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Low-Dimensional Analysis and Numerical Simulation of the Incompressible Flow Between Two Concentric Rotating Cylinders

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Abstract

There has been a very large number of experimental and theoretical studies of flow between concentric rotating cylinders in the century, since these pioneering works, the instability of Couette flow and Taylor vortex flow, now known as the “Couette–Taylor problem” is a paradigm of nonlinear problem. In order to explore the transition way of Couette–Taylor flow from laminar to turbulence and the characteristics of chaotic attractors in the turbulent regime of Couette–Taylor flow, dynamical behaviors and numerical simulation of Couette–Taylor flow have been studied by using low-dimensional model analysis method in this paper. Dynamical behaviors of a three-model Lorenz-like system of Couette–Taylor flow have been discussed, such as the stability of equilibrium, presence of limit cycles, occurrence of bifurcation and chaos, as well as the analysis of global stability etc. Moreover, using these results we explain successive transitions of Couette–Taylor flow from Laminar flow to turbulence in the experiment. By means of numerical simulation results of bifurcation diagram, Lyapunov exponent spectrum and Poincare map we analyze complex dynamic behaviors of the system from the bifurcation transition to chaos.

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References

  1. Chen, F., Hsien, D.Y.: A model study of stability of Couette flow. Commun. Appl. Math. Comput. 7, 22–33 (1987)

    MathSciNet  Google Scholar 

  2. Wang, H.: Lorenz systems for the incompressible flow between two concentric rotating, cylinders. J. Partial Differ. Equ. 23(3), 209–221 (2010)

    MATH  MathSciNet  Google Scholar 

  3. Wang, Heyuan: The chaos behavior and simulation of three model systems of Couette–Taylor flow. Acta Math. Sci. 352, 769–779 (2015)

    MATH  MathSciNet  Google Scholar 

  4. Gassa Feugainga, C.M., Crumeyrollea, O., Yangb, K.S., Mutabazia, I.: Destabilization of the Couette–Taylor flow by modulation of the inner cylinder rotation. Eur. J. Mech. B Fluids 44, 82–87 (2014)

    Article  Google Scholar 

  5. Stevens, RichardJAM, Grossmann, Siegfried, Verzicco, Roberto, Lohse, D.: Optimal Taylor–Couette flow: direct numerical simulations. J. Fluid Mech. 719, 14–46 (2013)

    Article  MATH  Google Scholar 

  6. Wang, Heyuan: Dynamical behaviors and numerical simulation of Lorenz systems for the incompressible Flow Between Two Concentric Rotating Cylinders. Int. J. Bifurc. Chaos 225, 56–73 (2012)

    Google Scholar 

  7. chossat, Pascal, Tooss, Gerard: The Couette–Taylor Problem. Springer, New York (1994)

    Book  Google Scholar 

  8. Swinney, H.L., Gollub, J.P.: Hydrodynamic, Instablilities and the Transition to Turbulence. Springer, New York, Berlin, Heidelberg (1981)

    Book  MATH  Google Scholar 

  9. Taylor, G.I.: Stability of a viscons liquid contained between two rotating cylinders. Philos. Trans. R. Soc. Lond. A 223, 289–343 (1923)

    Article  MATH  Google Scholar 

  10. Thomas, D.G., Khomami, B., Sureshkumar, R.: Nonlinear dynamics of viscoelastic Taylor–Couette flow, effect of elasticity on pattern selection, molecular conformation and drag. J. Fluid Mech. 620, 353C382 (2009)

    Article  MATH  Google Scholar 

  11. Anderreck, C.D., Liu, S.S., Swinney, H.L.: Flow regimes in a circular Couette system with independent rotating cylinders. J. Fluid Mech. 64, 155–183 (1986)

    Article  Google Scholar 

  12. Meseguera, A., Avilab, M., Mellibovskyc, F., Marquesd, F.: Solenoidal spectral formulations for the computation of secondary flows in cylindrical and annular geometries. Eur. Phys. J. 146, 249–259 (2007)

    Google Scholar 

Download references

Acknowledgements

This research works was supported by the National Nature Science Foundation of China (Grant Nos. 11572146; 11526105), the funds of education department of Liaoning Province (L2013248) and science and technology funds of Jinzhou city (13A1D32).

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Authors and Affiliations

  1. College of Sciences, Liaoning University of Technology, Jinzhou, 121001, China

    Heyuan Wang

  2. Liaoning University of Technology, Jinzhou, China

    Yukun Song & Jin Cui

Authors
  1. Heyuan Wang
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  2. Yukun Song
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  3. Jin Cui
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Correspondence to Heyuan Wang.

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Wang, H., Song, Y. & Cui, J. Low-Dimensional Analysis and Numerical Simulation of the Incompressible Flow Between Two Concentric Rotating Cylinders. Int. J. Appl. Comput. Math 3, 3021–3034 (2017). https://doi.org/10.1007/s40819-016-0282-4

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  • DOI: https://doi.org/10.1007/s40819-016-0282-4

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