinear Science, 32, 48 (2022)

    Article  MathSciNet  MATH  Google Scholar 

  22. VERRON, J. and SOMMERIA, J. Numerical simulation of a two-dimensional turbulence experiment in magnetohydrodynamics. Physics of Fluids, 30, 732 (1987)

    Article  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. School of Mathematics and Statistics, Shenzhen University, Shenzhen, Guangdong Province, 518060, China

    Zhimin Chen

  2. Ship Science, University of Southampton, Southampton, SO17 1BJ, UK

    W. G. Price

Authors
  1. Zhimin Chen
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. W. G. Price
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Zhimin Chen.

Additional information

Citation: CHEN, Z. M. and PRICE, W. G. Secondary steady-state and time-periodic flows from a basic flow with square array of odd number of vortices. Applied Mathematics and Mechanics (English Edition), 44(3), 447–458 (2023) https://doi.org/10.1007/s10483-023-2966-9

Project supported by the National Natural Science Foundation of China (No. 11571240) and the Shenzhen Natural Science Fund of China (the Stable Support Plan Program No. 20220805175116001)

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Chen, Z., Price, W.G. Secondary steady-state and time-periodic flows from a basic flow with square array of odd number of vortices. Appl. Math. Mech.-Engl. Ed. 44, 447–458 (2023). https://doi.org/10.1007/s10483-023-2966-9

Download citation

  • Received:

  • Revised:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s10483-023-2966-9

Key words

Chinese Library Classification

2010 Mathematics Subject Classification

Search

Navigation