Hostname: page-component-cd9895bd7-8ctnn Total loading time: 0 Render date: 2024-12-18T23:46:42.005Z Has data issue: false hasContentIssue false
  • English
  • Français

The continuous spectrum of the Orr-Sommerfeld equation. Part 1. The spectrum and the eigenfunctions

Published online by Cambridge University Press:  12 April 2006

Chester E. Grosch  and
Harold Salwen
Chester E. Grosch
Affiliation:
Institute of Oceanography, Old Dominion University, Norfolk, Virginia 23508
Harold Salwen
Affiliation:
Department of Physics and Engineering Physics, Stevens Institute of Technology, Hoboken, New Jersey 07030
Get access Rights & Permissions [Opens in a new window]

Abstract

It is shown that the Orr-Sommerfeld equation, which governs the stability of any mean shear flow in an unbounded domain which approaches a constant velocity in the far field, has a continuous spectrum. This result applies to both the temporal and the spatial stability problem. Formulae for the location of this continuum in the complex wave-speed plane are given. The temporal continuum eigenfunctions are calculated for two sample problems: the Blasius boundary layer and the two-dimensional laminar jet. The nature of the eigenfunctions, which are very different from the Tollmien-Schlichting waves, is discussed. Three mechanisms are proposed by which these continuum modes could cause transition in a shear flow while bypassing the usual linear Tollmien-Schlichting stage.

Type
Research Article
Information
Journal of Fluid Mechanics , Volume 87 , Issue 1 , 12 July 1978 , pp. 33 - 54
Copyright
© 1978 Cambridge University Press

