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Eddy cascade of instabilities and transition to turbulence

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Abstract

Numerical simulation is used to investigate a shear layer influenced by a constant external forcing in the theory of turbulence (Kolmogorov’s problem). The dynamics of flows developing in the case of various initial streamwise velocity profiles are studied. The transition from a two-dimensional laminar flow to a three-dimensional turbulent flow is considered. It is shown that developing hydrodynamic instabilities give rise to an eddy cascade, which, in the transition of the flow to a turbulent stage, corresponds to an eddy cascade developing in the energy and, then, inertial ranges.

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References

  1. L. F. Richardson, “The supply of energy from and to atmospheric eddies,” Proc. R. Soc. London A 97, 354–373 (1920).

    Article  Google Scholar 

  2. A. N. Kolmogorov, “The local structure of turbulence in incompressible viscous fluid for very large Reynolds numbers,” Dokl. Akad. Nauk SSSR 30, 299–303 (1941).

    Google Scholar 

  3. A. M. Obukhov, “Spectral Energy Distribution in Turbulent Flow,” Izv. Akad. Nauk SSSR, Ser. Geogr. Geofiz. 5, 453–466 (1941).

    Google Scholar 

  4. V. M. Ievlev, Numerical Simulation of Turbulent Flows (Nauka, Moscow, 1990) [in Russian].

    Google Scholar 

  5. O. M. Belotserkovskii, A. M. Oparin, and V. M. Chechetkin, Turbulence: New Approaches (Nauka, Moscow, 2002) [in Russian].

    Google Scholar 

  6. O. M. Belotserkovskii, Advances in Numerical Modeling (Nauka, Moscow, 2000) [in Russian].

    Google Scholar 

  7. S. O. Belotserkovskii, A. P. Mirabel’, and M. A. Chusov, “Construction of a postcritical regime for plane periodic flow,” Izv. Akad. Nauk SSSR Fiz. Atmos. Okeana 14, 11–20 (1978).

    Google Scholar 

  8. V. I. Yudovich, “Natural oscillations arising from loss of stability in parallel flows of a viscous liquid under long-wavelength periodic disturbances,” Izv. Akad. Nauk SSSR. Mekh. Zhidk. Gaza, No. 1, 32–35 (1973).

    Google Scholar 

  9. N. F. Bondarenko, M. Z. Gak, and F. V. Dolzhanskii, “Laboratory and theoretical models of plane periodic flow,” Izv. Akad. Nauk SSSR Fiz. Atmos. Okeana 15, 1017–1026 (1979).

    Google Scholar 

  10. O. M. Belotserkovskii and A. M. Oparin, Numerical Experiment: From Order to Chaos (Nauka, Moscow, 2000) [in Russian].

    Google Scholar 

  11. J. Hoffman and C. Johnson, “On transition to turbulence in shear flow”, Preprint of Royal Institute of Technology (Kunliga Tekniska, 2002).

    Google Scholar 

  12. M. M. Rogers and R. D. Moser, “The three-dimensional evolution of a plane mixing layer: The Kelvin-Helmholtz rollup,” J. Fluid Mech. 243, 183–226 (1992).

    Article  MATH  Google Scholar 

  13. R. D. Moser and M. M. Rogers, “The three-dimensional evolution of a plane mixing layer: Pairing and transition to turbulence,” J. Fluid Mech. 247, 275–320 (1993).

    Article  MATH  Google Scholar 

  14. O. M. Belotserkovskii, V. M. Chechetkin, S. V. Fortova, and A. M. Oparin, “The turbulence of free shear flows,” Proceedings of International Workshop on Hot Points in Astrophysics and Cosmology (Dubna, 2005), pp. 191–209.

    Google Scholar 

  15. O. M. Belotserkovskii and S. V. Fortova, “Macroscopic parameters of three-dimensional flows in free shear turbulence,” Comput. Math. Math. Phys. 50, 1071–1084 (2010).

    Article  MathSciNet  Google Scholar 

  16. S. V. Fortova, “Numerical simulation of the three-dimensional Kolmogorov flow in a shear layer,” Comput. Math. Math. Phys. 53, 311–319 (2013).

    Article  Google Scholar 

  17. A. M. Oparin, “Numerical simulation of problems related to intense development of hydrodynamic instabilities,” in Advances in Numerical Simulation: Algorithms, Numerical Experiments, and Results (Nauka, Moscow, 2000), pp. 63–90 [in Russian].

    Google Scholar 

  18. L. G. Loitsyanskii, Mechanics of Liquids and Gases (Pergamon, Oxford, 1972; Nauka, Moscow, 1978).

    MATH  Google Scholar 

  19. O. M. Belotserkovskii, V. A. Gushchin, and V. N. Kon’shin, “Splitting method for analysis of stratified fluid flows with free surface,” Zh. Vychisl. Mat. Mat. Fiz. 27, 594–609 (1987).

    MathSciNet  Google Scholar 

  20. S. V. Fortova, L. M. Kraginskii, A. V. Chikitkin, and E. I. Oparina, “Software package for the solution of the equations of hyperbolic type,” Zh. Mat. Model. 25(5), 123–135 (2013).

    MathSciNet  Google Scholar 

  21. G. K. Batchelor, The Theory of Homogeneous Turbulence (Cambridge Univ. Press, Cambridge, 1953; Inostrannaya Literatura, Moscow, 1955).

    MATH  Google Scholar 

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Authors and Affiliations

  1. Institute for Computer-Aided Design, Russian Academy of Sciences, ul. Vtoraya Brestskaya 19/18, Moscow, 123056, Russia

    S. V. Fortova

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  1. S. V. Fortova
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Correspondence to S. V. Fortova.

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Original Russian Text © S.V. Fortova, 2014, published in Zhurnal Vychislitel’noi Matematiki i Matematicheskoi Fiziki, 2014, Vol. 54, No. 3, pp. 536–544.

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Fortova, S.V. Eddy cascade of instabilities and transition to turbulence. Comput. Math. and Math. Phys. 54, 553–560 (2014). https://doi.org/10.1134/S0965542514030051

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  • DOI: https://doi.org/10.1134/S0965542514030051

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