Skip to main content
SpringerLink
Log in

Numerical Analysis of the Spectrum of Plane–Parallel Flows

  • Published:
Journal of Mathematical Sciences Aims and scope Submit manuscript

Based on the Galerkin method with Chebyshev polynomials, we numerically solve the Orr-Sommerfeld problem with high accuracy. We calculate the spectra of the planeparallel Poisseuille and Couette flows, the Blasius flow, and flows with pressure gradient above a flat semi-infinite plate for different values of the wave number and the Reynolds number.

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. C. C. Lin, The Theory of Hydrodynamic Stability, Cambridge Univ. Press, London etc. (1955).

    Google Scholar 

  2. P. J. Schmid and D. S. Henningson, Stability and Transition in Shear Flows, Springer, New York (2001).

    Book  Google Scholar 

  3. L. N. Thomas, “The stability of plane Poisseuille flow,” Phys. Rev. II 91, 780–783 (1953).

    Article  Google Scholar 

  4. L. M. Mack, “A numerical study of the temporal eigenvalue spectrum of the Blasius boundary layer,” J. Fluid Mech. 73, No. 3, 497–520 (1976).

    Article  Google Scholar 

  5. D. Gottlieb and S. A. Orszag, Numerical Analysis of Spectral Methods: Theory and Applications, SIAM, Philadelphia, PA (1977).

    Book  Google Scholar 

  6. J. M. Melenk, N. P. Kirchner, and C. Schwab, “Spectral Galerkin discredization for hydrodynamic stability problems,” Computing 65, No. 2, 97–118 (2000).

    Article  MathSciNet  Google Scholar 

  7. S. A. Orszag, “Accurate solution of the Orr-Sommerfeld stability equation,” J. Fluid Mech. 50, No. 4, 689–703 (1971).

    Article  Google Scholar 

  8. T. G. Darmaev, “Solving the Orr-Sommerfeld problem by the pseudo-spectral method using Chebyshev polynomials” [in Russian], Vychisl. Tekhnol. 13, No. 3, 38–44 (2008).

    Google Scholar 

  9. C. B. Moler and G. W. Stewart, “An algorithm for generalized matrix eigenproblems,” SIAM J. Numer. Anal. 10, 241–256 (1973).

    Article  MathSciNet  Google Scholar 

  10. P. G. Drazin, Introduction to Hydrodynamic Stability, Cambridge Univ. Press, Cambridge (2002).

    Book  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Buryat State University, 24a, Smolina St, Ulan-Ude, 670000, Russia

    Tumen Darmaev & Purbo Abiduev

Authors
  1. Tumen Darmaev
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Purbo Abiduev
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Tumen Darmaev.

Rights and permissions

Springer Nature or its licensor (e.g. a society or other partner) holds exclusive rights to this article under a publishing agreement with the author(s) or other rightsholder(s); author self-archiving of the accepted manuscript version of this article is solely governed by the terms of such publishing agreement and applicable law.

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Darmaev, T., Abiduev, P. Numerical Analysis of the Spectrum of Plane–Parallel Flows. J Math Sci 279, 776–781 (2024). https://doi.org/10.1007/s10958-024-07059-3

Download citation

  • Received:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s10958-024-07059-3

Search

Navigation