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

Batchelor, G. K. 1967 An Introduction to Fluid Dynamics. Cambridge University Press.
Bethe, H. A. 1947 Elementary Nuclear Theory. Wiley.
Brown, W. B. 1959 Numerical calculation of the stability of cross flow profiles in laminar boundary layers on a rotating disc and on a swept back wing and an exact calculation of the stability of the Blasius velocity profile. Northrop Aircraft, Inc., Rep. NAI 59–5.Google Scholar
Case, K. M. 1960 Stability of inviscid Couette flow. Phys. Fluids 3, 143.Google Scholar
Case, K. M. 1961 Hydrodynamic stability and the inviscid limit. J. Fluid Mech. 10, 420.Google Scholar
Corner, D., Houston, D. J. R. & Ross, M. A. S. 1976 Higher eigenstates in boundary-layer stability theory. J. Fluid Mech. 77, 81.Google Scholar
Davis, S. H. 1976 The stability of time-periodic flows. Ann. Rev. Fluid Mech. 8, 57.Google Scholar
Diprima, R. C. & Habetler, G. J. 1969 A completeness theorem for nonself-adjoint eigenvalue problems in hydrodynamic stability. Arch. Rat. Mech. Anal. 34, 218.Google Scholar
Friedman, B. 1956 Principles and Techniques of Applied Mathematics. Wiley.
Gallagher, A. P. & Mercer, A.MCD. 1964 On the behaviour of small disturbances in plane Couette flow. Part 2. The higher eigenvalues. J. Fluid Mech. 18, 350.Google Scholar
Grosch, C. E. & Salwen, H. 1968 The stability of steady and time-dependent plane Poiseuille flow. J. Fluid Mech. 34, 177.Google Scholar
Grosch, C. E. & Salwen, H. 1978 The spatial continuum revisited. Bull. Am. Phys. Soc. 23 (to appear).Google Scholar
Haupt, O. 1912 Über die Entwicklung einer willkürlichen Funktion nach den Eigenfunktionen des Turbulenzproblems. Sber. bayer. Akad. Wiss. 2, 289.Google Scholar
Howard, L. N. 1961 Note on a paper of John W. Miles. J. Fluid Mech. 10, 509.Google Scholar
Jordinson, R. 1971 Spectrum of eigenvalues of the Orr–Sommerfeld equation for Blasius flow. Phys. Fluids 14, 177.Google Scholar
Kaplan, R. E. 1964 The stability of laminar, incompressible boundary layers in the presence of compliant boundaries. Mass. Inst. Tech. Rep. ASRL TR 116–1.Google Scholar
Kurtz, E. F. & Crandall, S. H. 1962 Computer aided analysis of hydrodynamic stability. J. Math. & Phys. 41, 264.Google Scholar
Landahl, M. 1966 A time-shared program for the stability problem for parallel flows over rigid or flexible surfaces. Mass. Inst. Tech. Rep. ASRL TR 116–4.Google Scholar
Landau, L. D. & Lifshitz, E. M. 1959 Quantum Mechanics. Addison-Wesley.
Lessen, M. S., Sadler, G. & Liu, T. Y. 1968 Stability of pipe Poiseuille flow. Phys. Fluids 11, 1404.Google Scholar
Lin, C. C. 1955 The Theory of Hydrodynamic Stability. Cambridge University Press.
Lin, C. C. 1961 Some mathematical problems in the theory of the stability of parallel flows. J. Fluid Mech. 10, 430.Google Scholar
Mack, L. M. 1965 Computation of the stability of the laminar compressible boundary layer. In Methods of Computational Physics, vol. 4, p. 247. Academic Press.
Mack, L. M. 1976 A numerical study of the temporal eigenvalue spectrum of the Blasius boundary layer. J. Fluid Mech. 73, 497.Google Scholar
Nachtsheim, P. R. 1963 Stability of free-convection boundary layer flows. N.A.S.A. Tech. Note D-2089.Google Scholar
Orszag, S. A. 1971 Accurate solution of the Orr–Sommerfeld stability equation. J. Fluid Mech. 50, 689.Google Scholar
Radbill, J. R. & Van driest, E. R. 1966 A new method for the prediction of stability of laminar boundary layers. North Am. Aviation Rep. AD 633 978.Google Scholar
Rayleigh, Lord 1880 On the stability, or instability, of certain fluid motions. Scientific Papers, vol. 1, p. 474. Cambridge University Press.
Reynolds, O. 1883 An experimental investigation of the circumstances which determine whether the motion of water shall be direct or sinuous, and the law of resistance in parallel channels. Phil. Trans. Roy. Soc. 174, 935.Google Scholar
Rogler, H. L. 1975 The interaction between vortex-array representations of freestream turbulence and impermeable bodies. A.I.A.A. 13th Aerospace Sci. Meeting. Pasadena.Google Scholar
Rogler, H. L. & Reshotko, E. 1975 Disturbances in a boundary layer introduced by a low intensity array of vortices. SIAM J. Appl. Math. 28, 431.Google Scholar
Rosencrans, S. A. & Sattinger, D. H. 1966 On the spectrum of an operator occurring in the theory of hydrodynamic stability. J. Math. & Phys. 45, 289.Google Scholar
Salwen, H. & Grosch, C. E. 1972 Stability of Poiseuille flow in a circular pipe. J. Fluid Mech., 54, 93.Google Scholar
Saric, W. S. & Nayfeh, A. H. 1975 Nonparallel stability of boundary-layer flows. Phys. Fluids 18, 945.Google Scholar
Sattinger, D. H. 1967 On the Rayleigh problem in hydrodynamic stability. SIAM J. Appl. Math. 15, 419.Google Scholar
Schensted, I. V. 1960 Contributions to the theory of hydrodynamic stability. Ph.D. dissertation, University of Michigan.
Schlichting, H. 1951 Boundary Layer Theory. McGraw-Hill.
Schubauer, G. B. & Skramstad, H. K. 1941 Laminar boundary-layer oscillations and stability of laminar flow. N.A.C.A. Rep. W8.Google Scholar
Schubauer, G. B. & Skramstad, H. K. 1947 Laminar boundary-layer oscillations and stability of laminar flow. J. Aero. Sci. 14, 69.Google Scholar
Schubauer, G. B. & Skramstad, H. K. 1948 Laminar boundary-layer oscillations and stability of laminar flow. N.A.C.A. Rep. no. 909.Google Scholar
Squire, H. B. 1933 On the stability of the three-dimensional disturbances of viscous flow between parallel walls. Proc. Roy. Soc. A 142, 621.Google Scholar
Taylor, G. I. 1923 On the decay of vortices in a viscous fluid. Phil. Mag. 46, 671.Google Scholar
Tollmien, W. 1931 Über die Entstehung der Turbulenz. N.A.C.A. Tech. Memo. no. 609.Google Scholar
Wazzan, A. R., Okamura, T. T. & Smith, A. M. O. 1968 Spatial and temporal stability charts for the Falkner–Skan boundary layer profiles. Douglas Aircraft Co. Rep. DAC-67086.Google Scholar