Hostname: page-component-76fb5796d-zzh7m Total loading time: 0 Render date: 2024-04-26T06:17:01.380Z Has data issue: false hasContentIssue false
  • English
  • Français

Three-dimensional wavelike equilibrium states in plane Poiseuille flow

Published online by Cambridge University Press:  26 April 2006

U. Ehrenstein  and
W. Koch
U. Ehrenstein
Affiliation:
DLR Institute for Theoretical Fluid Mechanics, D-3400 Göttingen, Germany Present address: Institut Français du Pétrole, F-92506 Rueil Malmaison Cedex, France.
W. Koch
Affiliation:
DLR Institute for Theoretical Fluid Mechanics, D-3400 Göttingen, Germany
Get access Rights & Permissions [Opens in a new window]

Abstract

In the quest for a physically more realistic transition criterion, the prechaotic bifurcation behaviour of plane Poiseuille flow is studied. Various classes of nonlinear time-periodic equilibrium solutions are computed via Keller's pseudo-arclength continuation method. In particular, attention is focused on three-dimensional nonlinear travelling-wave type secondary bifurcation branches. These saturated equilibrium states originate on the nonlinear primary bifurcation surface from neutral, phase-locked secondary instability modes. Taking advantage of symmetries, only those nonlinear secondary branches which correspond to symmetric and antisymmetric linear secondary instability modes are investigated.

It appears that a new family of secondary bifurcation solutions which contains only even spanwise Fourier modes is particularly important. Dominated largely by the spanwise (0,2) mode and discovered by investigating bicritical secondary bifurcations, the mean quantities of these solutions show a certain resemblance to those observed in transitional flow during the ‘spike’ stage. The friction factor of this new solution branch is in the experimentally observed range and the critical Reynolds number, defined with the mean flow velocity, is reduced to about 1000 in general agreement with experiments.

Type
Research Article
Information
Journal of Fluid Mechanics , Volume 228 , July 1991 , pp. 111 - 148
Copyright
© 1991 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

