Hostname: page-component-cd9895bd7-hc48f Total loading time: 0 Render date: 2024-12-28T21:34:51.040Z Has data issue: false hasContentIssue false
  • English
  • Français

Transition to turbulence in plane Poiseuille and plane Couette flow

Published online by Cambridge University Press:  19 April 2006

Steven A. Orszag  and
Lawrence C. Kells
Steven A. Orszag
Affiliation:
Department of Mathematics, Massachusetts Institute of Technology, Cambridge, MA 02139
Lawrence C. Kells
Affiliation:
Department of Mathematics, Massachusetts Institute of Technology, Cambridge, MA 02139 Present address: 1929 Crisanto 907, Mountain View, CA 94040.
Get access Rights & Permissions [Opens in a new window]

Abstract

Direct numerical solutions of the three-dimensional time-dependent Navier-Stokes equations are presented for the evolution of three-dimensional finite-amplitude disturbances of plane Poiseuille and plane Couette flows. Spectral methods using Fourier series and Chebyshev polynomial series are used. It is found that plane Poiseuille flow can sustain neutrally stable two-dimensional finite-amplitude disturbances at Reynolds numbers larger than about 2800. No neutrally stable two-dimensional finite-amplitude disturbances of plane Couette flow were found.

Three-dimensional disturbances are shown to have a strongly destabilizing effect. It is shown that finite-amplitude disturbances can drive transition to turbulence in both plane Poiseuille flow and plane Couette flow at Reynolds numbers of order 1000. Details of the resulting flow fields are presented. It is also shown that plane Poiseuille flow cannot sustain turbulence at Reynolds numbers below about 500.

Type
Research Article
Information
Journal of Fluid Mechanics , Volume 96 , Issue 1 , 16 January 1980 , pp. 159 - 205
Copyright
© 1980 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

