Spatio-Temporal Instabilities in Closed and Open Flows

  • P. Huerre
Chapter
Part of the Mathematics and Its Applications book series (MAIA, volume 33)

Abstract

A review is given of the general theory describing the linear evolution of spatio-temporal instability waves in fluid media. According to the character of the impulse response, one can distinguish between absolutely unstable (closed) flows and convectively unstable (open) flows. These notions are then applied to several evolution models of interest in weakly nonlinear stability theory. It is argued that absolutely unstable flows, convectively unstable flows and mixed flows exhibit a very different sensitivity to external perturbations. Implications of these concepts to frequency selection mechanisms in mixed flows and the onset of chaos in convectively unstable flows are also discussed.

Keywords

Shear Layer Lyapunov Exponent Branch Point Convective Instability Absolute Instability 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.
This is a preview of subscription content, log in to check access.

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. [1]
    R. J. Briggs, Electron-Stream Interaction with Plasmas, Research Monograph No, 29, M.I.T. Press, Cambridge, Mass., 1964.Google Scholar
  2. [2]
    A. Bers. Linear Waves and Instabilities, in Physique des Plasmas, C. DeWitt & J. Peyraud (Ed.), Gordon & Breach, New York, 1975.Google Scholar
  3. [3]
    G. E. Mattingly & W. O. Criminale. J. Fluid Mech., 51, 233, 1972.MATHCrossRefGoogle Scholar
  4. [4]
    P. Huerre & P. A. Monkewitz. J. Fluid Mech., 159, 151, 1985.MathSciNetMATHCrossRefGoogle Scholar
  5. [5]
    P. A. Monkewitz & K. D. Sohn. Absolute Instability in Hot Jets and Their Control, AIAA Paper No. 86–1882, 1986.Google Scholar
  6. [6]
    P. Huerre. J. Méca Theor. Appl., Special Issue on Two-Dimensional Turbulence, 121–145, 1983.Google Scholar
  7. [7]
    D. W. Bechert. Excitation of Instability Waves, Proc. Symposium IUTAM Aero et Hydro-Acoustique, Lyon, 1985.Google Scholar
  8. [8]
    N. Bleistein & R.A. Handelsman. Asymptotic Expansions of Integrals, Holt, Rinehart & Winston, New York, 1975.MATHGoogle Scholar
  9. C. M. Bender & S. A. Orszag. Advanced Mathematical Methods for Scientists and Engineers, McGraw-Hill, New York, 1978,Google Scholar
  10. [10]
    R. T. Pierrehumbert. J. Atmos. Sci., 41, 2141, 1984.CrossRefGoogle Scholar
  11. [11]
    R. T. Pierrehumbert. Spatially Amplifying Modes of the Charney Baroclinic Instability Problem, Preprint, Princeton University, 1985.Google Scholar
  12. [12]
    A. C. Newell & J. A. Whitehead. J. Fluid Mech., 38, 279, 1969.MATHCrossRefGoogle Scholar
  13. [13]
    L. A. Segel. J. Fluid Mech., 38 203, 1969.MATHCrossRefGoogle Scholar
  14. [14]
    K. Stewartson & J. T. Stuart. J. Fluid Mech., 48, 529, 1971.MathSciNetMATHCrossRefGoogle Scholar
  15. [15]
    J. Pedlosky. J. Atmoz. Sci., 29, 680, 1972.CrossRefGoogle Scholar
  16. [16]
    M. A. Weissman. Phil. Trans. R. Soc. Lond., A290, 639, 1979.CrossRefGoogle Scholar
  17. [17]
    C. G. Lange & A. C. Newell. SIAM J. Appl. Math., 21, 605, 1971.MATHCrossRefGoogle Scholar
  18. [18]
    W. C. Thacker. Geopkys. Fluid Dyn., 7, 271, 1976.CrossRefGoogle Scholar
  19. [19]
    M. Abramowitz & I. A. Stegun. Handbook of Mathematical Functions, National Bureau of Standards, Washington D.C., 1964.MATHGoogle Scholar
