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Computational Mathematics and Mathematical Physics

, Volume 48, Issue 10, pp 1699–1714 | Cite as

Numerical spectral analysis of temporal stability of laminar duct flows with constant cross sections

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

Abstract

Problems related to the temporal stability of laminar viscous incompressible flows in ducts with a constant cross section are formulated, justified, and numerically solved. For the systems of ordinary differential and algebraic equations obtained by a spatial approximation, a new dimension reduction technique is proposed and substantiated. The solutions to the reduced systems are decomposed over subspaces of modes, which considerably improves the computational stability of the method and reduces the computational costs as compared with the usual decompositions over individual modes. The optimal disturbance problem is considered as an example. Numerical results for Poiseuille flows in a square duct are presented and discussed.

Keywords

duct flows temporal stability systems of ordinary differential and algebraic equations reduction spectral decompositions subspaces of modes optimal disturbances 

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References

  1. 1.
    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).zbMATHCrossRefMathSciNetGoogle Scholar
  2. 2.
    A. Gosset and S. Tavoularis, “Laminar Flow Instability in a Rectangular Channel with a Cylindrical Core,” Phys. Fluids 18(4), 044108.1–044108.8 (2006).CrossRefGoogle Scholar
  3. 3.
    S. J. Parker and S. Balachandar, “Viscous and Inviscid Instabilities of Flow along a Streamwise Corner,” Theor. Comput. Fluid Dyn. 13, 231–270 (1999).zbMATHCrossRefGoogle Scholar
  4. 4.
    T. Tatsumi and T. Yoshimura, “Stability of the Laminar Flow in a Rectangular Duct,” J. Fluid Mech. 212, 437–449 (1990).zbMATHCrossRefGoogle Scholar
  5. 5.
    B. Galletti and A. Bottaro, “Large-Scale Secondary Structure in Duct Flow,” J. Fluid Mech. 512, 85–94 (2004).zbMATHCrossRefGoogle Scholar
  6. 6.
    P. L. O’sullivan and K. S. Breuer, “Transient Growth in Circular Pipe Flow: I. Linear Disturbances,” Phys. Fluids 6, 3643–3651 (1994).zbMATHCrossRefGoogle Scholar
  7. 7.
    P. J. Schmid and D. S. Henningson, Stability and Transition in Shear Flows (Springer-Verlag, Berlin, 2000).Google Scholar
  8. 8.
    R. Temam, Navier-Stokes Equations: Theory and Numerical Analysis (North-Holland, Amsterdam, 1977; Mir, Moscow, 1981).zbMATHGoogle Scholar
  9. 9.
    E. Anderson, Z. Bai, C. Bischof, et al., LAPACK Users’ Guide (SIAM, Philadelphia, 1992).zbMATHGoogle Scholar
  10. 10.
    G. H. Golub and C. F. Van Loan, Matrix Computations (John Hopkins Univ. Press, London, 1991).Google Scholar
  11. 11.
    G. Stewart and J. Sun, Matrix Perturbation Theory (California Acad. Press, San Diego, 1990).zbMATHGoogle Scholar
  12. 12.
    S. K. Godunov, Modern Aspects of Linear Algebra (Am. Math. Soc., Providence, RI, 1998), Vol. 175.zbMATHGoogle Scholar
  13. 13.
    Yu. M. Nechepurenko, “Spectral Decompositions,” in Proceedings of Lobachevsky Center of Mathematics (Kazan. Mat. O-vo, Kazan, 2004), pp. 18–70 [in Russian].Google Scholar
  14. 14.
    G. Hechme and Yu. M. Nechepurenko, “Computing Reducing Subspaces of a Large Linear Matrix Pencil,” Russ. J. Numer. Anal. Math. Model. 21(3), 185–198 (2006).zbMATHCrossRefMathSciNetGoogle Scholar
  15. 15.
    C. Moler and C. F. Van Loan, “Nineteen Dubious Ways to Compute the Exponential of a Matrix, Twenty-Five Years Later,” SIAM Rev. 45(1), 3–49 (2003).zbMATHCrossRefMathSciNetGoogle Scholar
  16. 16.
    J. A. C. Weideman and S. C. Reddy, “A MATLAB Differentiation Matrix Suite,” ACM Trans. Math. Software 26, 465–519 (2000).CrossRefMathSciNetGoogle Scholar
  17. 17.
    J. Waldvogel, “Fast Construction of the Fejér and Clenshaw-Curtis Quadrature Rules,” BIT Numer. Math. 43(1), 1–18 (2003).CrossRefMathSciNetGoogle Scholar

Copyright information

© Pleiades Publishing, Ltd. 2008

Authors and Affiliations

  1. 1.Khristianovich Institute of Theoretical and Applied Mechanics, Siberian BranchRussian Academy of SciencesNovosibirskRussia
  2. 2.Institute of Numerical MathematicsRussian Academy of SciencesMoscowRussia

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