Agard Conference Proceedings 1977 Laminar—turbulent transition. AGARD-CP-224.
Bacher, E. V. & Smith, C. R. 1986 Turbulent boundary-layer modification by surface riblets. AIAA J. 24, 1382–1385.Google Scholar
Bayly, B. J., Orszag, S. A. & Herbert, T. 1988 Instability mechanisms in shear-flow transition. Ann. Rev. Fluid Mech. 20, 359–391.Google Scholar
Benney, D. J. 1964 Finite amplitude effects in an unstable laminar boundary layer. Phys. Fluids 7, 319–326.Google Scholar
Benney, D. J. & Lin, C. C. 1960 On the secondary motion induced by oscillations in a shear flow. Phys. Fluids 3, 656–657.Google Scholar
Bushnell, D. M., Malik, M. R. & Harvey, W. D. 1989 Transition prediction in external flows via linear stability theory. In IUTAM Symposium Transsonicum III (ed. J. Zierep & H. Oertel), pp. 225–242. Springer.
Canuto, C., Hussaini, M. Y., Quarteroni, A. & Zang, T. A. 1988 Spectral Methods in Fluid Dynamics. Springer.
Chen, T. S. & Joseph, D. D. 1973 Subcritical bifurcation of plane Poiseuille flow. J. Fluid Mech. 58, 337–351.Google Scholar
Craik, A. D. D. 1971 Nonlinear resonant instability in boundary layers. J. Fluid Mech. 50, 393–413.Google Scholar
Craik, A. D. D. 1985 Wave Interactions and Fluid Flows. Cambridge University Press.
Crouch, J. D. & Herbert, T. 1986 Perturbation analysis of nonlinear secondary instability in boundary layers. Bull. Am. Phys. Soc. 31, 1718.Google Scholar
Deissler, R. J. 1987 The convective nature of instability in plane Poiseuille flow. Phys. Fluids 30, 2303–2305.Google Scholar
Dhanak, M. R. 1983 On certain aspects of three-dimensional instability of parallel flows.. Proc. R. Soc. Lond. A 385, 53–84.Google Scholar
Doedel, E. 1981 Auto: a program for the automatic bifurcation analysis of autonomous systems. Cong. Num. 30, 265–284.Google Scholar
Drazin, P. G. & Reid, W. H. 1981 Hydrodynamic Stability. Cambridge University Press.
Eckelmann, H. 1970 Experimentelle Untersuchungen in einer turbulenten Kanalströmung mit starken viskosen Wandschichten. Mitt. MPI für Strömungsforschung und AVA Göttingen no. 48.Google Scholar
Ehrenstein, U. 1988 Lösung der linearen, inkompressiblen Stördifferentialgleichungen mittels Chebyshev–Kollokation. Internal DFVLR-Rep. IB 221–88 A 09.Google Scholar
Ehrenstein, U. & Koch, W. 1989 Nonlinear bifurcation study of plane Poiseuille flow. DLR Res. Rep. FB 89–42.Google Scholar
Ehrenstein, U. & Koch, W. 1991 Local and global analysis of secondary bifurcations in plane Poiseuille flow. To appear in the special issue of European J. Mech. B Fluids containing the Proc. IUTAM Symp. Nonlinear Hydrodynamic Stability and Transition, Sept. 3–7, 1990, Sophia-Antipolis, France.
Gilbert, H. 1988 Numerische Simulation der Transition von der laminaren in die turbulente Kanalströmung. DFVLR Res. Rep. FB 88–55.Google Scholar
Goldshtik, M. A., Lifshits, A. M. & Shtern, V. N. 1983 The Reynolds number in a transition in a plane channel. Sov. Phys. Dokl. 28, 923–925.Google Scholar
Golubitzky, M., Stewart, I. & Schaeffer, D. G. 1988 Singularities and Groups in Bifurcation Theory, vol. 2. Springer.
Gottlieb, D., Hussaini, M. Y. & Orszag, S. A. 1984 Theory and applications of spectral methods. In Spectral Methods for Partial Differential Equations (ed. R. G. Voigt, D. Gottlieb & M. Y. Hussaini), pp. 1–54. SIAM Philadelphia.
Grohne, D. 1969 Die Stabilität der ebenen Kanalströmung gegenüber dreidimensionalen Störungen von endlicher Amplitude. AVA Gottingen, Rep. 69 A 30.Google Scholar
Herbert, T. 1977 Die Neutralfläche der ebenen Poiseuilleströmung. 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. 1981 A secondary instability mechanism in plane Poiseuille flow. Bull. Am. Phys. Soc. 26, 1257.Google Scholar
Herbert, T. 1983 On perturbation methods in nonlinear stability theory. J. Fluid Mech. 126, 167–186.Google Scholar
Herbert, T. 1984 Modes of secondary instability in plane Poiseuille flow. In Turbulence and Chaotic Phenomena in Fluids (ed. T. Tatsumi), pp. 53–58. North-Holland.
Herbert, T. 1988 Secondary instability of boundary layers. Ann. Rev. Fluid Mech. 20, 487–526.Google Scholar
Herbert, T. 1991 Exploring transition by computer. Appl. Num. Maths. 7, 3–25.Google Scholar
Jepson, A. D. & Keller, H. B. 1984 Steady state and periodic solution paths: their bifurcations and computations. In Numerical Methods for Bifurcation Problems (ed. T. Küpper, H. D. Mittelmann & H. Weber), pp. 219–246. Birkhäuser.
Jiménez, J. 1987a Coherent structures and dynamical systems In Proc. 1987 Summer Program of the NASA Stanford Center for Turbulence Res. Rep. CTR-S87, pp. 323–324.
Jiménez, J. 1987b Bifurcations and bursting in two-dimensional Poiseuille flow. Phys. Fluids 30, 3644–3646.Google Scholar
Jiménez, J. 1990 Bifurcations and turbulence in plane Poiseuille flow. In Near-Wall Turbulence. 1988 Zoran Zaric Memorial Conference (ed. S. J. Kline & N. H. Afgan), pp. 28–44. Hemisphere.
