Eigen modes of Orr-Sommerfeld equation in the developed turbulent boundary layer


Selim R. S.

Moscow Institute of Physics and Technology, 9, Institutskiy per., Dolgoprudny, Moscow region, 141701, Russia

e-mail: selim.rs@phystech.edu


Spectral methods for solving mathematical physics problems have been intensively studied in the last two decades due to their good approximation properties. Estimation of complex eigenvalues of the fourth order ordinary Orr-Sommerfeld differential equation, which has attracted attention of a number of mathematicians in recent decades, is stipulated by its relevance in the problems of hydrodynamic stability. These methods are considered to be extremely effective for hydrodynamic stability problems where high-precision results are needed. Orr-Sommerfeld type equations are essential for the shear flows analysis, which are important in many fields. One of such areas is climate modeling, when an attempt is made to explain the origin of a mid-latitude cyclone, which in its turn is responsible for creating areas of high and low pressure from which variable weather conditions arise. On the problem of transition from laminar to turbulent flow, all the ideas involved, physical mechanisms, methods used and results obtained are considered, and, if possible, the theory is correlated with experimental and numerical results. It is well known that classical Orr-Sommerfeld eigenvalue problem, which occurs in the linear analysis of the hydrodynamic stability of some basic gas flows, was first solved using the direct spectral method. A numerical study of the eigenvalues of the Orr-Sommerfeld equation for the Blasius boundary layer in the time setting is conducted, and the spectrum of the Orr-Sommerfeld operator is also investigated by Reddy, Schmid and Henningson. In a general mathematical framework, spectral methods for hydrodynamic stability problems are considered, and the Orr-Sommerfeld equation is analyzed, posed with homogeneous boundary conditions that contain a derivative up to the first order.

The collocation method (pseudospectral) based on Chebyshev polynomials is debugged on the Blasius profile and compared with the results of the work Mack.

It is obvious that the increase in the number of Chebyshev polynomials has a significant impact on the accuracy of determining the eigenvalues for the Blasius and Musker profiles. The effect of Chebyshev polynomial degrees on the accuracy of determining the real and imaginary part of the eigenvalues of the first mode for laminar and turbulent boundary layers was considered. The real and imaginary parts of the eigenvalues depending on the wave number for the first three modes of the Orr-Sommerfeld equation for the profile of a developed turbulent boundary layer were obtained This is important for the study of the dynamics (including nonlinear) of Tollmien-Schlichting waves.


collocation method, Chebyshev polynomials, incompressible viscous fluid, turbulent boundary layer, Orr-Sommerfeld equation, Tollmien-Schlichting wave


  1. Trefethen L.N. Spectral Methods in MATLAB, SIAM, 2000, 186 p.

  2. Peyret R. Spectral methods for incompressible viscous flow. Applied Mathematical Sciences, Springer-Verlag, New York, 2002, 434 p.

  3. Gheorghiu C.I. Spectral Methods for Non-Standard Eigenvalue Problems: Fluid and Structural Mechanics and Beyond, Springer, 2014, 120 p.

  4. Schmid P.J., Henningson D.S. Stability and transition in shear flows, Springer, 2001, 572 p.

  5. Boiko A.V., Dovgal A.V., Grek G.R. and Kozlov V.V. Physics of Transitional Shear, New York, Springer, 2012. DOI: 10.1007/978-94-007-2498-3.

  6. Drazin P.G. and Reid W.H. Hydrodynamic Stability, Cambridge University Press, 2004, 619 p.

  7. Shlikhting G. Boundary layer theory, London, Pergamon Press, 1955, 535 p.

  8. Orszag S.A. Accurate solution of the Orr-Sommerfeld stability equation, Journal of Fluid Mechanics, 1971, vol. 50, issue 4, pp. 689 – 703.

  9. Mack L.M. A numerical study of the temporal eigenvalues spectrum of the Blasius boundary layer, Journal of Fluid Mechanics, 1976, vol. 73, pp. 497 – 520.

  10. Stewart G.W. and Sun J.G. Matrix Perturbation Theory, Academic Press, Boston, 1990, 188 p.

  11. McFadden G.B., Murray B.T., Boisvert R.F. Elimination of spurious eigenvalues in the Chebyshev tau spectral method, Journal Computational Physics, 1990, vol. 91, pp. 228 – 239.

  12. Reddy S.C., Schmid P.J., Henningson D. Pseudospectra of Orr-Sommerfeld operator, SIAM Journal on Applied Mathematics, 1993, vol. 53, issue 1, pp. 15 – 47.

  13. Dongarra J.J., Straughan B., Walker D.W. Chebyshev tau-QZ algorithm methods for calculating spectra of hydrodynamic stability problems, Applied Numerical Mathematics, 1996, vol. 22, pp. 399 – 434.

  14. Melenk J.M., Kirchner N.P., Schwab C. Spectral Galerkin discretization for hydrodynamic stability problems, Computing, 2000, vol. 65, issue 2, pp. 97 – 118.

  15. Boyd J.P. Chebyshev and Fourier Spectral Methods, Dover, Mineola, New York, 2001, 690 p.

  16. She J, Tang. T. Spectral and High-Order Methods with Applications, Science Press, Beijing, 2006, 326 p.

  17. Kha L.V. Trudy MAI, 2016, no. 87, available at: http://trudymai.ru/eng/published.php?ID=69519

  18. Khatuntseva O.N. Trudy MAI, 2019, no. 106, available at: http://trudymai.ru/eng/published.php?ID=105673

  19. Khatuntseva O.N. Trudy MAI, 2019, no. 104, available at: http://trudymai.ru/eng/published.php?ID=102068

  20. Zharov V.A. Uchenye zapiski TsAGI, 2014, vol. XLV, no. 5, pp. 33 – 46.

  21. Zharov V.A., Kha L.V. Uchenye zapiski TsAGI, 2016, vol. XLV, no. 8, pp. 50 – 61.

  22. Mason J.C., Handscomb D.C. Chebyshev Polynomials, Chapman and Hall, CRC, New York, 2003, 360 p.

  23. Mathematica 5.0, User’s Guide, Wolfram Research, 2003, 1301 p.

  24. Musker A.J. Explicit expression for the smooth wall velocity distribution in turbulent boundary layer, AIAA Journal, 1979, vol. 17(6), pp. 655 – 657.


mai.ru — informational site MAI

Copyright © 2000-2020 by MAI