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Uniqueness and Stability

  • Kolumban HutterEmail author
  • Yongqi Wang
Chapter
Part of the Advances in Geophysical and Environmental Mechanics and Mathematics book series (AGEM)

Abstract

This chapter on uniqueness and stability provides a first flavor into the subject of laminar-turbulent transition. Two different theoretical concepts are in use and both assume that the laminar-turbulent transition is a question of loss of stability of the laminar motion. With the use of the energy method one tries to find conditions for the laminar flow to be stable. Energy stability criteria operate with the construction of upper bounds of the rate of the perturbed kinetic energy K(t) of the fluid system, in order to obtain by time integration an inequality of the form \(K(t) < K(0)\mathrm {exp}\,(-t/\tau )\). Here, \(\tau > 0\) guarantees decay and \(\tau < 0\) growth rates of the perturbed energy, \(\tau = 0\) guarantees neutral stability of the perturbation flows. The difficulty of the method is that the condition \(\tau = 0\) generally provides poor, i.e., very safe estimates for stability. More successful for pinpointing the laminar-turbulent transition has been the method of linear instability analysis, in which a lowest bound, is searched for, at which the onset of deviations from the laminar flow is taking place. For plane channel flows the Rayleigh and OrrSommerfeld equations with associated boundary conditions for an ideal and viscous fluid, respectively, are derived and the associated eigenvalue problems are discussed, which leads to the stability chart, separating Reynolds number dependent stable and unstable flow regimes.

Keywords

Kinetic energy of the difference motion Uniqueness Energy stability of laminar channel flows Rayleigh equation OrrSommerfeld equation 