Antar, B. N. & Collins, F. G. 1975 Numerical calculation of finite amplitude effects in unstable laminar boundary layers. Phys. Fluids 18, 289.Google Scholar
Benney, D. J. 1961 A non-linear theory for oscillations in a parallel flow. J. Fluid Mech. 10, 209.Google Scholar
Benney, D. J. 1964 Finite amplitude effects in an unstable laminar boundary layer. Phys. Fluids 7, 319.Google Scholar
Benney, D. J. & Lin, C. C. 1960 On the secondary motion induced by oscillations in a shear flow. Phys. Fluids 3, 656.Google Scholar
Coffee, T. 1977 Finite amplitude instability of plane Couette flow. J. Fluid Mech. 83, 401.Google Scholar
Comte-Bellot, G. 1965 Écoulement turbulent entre deux parois parallèles. Publications Scientifiques et Techniques du Ministère de l'Air, no. 419.
Davey, A. 1973 On the stability of plane Couette flow to infinitesimal disturbances. J. Fluid Mech. 57, 369.Google Scholar
Davey, A. 1978 On Itoh's finite amplitude stability theory for pipe flow. J. Fluid Mech. 86, 695.Google Scholar
Davey, A., Hocking, L. M. & Stewartson, K. 1974 On the nonlinear evolution of three-dimensional disturbances in plane Poiseuille flow. J. Fluid Mech. 63, 529.Google Scholar
Davey, A. & Nguyen, H. P. F. 1971 Finite-amplitude stability of pipe flow. J. Fluid Mech. 45, 701.Google Scholar
Davies, S. J. & White, C. M. 1928 An experimental study of the flow of water in pipes of rectangular section. Proc. Roy. Soc. A 119, 92.Google Scholar
Ellingsen, T., Gjevik, B. & Palm, E. 1970 On the non-linear stability of plane Couette flow. J. Fluid Mech. 40, 97.Google Scholar
Eckhaus, W. 1965 Studies in Non-Linear Stability Theory. Springer.
Fasel, H. 1976 Investigation of the stability of boundary layers by a finite-difference model of the Navier-Stokes equations. J. Fluid Mech. 78, 355.Google Scholar
Fasel, H., Bestek, H. & Schefenacker, R. 1977 Numerical simulation studies of transition phenomena in incompressible, two-dimensional flows. In Laminar-Turbulent Transition, AGARD Conf. Proc. no. 224, p. 14–1.
George, W. D., Hellums, J. D. & Martin, B. 1974 Finite-amplitude neutral disturbances in plane Poiseuille flow. J. Fluid Mech. 63, 765.Google Scholar
Gottlieb, D. & Orszag, S. A. 1977 Numerical Analysis of Spectral Methods: Theory and Applications. NSF-CBMS Monograph no. 26, Soc. Ind. App. Math., Philadelphia.
Grohne, D. 1969 Die Stabilität der ebenen Kanalströmung gegenüber dreidimensionalen Störungen von endlicher Amplitude. A. V. A. Göttingen Rep. 69-A-30.
Herbert, T. 1976 Periodic secondary motions in a plane channel. In Proc. 5th Int. Conf. on Numerical Methods in Fluid Dynamics (ed. A. I. van de Vooren and P. J. Zandbergen), p. 235. Springer.
Herbert, T. 1977 Finite amplitude stability of plane parallel flows. In Laminar-Turbulent Transition, AGARD Conf. Proc. no. 224, p. 3–1.
Hocking, L. M., Stewartson, K. & Stuart, J. T. 1972 A nonlinear instability burst in plane parallel flow. J. Fluid Mech. 51, 705.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.Google Scholar
Itoh, N. 1978 Three-dimensional growth of finite wave disturbances in plane Poiseuille flow. To be published.
Kao, T. W. & Park, C. 1970 Experimental investigations of the stability of channel flows. Part 1. Flow of a single liquid in a rectangular channel. J. Fluid Mech. 43, 145.Google Scholar
Kells, L. C. 1978 Numerical calculations of turbulent and transition flows. Ph.D. thesis, M.I.T, Cambridge, MA.
Klebanoff, P. S., Tidstrom, K. D. & Sargent, L. M. 1962 The three-dimensional nature of boundary-layer instability. J. Fluid Mech. 12, 1.Google Scholar
Kuwabara, S. 1967 Nonlinear instability of plane Couette flow. Phys. Fluids 10, S115.Google Scholar
Lakin, W. D., Ng, B. S. & Reid, W. H. 1978 Approximations to the eigenvalue relation for the Orr-Sommerfeld problem. Phil. Trans. Roy. Soc. A 289, 347.Google Scholar
Landahl, M. T. 1972 Wave mechanics of breakdown. J. Fluid Mech. 56, 775.Google Scholar
Laufer, J. 1951 Investigation of turbulence in a two-dimensional channel. N.A.C.A. Tech. Rep. no. 1053.Google Scholar
Lessen, M. & Cheifetz, M. G. 1975 Stability of plane Couette flow with respect to finite two-dimensional disturbances. Phys. Fluids 18, 939.Google Scholar
Lin, C. C. 1955 The Theory of Hydrodynamic Stability. Cambridge University Press.
McLaughlin, J. & Orszag, S. A. 1979 To be published.
Meksyn, D. 1964 Stability of laminar flow between parallel planes for two- and three-dimensional finite disturbances. Z. Phys. 178, 159.Google Scholar
Meksyn, D. & Stuart, J. T. 1951 Stability of viscous motion between parallel planes for finite disturbances. Proc. Roy. Soc. A 208, 517.Google Scholar
Metcalfe, R. W. 1974 Spectral methods for boundary value problems in fluid mechanics. Ph.D. thesis, M.I.T., Cambridge, MA.
Michael, D. H. 1961 Note on the stability of plane parallel flows. J. Fluid Mech. 10, 525.Google Scholar
Murdock, J. 1977 A numerical study of nonlinear effects on boundary-layer stability. A.I.A.A. 15th Aerospace Sciences Meeting, Paper 77-127.Google Scholar
Murdock, J. & Taylor, T. D. 1977 Numerical investigation of nonlinear wave interaction in a two-dimensional boundary layer. In Laminar-Turbulent Transition, AGARD Conf. Proc. no. 224, p. 4–1.
Nishioka, M., Iida, S. & Ichikawa, Y. 1975 An experimental investigation of the stability of plane Poiseuille flow. J. Fluid Mech. 72, 731.Google Scholar
Orszag, S. A. 1971a Numerical simulation of incompressible flows within simple boundaries. I. Galerkin (spectral) representations. Stud. Appl. Math. 50, 293.Google Scholar
Orszag, S. A. 1971b Accurate solution of the Orr-Sommerfeld stability equation. J. Fluid Mech. 50, 689.Google Scholar
Orszag, S. A. 1976 Turbulence and transition: a progress report. Proc. 5th Int. Conf. on Numerical Methods in Fluid Dynamics (ed. A. I. van de Vooren and P. J. Zandbergen), p. 32. Springer.
Orszag, S. A. & Patterson, G. S. 1972 Numerical simulation of three-dimensional homogeneous isotropic turbulence. Phys. Rev. Letters 28, 76.Google Scholar
Patel, V. & Head, M. R. 1969 Some observations on skin friction and velocity profiles in fully developed pipe and channel flows. J. Fluid Mech. 38, 181.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.Google Scholar
Pekeris, C. L. & Shkoller, B. 1969 Stability of plane Poiseuille flow to periodic disturbances of finite amplitude. J. Fluid Mech. 39, 611.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.Google Scholar
Reichardt, H. 1959 Gesetzmässigkeiten der geradlinigen turbulenten Couetteströmung. Mitt. Max-Planck-Inst. StrömForsch. 22, Göttingen.Google Scholar
Reynolds, W. C. & Potter, M. C. 1967 Finite-amplitude instability of parallel shear flows. J. Fluid Mech. 27, 465.Google Scholar
Rosenblat, S. & Davis, S. H. 1978 Bifurcation from infinity. To be published.
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.CrossRefGoogle Scholar
Stuart, J. T. 1961 On three-dimensional nonlinear effects in the stability of parallel flows. Adv. Aeronaut. Sci. 3, 121.Google Scholar
Stuart, J. T. 1971 Nonlinear stability theory. Ann. Rev. Fluid Mech. 3, 347.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 for plane Couette flow. J. Fluid Mech. 9, 371.CrossRefGoogle Scholar
Watson, J. 1962 On spatially growing finite disturbances in plane Poiseuille flow. J. Fluid Mech. 14, 211.Google Scholar
Zahn, J.-P., Toomre, J., Spiegel, E. A. & Gough, D. O. 1974 Nonlinear cellular motions in Poiseuille channel flow. J. Fluid Mech. 64, 319.Google Scholar