Hostname: page-component-77c89778f8-n9wrp Total loading time: 0 Render date: 2024-07-19T07:54:57.655Z Has data issue: false hasContentIssue false
  • English
  • Français

Transition to turbulence in two-dimensional Poiseuille flow

Published online by Cambridge University Press:  26 April 2006

Javier Jiménez
Javier Jiménez
Affiliation:
School of Aeronautics, Universidad Politécnica Madrid, P. Cardenal Cisneros 3, 28040 Madrid, SpainandIBM Madrid Scientific Centre
Get access Rights & Permissions [Opens in a new window]

Abstract

The transition of strictly two-dimensional Poiseuille flow from laminar to chaotic behaviour is studied through full numerical simulation of spatially periodic channels with fairly large longitudinal aspect ratios. The successive bifurcations are studied in detail and their physical mechanism is elucidated. The Liapunov exponents of the flow are measured and shown to be positive at large Reynolds numbers. Isolated, permanent patches of unsteady behaviour, resembling the turbulent ‘puffs’ observed in circular pipes, are found at low Reynolds numbers and shown to be important for the transition to chaos. The flow exhibits several other phenomena present in natural three-dimensional flows, including wall sweeps, ejections, and intermittency.

Type
Research Article
Information
Journal of Fluid Mechanics , Volume 218 , September 1990 , pp. 265 - 297
Copyright
© 1990 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

Andereck, C. D., Liu, S. S. & Swinney, H. L., 1986 Flow regimes in a circular Couette flow system with independently rotating cylinders. J. Fluid Mech. 164, 155–183.Google Scholar
Bandyopadhyay, P. R.: 1986 Aspects of the equilibrium puff in transitional pipe flow. J. Fluid Mech. 163, 439–458.Google Scholar
Batchelor, G. K.: 1969 Computation of the energy spectrum in homogeneous turbulence. Phys. Fluids Suppl. II, 233–239.Google Scholar
Benzi, R., Paladin, G., Parisi, G. & Vulpiani, A., 1985 Characterisation of intermittency in chaotic systems. IBM Rome Scientific Centre Rep. G513-4066.Google Scholar
Benzi, R., Patarnelo, S. & Santangelo, P., 1987 On the statistical properties of two-dimensional decaying turbulence. Europhys. Lett. 3, 811–818.Google Scholar
Cantwell, B. J.: 1981 Organized motion in turbulent flow. Ann. Rev. Fluid Mech. 13, 457–515.Google Scholar
Coles, D.: 1965 Transition in circular Couette flow. J. Fluid Mech. 21, 385–425.Google Scholar
Dean, R. B.: 1976 Reynolds number dependence of skin friction and other bulk flow variables in two-dimensional rectangular duct flow. Trans. ASME I: J. Fluids Engng 100, 215–223.Google Scholar
Deville, M., Kleiser, L. & Montigny-Rannou, F. 1984 Pressure and time treatment for Chebyshev spectral solution of a Stokes problem. Intl J. Numer. Methods Fluids 4, 1149–1163.Google Scholar
Gottlieb, D. & Orszag, S. A., 1977 Numerical Analysis of Spectral Methods, p. 119. Philadelphia: SIAM.
Guckenheimer, J. & Holmes, P., 1983 Nonlinear Oscillations Dynamical Systems and Bifurcations of Vector Fields. Springer.
Henshaw, W. D., Kreiss, H. O. & Reyna, L. G., 1989 On the smallest scales for the incompressible Navier–Stokes equations. J. Theor. Comput. Fluid Mech. 1, 1–32.Google Scholar
Herbert, T.: 1976 Periodic secondary 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), pp. 235–240. Springer.
Herbert, T.: 1983 Secondary instability of plane channel flow to subharmonic disturbances. Phys. Fluids 26, 871–874.Google Scholar
Jiménez, J.: 1987 Bifurcations and bursting in two dimensional Poiseuille flow. Phys. Fluids 30, 3644–3646.Google Scholar
Jiménez, J.: 1989 Bifurcations in Poiseuille flow and wall turbulence. In Fluid Dynamics of 3D Shear Flows and Transition, AGARD-CP 438, pp. 14.1–14.11.
Jiménez, J., Moin, R., Moser, R. & Keefe, L., 1988 Ejection mechanisms in the sublayer of a turbulent channel. Phys. Fluids 31, 1311–1313.Google Scholar
Keefe, L., Moin, P. & Kim, J., 1987 The dimension of an attractor in turbulent Poiseuille flow. Bull. Am. Phys. Soc. 32, 2026.Google Scholar
Kim, J., Moin, P. & Moser, R., 1987 Turbulence statistics in fully developed channel flow at low Reynolds number. J. Fluid Mech. 177, 133–166.Google Scholar
Kraichnan, H. R.: 1967 Inertial ranges in two dimensional turbulence. Phys. Fluids 10, 1417–1423.Google Scholar
Orszag, S. A.: 1971 Numerical simulation of incompressible flows with simple boundaries. I. Galerkin (spectral) representations. Stud. Appl Maths. 50, 293–327.Google Scholar
Orszag, S. A. & Patera, A. T., 1983 Secondary instability of wall bounded shear flows. J. Fluid Mech. 128, 347–385.Google Scholar
Pomeau, Y.: 1986 Front motion, metastability and subcritical bifurcations in hydrodynamics. Physica 23D, 3–11.Google Scholar
Pugh, J. D.: 1987 Finite amplitude waves in plane Poiseuille flow. Ph.D. thesis, Caltech, Pasadena.
Rozhdestvensky, B. L. & Simakin, I. N., 1984 Secondary flows in a plane channel: their relationship and comparison with turbulent flows. J. Fluid Mech. 147, 261–289.Google Scholar
Rubin, Y., Wygnanski, I. & Haritonidis, J. H., 1980 Further observations on transition in a pipe. In Laminar–Turbulent Transition (ed. R. Eppler and H. Fasel), pp. 17–26. Springer.
Saffman, P. G.: 1971 On the spectrum and decay of random two dimensional vorticity distributions at large Reynolds numbers. Stud. Appl. Maths. 50, 377–383.Google Scholar
Saffman, P. D.: 1983 Vortices, stability and turbulence. Ann. N. Y. Acad. Sci. 404, 12–24.Google Scholar
Stewartson, K. & Stuart, J. T., 1971 A nonlinear stability theory for a wave system in plane Poiseuille flow. J. Fluid Mech. 48, 529–545.Google Scholar
Thual, O. & Fauve, S., 1988 Localized structures generated by subcritical instabilities. J. Phys. Paris 49, 1829–1833.Google Scholar
Van Atta, C. 1966 Exploratory measurements in spiral turbulence. J. Fluid Mech. 25, 495–512.Google Scholar
Vastano, J. A. & Kostelich, E. J., 1986 Comparison of algorithms for determining Liapunov exponents from experimental data. In Dimensions and Entropies in Chaotic Systems (ed. G. Mayer-Kress), pp. 100–107. Springer.
Wolf, A., Swift, J., Swinney, H. L. & Vastano, J., 1985 Determining Liapunov exponents from a time series. Physica 16D, 285–317.Google Scholar
Wygnanski, I. J. & Champagne, F. H., 1973 On transition in a pipe. Part 1. The origin of puffs and slugs and the flow in a turbulent slug. J. Fluid Mech. 59, 281–335.Google Scholar
Wygnanski, I. J., Sokolov, M. & Friedman, D., 1975 On transition in a pipe. Part 2. The equilibrium puff. J. Fluid Mech. 69, 283–304.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, 319–345.Google Scholar