  20. [20]
    F. Roddier. Distributions et Trans formation de Fourier, McGraw-Hill, Paris, 1978.Google Scholar
  21. [21]
    A. Michalke. J. Fluid Mech., 23, 521, 1965.MathSciNetCrossRefGoogle Scholar
  22. [22]
    P. A. Monkewitz & P. Huerre. Thys. Fluids, 25, 1137, 1982.CrossRefGoogle Scholar
  23. [23]
    T. F. Balsa. Three-Dimensional Wavepackets and Instability Waves in Free Shear Layers, University of Arizona preprint, 1985.Google Scholar
  24. [24]
    C. M. Ho & P. Huerre. Ann. Rev. Fluid Mech., 16, 365, 1984.CrossRefGoogle Scholar
  25. [25]
    M. Gaster. J. Fluid Mech., 22, 433, 1965.MathSciNetMATHCrossRefGoogle Scholar
  26. [26]
    M. Gaster. Phys. Fluids, 11, 723, 1968.CrossRefGoogle Scholar
  27. [27]
    M. Gaster. Proc. R. Soc. Lond., A347, 271, 1975.Google Scholar
  28. [28]
    M. Gaster. The propagation of wavepackets in laminar boundary layers: asymptotic theory for non-conservative wave systems, Preprint, National Maritime Institute, Teddington, England, 1980.Google Scholar
  29. [29]
    P. G. Drazin & W. H. Reid. Hydrodynamic Stability, Cambridge University Press, Cambridge, 1981.Google Scholar
  30. [30]
    A. Michalke. Z. Flugwiss, 19, 319, 1971.MATHGoogle Scholar
  31. [31]
    F. H. Busse, in Topics in Applied Physics, 45, Springer-Verlag, Berlin, 1981.Google Scholar
  32. [32]
    P. Hall. J. Fluid Mech., 130, 41, 1983.MathSciNetMATHCrossRefGoogle Scholar
  33. [33]
    S. A. Thorpe. J. Fluid Mech., 46, 299, 1971.CrossRefGoogle Scholar
  34. [34]
    A. Michalke. J. Fluid Mech., 19, 543, 1964.MathSciNetMATHCrossRefGoogle Scholar
  35. [35]
    W. Koch. J. Sound and Vib., 99, 53, 1985.CrossRefGoogle Scholar
  36. [36]
    L. O. Merkine. Geophys. Astrophys. Fluid Dyn., 9, 129, 1977.MATHCrossRefGoogle Scholar
  37. [37]
    D. Rockwell & E. Naudascher. Ann. Rev. Fluid Mech., 11, 67, 1979.CrossRefGoogle Scholar
  38. [38]
    L. N. Nguyen. Frequency Selection in Jets and Wakes, Master of Science Thesis, U.C.L.A., Los Angeles, California, 1986.Google Scholar
  39. [39]
    S. J. Leib & M. E. Goldstein, Phys. Fluids, 29, 952, 1986.CrossRefGoogle Scholar
  40. [40]
    R. J. Deissler. J. Stat. Phys., 40, 371, 1985a.MathSciNetMATHCrossRefGoogle Scholar
  41. [41]
    R. J. Deissler. Spatially Growing Waves, Intermittency and Convective Chaos in an Open-Flow System, Los Alamos Report LA-UR-85–4211, 1985b.Google Scholar
  42. [42]
    R. J. Deissler & K. Kaneko. Velocity-Dependent Liapunov Exponents as a Measure of Chaos for Open Flow Systems, Los Alamos Report LA-UR-85–3249, 1985.Google Scholar
  43. [43]
    P. Bergé, Y. Pomeau & C. Vidal. L’Ordre dans le Chaos, Herman, Paris, 1984.MATHGoogle Scholar
  44. [44]
    Y. Kuramoto. Prog. Theor. Phys. Suppl., 64, 346, 1978.CrossRefGoogle Scholar
  45. [45]
    H. T. Moon, P. Huerre & L. G. Redekopp. Physica D7, 135, 1983.MathSciNetGoogle Scholar
  46. [46]
    L. R. Keefe. Stud. Appl. Math., 73, 91, 1985.MathSciNetMATHGoogle Scholar
  47. [47]
    J. T. Stuart & R. C. DiPrimaProc. R. Soc. London, A362, 27, 1978.Google Scholar

Copyright information

© D. Reidel Publishing Company 1987

Authors and Affiliations

  • P. Huerre
    • 1
  1. 1.Department of Aerospace EngineeringUniversity of Southern CaliforniaLos AngelesUSA

Personalised recommendations