Hostname: page-component-77c89778f8-m42fx Total loading time: 0 Render date: 2024-07-16T21:19:39.406Z Has data issue: false hasContentIssue false
  • English
  • Français

The instability of oscillatory plane Poiseuille flow

Published online by Cambridge University Press:  20 April 2006

Christian H. Von Kerczek
Christian H. Von Kerczek
Affiliation:
Mechanical Engineering Department, The Catholic University of America, Washington, D.C.
Get access Rights & Permissions [Opens in a new window]

Abstract

The instability of oscillatory plane Poiseuille flow, in which the pressure gradient is time-periodically modulated, is investigated by a perturbation technique. The Floquet exponents (i.e. the complex growth rates of the disturbances to the oscillatory flow) are computed by series expansions, in powers of the oscillatory to steady flow velocity amplitude ratio, about the values of the growth rates of the disturbances of the steady flow. It is shown that the oscillatory flow is more stable than the steady flow for values of Reynolds number and disturbance wave number in the vicinity of the steady flow critical point and for values of frequencies of imposed oscillation greater than about one tenth of the frequency of the steady flow neutral disturbance. At very high and low values of imposed oscillation frequency, the unsteady flow is slightly less stable than the steady flow. These results hold for the values of the velocity amplitude ratio at least up to 0·25.

Type
Research Article
Information
Journal of Fluid Mechanics , Volume 116 , March 1982 , pp. 91 - 114
Copyright
© 1982 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

Davis, S. H. 1976 The stability of time-periodic flows. Ann. Rev. Fluid Mech. 8, 57–74.Google Scholar
Coddington, E. A. & Levinson, N. 1955 Theory of Ordinary Differential Equations, pp. 78–81. McGraw-Hill.
Finucane, R. G. & Kelly, R. E. 1976 Onset of instability in a fluid layer heated sinusoidally from below. Int. J. Heat Mass Transfer 19, 71–85.Google Scholar
Gaunt, D. S. & Guttmann, A. T. 1974 Asymptotic analysis of coefficients. In Phase Transitions and Critical Points (ed. C. Domb & M. S. Green), vol. 3, pp. 181–243. Academic.
Gottlieb, D. & Orszag, S. A. 1978 Numerical Analysis of Spectral Methods: Theory and Applications, pp. 143–148. Soc. for Industrial and Appl. Math.
Grosch, C. E. & Salwen, H. 1968 The stability of steady and time-dependent plane Poisueille flow. J. Fluid Mech. 34, 177–205.Google Scholar
Hall, P. 1975 The stability of Poiseuille flow modulated at high frequencies. Proc. R. Soc. Lond. A 344, 453–464.Google Scholar
Herbert, D. M. 1972 The energy balance in modulated plane Poiseuille flow. J. Fluid Mech. 56, 73–80.Google Scholar
Kerczek, C. Von & Davis, S. H. 1974 Linear stability theory of oscillatory Stokes layers. J. Fluid Mech. 62, 689–703.Google Scholar
Miller, J. A. & Fejer, A. A. 1964 Transition phenomena in oscillating boundary layer flows. J. Fluid Mech. 18, 438–448.Google Scholar
Obremski, H. J. & Fejer, A. A. 1967 Transition in oscillating boundary layer flows. J. Fluid Mech. 29, 93–111.Google Scholar
Orszag, S. A. 1971 Accurate solution of the Orr — Sommerfeld stability equation. J. Fluid Mech. 50, 689–703.Google Scholar
Rosenblat, S. & Herbert, D. M. 1970 Low-frequency modulation of thermal instability. J. Fluid Mech. 43, 385–398.Google Scholar
Shanks, D. 1955 Non-linear transformation of divergent and slowly convergent sequences. J. Math. & Phys. 34, 1–42.Google Scholar
Vanden-Broeck, J. M. & Schwartz, L. W. 1979 A one-parameter family of sequence transformations. SIAM J. Math. Anal. 10, 658–666.Google Scholar
Van Dyke, M. 1974 Analysis and improvement of perturbation series. Quart. J. Mech. Appl. Math. 27, 423–450.Google Scholar
Yakubovich, V. A. & Starzhinskii, V. M. 1975 Linear differential equations with periodic coefficients (transl. D. Louvish), cha. 4. Wiley.