Kachanov, Yu. S. & Levchenko, V. Ya. 1984 The resonant interaction of disturbances at laminar—turbulent transition in a boundary layer. J. Fluid Mech. 138, 209–247.Google Scholar
Keller, H. B. 1977 Numerical solution of bifurcation and nonlinear eigenvalue problems. In Applications of Bifurcation Theory (ed. P. H. Rabinowitz), pp. 359–384. Academic Press.
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
Klebanoff, P. S., Tidstrom, K. D. & Sargent, L. M. 1962 The three-dimensional nature of boundary-layer instability. J. Fluid Mech. 12, 1–34.Google Scholar
Kleiser, L. & Zang, T. A. 1991 Numerical simulation of transition in wall-bounded shear flows. Ann. Rev. Fluid Mech. 23, 495–537.Google Scholar
Koch, W. 1988 Nonlinear limit-cycle solutions — a rational method for transition prediction in shear flows? In Propagation in Systems far from Equilibrium (ed. J. E. Wesfreid, H. R. Brand, P. Manneville, G. Albinet & N. Boccara), pp. 360–368. Springer.
Kubiček, M. & Marek, M. 1983 Computational Methods in Bifurcation Theory and Dissipative Structures. Springer.
Liu, J. T. C. 1989 Coherent structures in transitional and turbulent free shear flows. Ann. Rev. Fluid Mech. 21, 285–315.Google Scholar
May, C. L. & Kleiser, L. 1985 Numerical simulation of subharmonic transition in plane Poiseuille flow. Bull. Am. Phys. Soc. 30, 1748.Google Scholar
Meksyn, D. 1964 Stability of laminar flow between parallel planes for two- and three-dimensional finite disturbances. Z. Phys. 178, 159–172.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
Meyer-Spasche, R. & Keller, H. B. 1980 Computations of the axisymmetric flow between rotating cylinders. J. Comp. Phys. 35, 100–109.Google Scholar
Milinazzo, F. A. & Saffman, P. G. 1985 Finite-amplitude waves in plane viscous shear flows. J. Fluid Mech. 160, 281–295.Google Scholar
Morkovin, M. V. 1988 Recent insights into instability and transition to turbulence in open-flow systems. AIAA paper 88–3675 (also ICASE Rep. 88–44).Google Scholar
Nishioka, M. & Asai, M. 1985 Some observations of the subcritical transition in plane Poiseuille flow. J. Fluid Mech. 150, 441–450.Google Scholar
Nishioka, M., Iida, S. & Ichikawa, Y. 1975 An experimental investigation of the stability of plane Poiseuille flow. J. Fluid Mech. 72, 731–751.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 Subcritical transition to turbulence in plane channel flows. Phys. Rev. Lett. 45, 989–993.Google Scholar
Orszag, S. A. & Patera, A. T. 1983 Secondary instability of wall-bounded shear flows. J. Fluid Mech. 128, 347–385.Google Scholar
Patel, V. C. & Head, M. R. 1969 Some observations on skin friction and velocity profiles in fully developed pipe and channel flows. J. Fluid Mech. 38, 181–201.Google Scholar
Pekeris, C. L. & Shkoller, B. 1969 The neutral curves for periodic perturbations of finite amplitude of plane Poiseuille flow. J. Fluid Mech. 39, 629–639.Google Scholar
Pugh, J. D. 1987 Finite amplitude waves in plane Poiseuille flow. Ph.D. Thesis, California Institute of Technology.
Pugh, J. D. & Saffman, P. G. 1988 Two-dimensional superharmonic stability of finite amplitude waves in plane Poiseuille flow. J. Fluid Mech. 194, 295–307.Google Scholar
Reed, H. L. & Saric, W. S. 1989 Stability of three-dimensional boundary layers. Ann. Rev. Fluid Mech. 21, 235–284.Google Scholar
Reshotko, E. 1976 Boundary-layer stability and transition. Ann. Rev. Fluid Mech. 8, 311–349.Google Scholar
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
Saffman, P. G. 1983 Vortices, stability and turbulence. Ann. N.Y. Acad. Sci. 404, 12–24.Google Scholar
Saric, W. S. & Thomas, A. S. W. 1984 Experiments on the subharmonic route to turbulence in boundary layers. In Turbulence and Chaotic Phenomena in Fluids (ed. T. Tatsumi), pp. 117–122. North-Holland.
Schlichting, H. 1933 Zur Entstehung der Turbulenz bei der Plattenströmung. Nachr. Ges. Wiss. Göttingen, Math. Phys. Kl. 182–208.Google Scholar
Schlichting, H. 1958 Grenzschicht-Theorie (3rd edn). Braun.
Schrauf, G. 1986 The first instability in spherical Taylor–Couette flow. J. Fluid Mech. 166, 287–303.Google Scholar
Schubauer, G. B. & Skramstad, H. K. 1947 Laminar boundary-layer oscillations and transition on a flat plate. J. Res. Natl Bur. Stand. 38, 251–292 (reprint of NACA Adv. Conf. Rep. 1943).Google Scholar
Sen, P. K. & Venkateswarlu, D. 1983 On the stability of plane Poiseuille flow to finite-amplitude disturbances, considering the higher-order Landau constants. J. Fluid Mech. 133, 179–206.Google Scholar
Shtern, V. N. 1976 Instability due to three-dimensional disturbances. Izv. Akad. Nauk SSSR Mekh. Zhid. i Gaza, 5, 29–34.Google Scholar
Squire, H. B. 1933 On the stability of three-dimensional disturbances of viscous flow between parallel walls.. