Hostname: page-component-848d4c4894-tn8tq Total loading time: 0 Render date: 2024-06-21T02:11:47.649Z Has data issue: false hasContentIssue false
  • English
  • Français

On the stability of Poiseuille-Couette flow: a bifurcation from infinity

Published online by Cambridge University Press:  20 April 2006

S. J. Cowley  and
F. T. Smith
S. J. Cowley
Affiliation:
Mathematics Department, University College, Gower Street, London WC1E 6BT
F. T. Smith
Affiliation:
Mathematics Department, University College, Gower Street, London WC1E 6BT
Get access Rights & Permissions [Opens in a new window]

Abstract

The linear and weakly nonlinear stability of Poiseuille–Couette flow is considered for various values of the relative wall velocity 2uw, An account is given first of the asymptotic upper and lower branches of the linear neutral curve (s), followed by their disappearance, as uw is increased. Two main (and one minor) neutral curves are found to exist for smaller O(1) (or lesser) values of uw, then one for moderate O(1) values of uw, and none for larger O(1) values of uw. The cut-off velocity at which each main neutral curve disappears is determined, and in each case the whole neutral curve for uw just below the cut-off value is determined in closed form. Secondly, weakly nonlinear solutions are found to bifurcate subcritically from the neutral curve for uw just below cut-off, but to ‘bifurcate from infinity’ just above cut-off. This identifies a minimum threshold amplitude at the entry to the regime where no linear neutral curve exists.

Type
Research Article
Information
Journal of Fluid Mechanics , Volume 156 , July 1985 , pp. 83 - 100
Copyright
© 1985 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