References

  1. 1.
    Betchov, R., & Criminale, W. O. (1967). It Stability of Parallel Flows. New York: Academic Press.Google Scholar
  2. 2.
    Chandrasekhar, S. (1981). Hydrodynamic and Hydromagnetic Stability. New York: Dover. ISBN 0-486-64071-X.Google Scholar
  3. 3.
    Charru, F. (2011). Hydrodynamic Instabilities. Cambridge: Cambridge University Press. ISBN 1139500546.CrossRefGoogle Scholar
  4. 4.
    Drazin, P. G. (2002). Introduction to Hydrodynamic Stability. Cambridge: Cambridge University Press. ISBN 0-521-00965-0.CrossRefGoogle Scholar
  5. 5.
    Drazin, P. G., & Howard, L. N. (1966). Hydrodynamic stability of parallel flow of inviscid fluid. Adv. Appl. Mech., 9, 1–89.CrossRefGoogle Scholar
  6. 6.
    Drazin, P. G., & Reid, W. H. (1981). Hydrodynamic Stability. Cambridge: Cambridge University Press. ISBN 0-521-28980-7.Google Scholar
  7. 7.
    Gersting, J. M., & Jankowski, D. F. (1972). Numerical methods for Orr-Sommerfeld problems. Int. J. Numer. Methods Eng., 4, 195–206.CrossRefGoogle Scholar
  8. 8.
    Godreche, C., & Manneville, P. (1998). Hydrodynamics and Nonlinear Instabilities. Cambridge: Cambridge University Press. ISBN 0521455030.CrossRefGoogle Scholar
  9. 9.
    Heisenberg, W. (1924). Über stabilität und turbulenz von flüssigkeitsströmen. Annu. d. Physik., 74, 577–627.CrossRefGoogle Scholar
  10. 10.
    Joseph, D.D.: Stability of Fluid Motions I. In: Tracts in Natural Philosophy, vol. 27, Springer, Berlin (1976). ISBN 3-540-07514-3Google Scholar
  11. 11.
    Joseph, D.D. (1976), Joseph, D.D.: Stability of Fluid Motions II. In: Tracts in Natural Philosophy, vol. 28, Springer, Berlin (1976). ISBN 3-540-07516-XGoogle Scholar
  12. 12.
    Lin, C.C.: The Theory of Hydrodynamic Stability (corrected ed.). Cambridge University Press, Cambridge (1966). OCLC 952854Google Scholar
  13. 13.
    Lorentz, H.A.: Abhandlung über theoretische Physik I, 43.71. Leipzig (1907). Revision of a paper published by Zittingsverlag, Akad. V. Wet. Amsterdam, 6, 28 (1897)Google Scholar
  14. 14.
    Orr, W.M.F.: The stability or instability of steady motions of a perfect liquid and of a viscous liquid. Part I: A perfect liquid, Part II: A viscous liquid. Proc. Roy. Irish. Acad. 27, 9–38 and 69–138 (1907)Google Scholar
  15. 15.
    Orr, W. M. F. (1907). The stability or instability of steady motions of a liquid. Part I. Proc. Royal Irish Academy, A27, 9–68.Google Scholar
  16. 16.
    Orr, W. M. F. (1907). The stability or instability of steady motions of a liquid. Part II Proc. Royal Irish Academy, A27, 69–138.Google Scholar
  17. 17.
    Rayleigh, L.: On the stability or instability of certain fluid motions. Proc. London Math. Soc. 11, 57 (1880) and 19, 67 (1887)Google Scholar
  18. 18.
    Schlichting, H., Gersten, K.: Boundary Layer Theory. 8th revised and enlarged edition. pp. 799. Springer, Berlin (2000)Google Scholar
  19. 19.
    Sommerfeld, A.: Ein Beitrag zur hydrodynamischen Erklärung der turbulenten Flüssigkeitsbewegungen. In: Proceedings of the 4th International Congress Mathematics III, pp. 116–124. Roma (1908)Google Scholar
  20. 20.
    Squire, H. B. (1933). On the stability of three-dimensional distribution of viscous fluid between parallel walls. Proc. Roy. Soc. London, A142, 621–628.CrossRefGoogle Scholar
  21. 21.
    Sritharan, S.S.: Invariant Manifold Theory for Hydrodynamic Transition. Pitman Research Notes in Mathematics Series 241, Wiley, New York (1990). ISBN 0-582-06781-2[1]Google Scholar
  22. 22.
    Swinney, H.L., Gollub, J.P., (eds.): Hydrodynamic Instabilities and the Transition to Turbulence (2nd ed.). Springer, Berlin (1985). ISBN 978-3-540-13319-3Google Scholar
  23. 23.
    Tietjens, O.: Beiträge zur Entstehung der Turbulenz. Dissertation Universität Göttingen (1922) see also ZAMM, Zeitschr. Angew. Math. Mech. 5, 200–217 (1925)Google Scholar
  24. 24.
    Tollmien, W.: Über die Entstehung der TurbulenzKlasse. Mitteilung, Nachr. Ges. Wiss. Göttingen , Math. Phys. Klasse Engl. Translation in NACA-TM-609 (1931)Google Scholar
  25. 25.
    Tollmien, W.: Ein allgemeines Kriterium der Instabilität der laminaren Geschwindigkeitsverteilungen. Nachr. Ges. Wiss. Göttingen, math. Phys. Klasse, Fachgruppe I, 1, 79–114 (1935). Engl. Translation in NACA-TM-792 (1936)Google Scholar
  26. 26.
    Tollmien, W.: Asymptotische Integration der Störungsdifferentialgleichungen ebener laminarer Strömungen bei hohen Reynolsschen Zahlen. ZAMM, Z. angew. Math. Mech. 25, 33–45 and 27, 70–83 (1947)Google Scholar

Copyright information

© Springer International Publishing Switzerland 2016

Authors and Affiliations

  1. 1.c/o Versuchsanstalt für Wasserbau, Hydrologie und GlaziologieETH ZürichZürichSwitzerland
  2. 2.Department of Mechanical EngineeringTechnische Universität DarmstadtDarmstadtGermany

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