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Numerical Analysis of Spatial Hydrodynamic Stability of Shear Flows in Ducts of Constant Cross Section

  • A. V. Boiko
  • K. V. Demyanko
  • Yu. M. Nechepurenko
Article

Abstract

A technique for analyzing the spatial stability of viscous incompressible shear flows in ducts of constant cross section, i.e., a technique for the numerical analysis of the stability of such flows with respect to small time-harmonic disturbances propagating downstream is described and justified. According to this technique, the linearized equations for the disturbance amplitudes are approximated in space in the plane of the duct cross section and are reduced to a system of first-order ordinary differential equations in the streamwise variable in a way independent of the approximation method. This system is further reduced to a lower dimension one satisfied only by physically significant solutions of the original system. Most of the computations are based on standard matrix algorithms. This technique makes it possible to efficiently compute various characteristics of spatial stability, including finding optimal disturbances that play a crucial role in the subcritical laminar–turbulent transition scenario. The performance of the technique is illustrated as applied to the Poiseuille flow in a duct of square cross section.

Keywords

duct flows spatial stability spectral reduction optimal disturbances 

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References

  1. 1.
    P. J. Schmid and D. S. Henningson, Stability and Transition in Shear Flows (Springer, Berlin, 2000).MATHGoogle Scholar
  2. 2.
    P. G. Drazin, Introduction to Hydrodynamic Stability (Cambridge Univ. Press, Cambridge, 2002).CrossRefMATHGoogle Scholar
  3. 3.
    A. V. Boiko, A. V. Dovgal, G. R. Grek, and V. V. Kozlov, Physics of Transitional Shear Flows (Springer, Berlin, 2012).CrossRefGoogle Scholar
  4. 4.
    V. Theofilis, “Advances in global linear instability analysis of nonparallel and three-dimensional flows,” Progr. Aerospace Sci. 39 (4), 249–315 (2003).CrossRefGoogle Scholar
  5. 5.
    A. V. Boiko and Yu. M. Nechepurenko, “Numerical spectral analysis of temporal stability of laminar duct flows with constant cross sections,” Comput. Math. Math. Phys. 48 (10), 1699–1714 (2008).MathSciNetCrossRefMATHGoogle Scholar
  6. 6.
    A. V. Boiko and Yu. M. Nechepurenko, “Technique for the numerical analysis of the riblet effect on temporal stability of plane flows,” Comput. Math. Math. Phys. 50 (6), 1055–1070 (2010).MathSciNetCrossRefMATHGoogle Scholar
  7. 7.
    K. V. Demyanko and Yu. M. Nechepurenko, “Dependence of the linear stability of Poiseuille flows in a rectangular duct on the cross-sectional aspect ratio,” Dokl. Phys. 56 (10), 531–533 (2011).CrossRefGoogle Scholar
  8. 8.
    K. V. Demyanko and Yu. M. Nechepurenko, “Linear stability analysis of Poiseuille flow in a rectangular duct,” Russ. J. Numer. Anal. Math. Model. 28 (2), 125–148 (2013).MathSciNetCrossRefMATHGoogle Scholar
  9. 9.
    K. V. Demyanko, Candidate’s Dissertation in Mathematics and Physics (Moscow Inst. of Physics and Technology, Moscow, 2014).Google Scholar
  10. 10.
    A. V. Boiko, Yu. M. Nechepurenko, and M. Sadkan, “Fast computation of optimal disturbances with a given accuracy for duct flows,” Comput. Math. Math. Phys. 50 (11), 1914–1924 (2010).MathSciNetCrossRefGoogle Scholar
  11. 11.
    A. V. Boiko, Yu. M. Nechepurenko, and M. Sadkan, “Computing the maximum amplification of the solution norm of differential-algebraic systems,” Comput. Math. Model. 23 (2), 216–227 (2012).MathSciNetCrossRefMATHGoogle Scholar