Chen, T. S. & Joseph, D. D. 1973 Subcritical bifurcation of plane Poiseuille flow. J. Fluid Mech. 58, 337–351.Google Scholar
Coffee, T. 1977 Finite-amplitude instability of plane Couette flow. J. Fluid Mech. 83, 401–414.Google Scholar
Davey, A. 1978a On Itoh's amplitude stability theory for pipe flow. J. Fluid Mech. 86, 695–703.Google Scholar
Davey, A. 1978b On the stability of flow in an elliptic pipe which is nearly circular. J. Fluid Mech. 87, 233–241.Google Scholar
Davey, A. & Nguyen, H. P. F. 1971 Finite-amplitude stability of pipe flow. J. Fluid Mech. 45, 701–720.Google Scholar
Ellingsen, T., Gjevick, B. & Palm, E. 1970 On the nonlinear stability of plane Couette flow. J. Fluid Mech. 40, 97–112.Google Scholar
Hains, F. D. 1967 Stability of plane Couette-Poiseuille flow. Phys. Fluids 10, 2079–2080.Google Scholar
Heisenberg, W. 1924 Über Stabilität und Turbulenz von Flüssigkeitsströmen. Ann. der Phys. 74, 577–627.Google Scholar
Herbert, T. 1976 Periodic motions in a plane channel. In Proc. 5th Intl Conf. on Numerical Methods in Fluid Dynamics (ed. A. I. van de Vooren & P. J. Zandbergen). Lecture Notes in Physics, vol. 59. Springer.
Herbert, T. 1977 Die neutrale Fläche der ebenen Poiseuille-Strömung. Habilitationsschrift, Universität Stuttgart.
Herbert, T. 1980 Nonlinear stability of parallel flows by high-order amplitude expansions. AIAA J. 18, 243–248.Google Scholar
Herbert, T. 1981a Stability of plane Poiseuille flow. Theory and experiment. In Proc. XV Symp. on Advanced Problems and Methods in Fluid Mechanics, Jachranka, Poland. See also Fluid Dyn. Trans. 11 (1983), 77–126, and VPI-E-81-35.
Herbert, T. 1981b A secondary instability mechanism in plane Poiseuille flow. Bull. Am. Phys. Soc. 26, 1257.Google Scholar
Herbert, T. 1983a Secondary instability of plane channel flow to subharmonic three-dimensional disturbances. Phys Fluids 26, 871–874.Google Scholar
Herbert, T. 1983b Subharmonic three-dimensional disturbances in unstable plane shear flows. AIAA-83-1759.
Hocking, L. M. 1977 The stability of flow in an elliptic pipe with large aspect ratio. Q. J. Mech. Appl. Maths 30, 343–353.Google Scholar
Itoh, N. 1977 Nonlinear stability of parallel flows with subcritical Reynolds numbers. Part 1. An asymptotic theory valid for small-amplitude disturbances. J. Fluid Mech. 82, 455–467.Google Scholar
Lin, C. C. 1945 On the stability of two-dimensional parallel flows. Q. Appl. Maths 3, 117–142, 218–234, 277–301.Google Scholar
Lin, C. C. 1955 The Theory of Hydrodynamic Stability. Cambridge University Press.
Mackrodt, P.-A. 1976 Stability of Hagen-Poiseuille flow with superimposed rigid rotation. J. Fluid Mech. 73, 153–164.Google Scholar
Meksyn, D. & Stuart, J. T. 1951 Stability of viscous motion between parallel planes for finite disturbances. Proc. R. Soc. Lond. A 208, 517–526.Google Scholar
Miles, J. W. 1960 The hydrodynamic stability of a thin film of liquid in uniform shearing motion. J. Fluid Mech. 8, 593–610.Google Scholar
Orszag, S. A. 1971 Accurate solution of the Orr-Sommerfeld stability equation. J. Fluid Mech. 50, 689–703.Google Scholar
Orszag, S. A. & Kells, L. C. 1980 Transition to turbulence in plane Poiseuille and plane Couette flow. J. Fluid Mech. 96, 159–206.Google Scholar
Orszag, S. A. & Patera, A. T. 1980 Suberitical transition to turbulence in plane channel flows. Phys. Rev. Lett. 45, 989–993.Google Scholar
Potter, M. C. 1966 Stability of plane Couette-Poiseuille flow. J. Fluid Mech. 24, 609–619.Google Scholar
Reid, W. H. 1965 The stability of parallel flows. In Basic Developments in Fluid Dynamics, vol. 1 (ed. M. Holt). Academic.
Reynolds, W. C. & Potter, M. C. 1967 Finite-amplitude instability of parallel shear flows. J. Fluid Mech. 27, 465–492.Google Scholar
Romanov, V. A. 1973 Stability of plane-parallel Couette flow. Funct. Anal. Applics 7, 137–146.Google Scholar
Rosenblat, S. & Davis, S. H. 1979 Bifurcation from infinity. SIAM J. Appl. Maths 37, 1–19.Google Scholar
Smith, F. T. 1974 Boundary layer flow near a discontinuity in wall conditions. J. Inst. Maths Applics 13, 127–145.Google Scholar
Smith, F. T. 1979 Instability of flow through pipes of general cross-section. Parts I and II. Mathematika 26, 187–210, 211–223.Google Scholar
Smith, F. T. & Bodonyi, R. J. 1982 Amplitude-dependent neutral modes in the Hagen-Poiseuille flow through a circular pipe. Proc. R. Soc. Lond. A 384, 463–489.Google Scholar
Smith, F. T., Papageorgiou, D. & Elliott, J. W. 1984 An alternative approach to linear and nonlinear stability calculations at finite Reynolds numbers. J. Fluid Mech. 146, 313–330.Google Scholar
Stewartson, K. & Stuart, J. T. 1971 A nonlinear instability theory for a wave system in plane Poiseuille flow. J. Fluid Mech. 48, 529–545.Google Scholar
Stuart, J. T. 1960 On the non-linear mechanics of wave disturbances in stable and unstable parallel flows. Part 1. J. Fluid Mech. 9, 353–370.Google Scholar
Stuart, J. T. 1963 Hydrodynamic stability. In Laminar Boundary Layers (ed. L. Rosenhead). Clarendon.
Stuart, J. T. 1981 Instability and transition in pipes and channels. In Transition and Turbulence (ed. R. E. Meyer). Academic.
Zahn, J.-P., Toomre, J., Spiegel, E. A. & Gough, D. O. 1974 Nonlinear cellular motions in Poiseuille channel flow. J. Fluid Mech. 64, 319–345.Google Scholar