An Eigenfunction Expansion of Localized Disturbances

  • Dan S. Henningson

Abstract

Turbulence inherently involves complicated three-dimensional motions, whereas laminar flow in many instances is two-dimensional. In investigations of the transition process from laminar to turbulent flow, the onset of three-dimensionality has therefore played a major role. The pioneering work of Klebanoff, Tidstrom & Sargent (1962) on the three-dimensional nature of transition in boundary layers started a process which led to the theory of secondary instability. Secondary instability, recently reviewed by Herbert (1988), is able to predict the onset of three-dimensionality as an instability of two-dimensional finite amplitude waves to infinitesimal oblique disturbances. The primary wave is usually taken as the least stable Orr-Sommerfeld (O-S) mode, which is two-dimensional in the Reynolds number range around the onset of growth, i.e. around the critical Reynolds number

Keywords

Vortex Vorticity 

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. DiPrima, R.C. and Habetler, G.J. 1969 A completeness theorem for non-selfadjoint eigen-value problems in hydrodynamic stability. Arch. Rat. Mech. Anal. 34, 218–227.CrossRefMathSciNetGoogle Scholar
  2. Gustaysson, L.H. and Hultgren, L.S. 1980 A resonance mechanism in plane Couette flow. J. Fluid Mech. 98, 149–159.CrossRefADSMathSciNetGoogle Scholar
  3. Henningson, D.S. 1988 The inviscid initial value problem for a piecewise linear mean flow. Stud. Appi. Math. 78, 31–56.MATHMathSciNetGoogle Scholar
  4. Henningson, D.S., Johansson, A.V. and Lundbladh, A. 1990 On the evolution of localized disturbances in laminar shear flows. In Laminar — Turbulent Transition 3, ( Eds. Michel, R. and Arnal, D. ), Springer.Google Scholar
  5. Herbert, T. 1988 Secondary instability of boundary layers. Ann. Rev. Fluid Mech. 20, 487–526.CrossRefADSGoogle Scholar
  6. Klebanoff, P.S., Tidstrom, K.D. and Sargent, L.M. 1962 The three dimensional nature of boundary layer transition. J. Fluid Mech. 12, 1–34.CrossRefMATHADSGoogle Scholar
  7. Landahl, M.T. 1980 A note on an algebraic instability of inviscid parallel shear flows. J. Fluid Mech. 98, 243–251.CrossRefMATHADSMathSciNetGoogle Scholar
  8. Mack, L.M. 1976 A numerical study of the temporal eigenvalue spectrum of the Blasius boundary layer. J. Fluid Mech. 73, 497–520.CrossRefMATHADSGoogle Scholar
  9. Schensted, I.V. 1961 Contributions to the theory of hydrodynamic stability. Ph.D. Thesis, Univ. Michigan, Ann Arbor.Google Scholar

Copyright information

© Springer-Verlag Berlin, Heidelberg 1991

Authors and Affiliations

  • Dan S. Henningson
    • 1
  1. 1.Department of MathematicsMITCambridgeUSA

Personalised recommendations