  12. 12.
    Yu. M. Nechepurenko and M. Sadkan, “A low-rank approximation for computing the matrix exponential norm,” SIAM J. Matr. Anal. Appl. 32 (2), 349–363 (2011).MathSciNetCrossRefMATHGoogle Scholar
  13. 13.
    A. V. Boiko, N. V. Klyushnev, and Yu. M. Nechepurenko, “On stability of Poiseuille flow in grooved channels,” Europhys. Lett. 111, 14001 (2015).CrossRefGoogle Scholar
  14. 14.
    A. V. Boiko, I. V. Klyushnev, and Yu. M. Nechepurenko, Preprint, IPM RAN (Keldysh Inst. of Applied Mathematics, Russian Academy of Sciences, Moscow, 2016).Google Scholar
  15. 15.
    V. Theofilis, P. W. Duck, and J. Owen, “Viscous linear stability analysis of rectangular duct and cavity flow,” J. Fluid Mech. 505, 249–286 (2004).MathSciNetCrossRefMATHGoogle Scholar
  16. 16.
    T. Tatsumi and T. Yoshimura, “Stability of the laminar flow in a rectangular duct,” J. Fluid Mech. 212, 437–449 (1990).CrossRefMATHGoogle Scholar
  17. 17.
    B. Galletti and A. Bottaro, “Large-scale secondary structures in duct flow,” J. Fluid Mech. 512, 85–94 (2004).CrossRefMATHGoogle Scholar
  18. 18.
    P. L. O’Sullivan and K. S. Breuer, “Transient growth in circular pipe flow: I. Linear disturbances,” Phys. Fluids 6 (11), 3643–3651 (1994).CrossRefMATHGoogle Scholar
  19. 19.
    A. Tumin and E. Reshotko, “Spatial theory of optimal disturbances in a circular pipe flow,” Phys. Fluids 13 (4), 991–996 (2001).MathSciNetCrossRefMATHGoogle Scholar
  20. 20.
    A. Gosset and S. Tavoularis, “Laminar flow instability in a rectangular channel with a cylindrical core,” Phys. Fluids 18 (4), 8 (2006).CrossRefGoogle Scholar
  21. 21.
    S. J. Parker and S. Balachandar, “Viscous and inviscid instabilities of flow along a streamwise corner,” Theor. Comput. Fluid Dyn. 13, 231–270 (1999).CrossRefMATHGoogle Scholar
  22. 22.
    G. H. Golub and C. F. van Loan, Matrix Computations (John Hopkins University Press, London, 1991).MATHGoogle Scholar
  23. 23.
    G. Canuto, M. Y. Hussaini, A. Quarteroni, and T. A. Zang, Spectral Methods: Fundamentals in Single Domains (Springer, Berlin, 2006).MATHGoogle Scholar
  24. 24.
    G. Canuto, M. Y. Hussaini, A. Quarteroni, and T. A. Zang, Spectral Methods: Evolution to Complex Geometries and Applications to Fluid Dynamics (Springer, Berlin, 2007).MATHGoogle Scholar
  25. 25.
    J. A. G. Weideman and S. C. Reddy, “A MATLAB differentiation: Matrix suite,” ACM Trans. Math. Software 26 (4), 465–519 (2000).MathSciNetCrossRefGoogle Scholar
  26. 26.
    G. El. Khoury, Yu. M. Nechepurenko, and M. Sadkane, “Acceleration of inverse subspace iteration with Newton’s method,” J. Comput. Appl. Math. 259, 205–215 (2014).MathSciNetCrossRefMATHGoogle Scholar
  27. 27.
    E. Anderson, Z. Bai, G. Bischof, J. Demmel, J. Dongarra, J. Du Croz, A. Greenbaum, S. Hammarling, A. McKenney, S. Ostrouchov, and D. Sorensen, LAPACK Users Guide (SIAM, Philadelphia, 1992).MATHGoogle Scholar

Copyright information

© Pleiades Publishing, Ltd. 2018

Authors and Affiliations

  • A. V. Boiko
    • 1
  • K. V. Demyanko
    • 2
    • 3
  • Yu. M. Nechepurenko
    • 2
    • 3
  1. 1.Khristianovich Institute of Theoretical and Applied Mechanics, Siberian BranchRussian Academy of SciencesNovosibirskRussia
  2. 2.Institute of Numerical MathematicsRussian Academy of SciencesMoscowRussia
  3. 3.Keldysh Institute of Applied MathematicsRussian Academy of SciencesMoscowRussia

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