Hostname: page-component-6dbcb7884d-g628x Total loading time: 0 Render date: 2025-02-14T08:22:12.175Z Has data issue: false hasContentIssue false
  • English
  • Français

Nonlinear cellular motions in Poiseuille channel flow

Published online by Cambridge University Press:  29 March 2006

J.-P. Zahn ,
Juri Toomre ,
E. A. Spiegel  and
D. O. Gough
J.-P. Zahn
Affiliation:
Department of Mathematics, New York University and Astronomy Department, Columbia University, New York 10027[dagger]
Juri Toomre
Affiliation:
Department of Mathematics, New York University and Goddard Institute for Space Studies, New York 10025
E. A. Spiegel
Affiliation:
Astronomy Department, Columbia University, New York 10027
D. O. Gough
Affiliation:
Goddard Institute for Space Studies, New York 10025
Get access Rights & Permissions [Opens in a new window]

Abstract

We expand the equations describing plane Poiseuille flow in Fourier series in the co-ordinates in the plane parallel to the bounding walls. There results an infinite system of equations for the amplitudes, which are functions of time and of the cross-stream co-ordinate. This system is drastically truncated and the resulting set of equations is solved accurately by a finite difference method. Three truncations are considered: (I) a single mode with dependence only on the downstream co-ordinate and time, (II) the mode of (I) plus its first harmonic, (III) a single three-dimensional mode. For all three cases, for a variety of initial conditions, the solutions evolve to a steady state as seen in a particular moving frame of reference. No runaways are encountered.

Type
Research Article
Information
Journal of Fluid Mechanics , Volume 64 , Issue 2 , 19 June 1974 , pp. 319 - 346
Copyright
© 1974 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

Bergeron, R. F. 1970 Some inviscid ‘cat's eye’ flows. Studies Appl. Math. 49, 197–203.Google Scholar
Busse, F. H. 1967 On the stability of two-dimensional convection in a layer heated from below. J. Math. & Phys. 46, 140–150.Google Scholar
Chen, T. S. & Joseph, D. D. 1973 Subcritical bifurcation of plane Poiseuille flow. J. Fluid Mech. 58, 337–351.Google Scholar
Comte-Bellot, G. 1965 Écoulement turbulent entre deux parois parallèles. Ph.D. thesis, University of Grenoble.
Dowell, E. H. 1969 Non-linear theory of unstable plane Poiseuille flow. J. Fluid Mech. 38, 401–414.Google Scholar
Eckhaus, W. 1965 Studies in Non-linear Stability Theory. Springer.
George, W. D. & Hellums, J. D. 1972 Hydrodynamic stability of plane Poiseuille flow with finite amplitude disturbances. J. Fluid Mech. 51, 687–704.Google Scholar
Grohne, D. 1969 Die Stabilität der ebenen Kanalströmung gegenüber dreidimensionalen Störungen von endlicher Amplitude. A V A Göttingen Rep. 69-A-30.Google Scholar
Henrici, P. 1962 Discrete Variable Methods in Ordinary Differential Equations. Wiley.
Landahl, M. T. 1972 Wave mechanics of breakdown. J. Fluid Mech. 56, 775–802.Google Scholar
Landau, L. 1944 On the problem of turbulence. C. R. Acad. Sci. U.R.S.S. 44, 311–314.Google Scholar
Laufer, J. 1951 Investigation of turbulence in a two-dimensional channel. N.A.C.A. Tech. Rep. no. 1053.Google Scholar
Lin, C. C. 1955 The Theory of Hydrodynamic Stability. Cambridge University Press.
Meksyn, D. & Stuart, J. T. 1951 Stability of viscous motion between parallel planes for finite disturbances. Proc. Roy. Soc. A 208, 517–526.Google Scholar
Monin, A. S. & Yaglom, A. M. 1971 Statistical Fluid Mechanics (ed. J. L. Lumley). M.I.T. Press.
Orszag, S. A. 1971 Accurate solution of the Orr—Sommerfeld stability equation. J. Fluid Mech. 50, 689–703.Google Scholar
Pekeris, C. L. & Shkoller, B. 1967 Stability of plane Poiseuille flow to periodic disturbances of finite amplitude in the vicinity of the neutral curve. J. Fluid Mech. 29, 31–38.Google Scholar
Pekeris, C. L. & Shkoller, B. 1969a Stability of plane Poiseuille flow to periodic disturbances of finite amplitude. J. Fluid Mech. 39, 611–627.Google Scholar
Pekeris, C. L. & Shkoller, B. 1969b The neutral curves for periodic disturbances of finite amplitude of Poiseuille flow. J. Fluid Mech. 39, 629–639.Google Scholar
Pekeris, C. L. & Shkoller, B. 1971 Stability of plane Poiseuille flow to periodic disturbances of finite amplitude. II. Proc. Nat. Acad. Sci. 68, 197–199.Google Scholar
Reynolds, W. C. & Potter, M. C. 1967 Finite-amplitude instability of parallel shear flow. J. Fluid Mech. 27, 465–492.Google Scholar
Stewartson, K. & Stuart, J. T. 1971 A non-linear 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. The basic behaviour in plane Poiseuille flow. J. Fluid Mech. 9, 353–370.CrossRefGoogle Scholar
Stuart, J. T. 1971 Nonlinear stability theory. Ann. Rev. Fluid Mech. 3, 347–370.Google Scholar
Thomas, L. H. 1953 The stability of plane Poiseuille flow. Phys. Rev. 91, 780–783.Google Scholar
Veronis, G. 1966 Motions at subcritical values of the Rayleigh number in a rotating fluid. J. Fluid Mech. 24, 545–554.Google Scholar
Veronis, G. 1968 Effect of a stabilizing gradient of solute on thermal convection. J. Fluid Mech. 34, 315–336.Google Scholar
Watson, J. 1960 On the non-linear mechanics of wave disturbances in stable and unstable parallel flows. Part 2. The development of a solution for plane Poiseuille flow and plane Couette flow. J. Fluid Mech. 9, 371–389.CrossRefGoogle Scholar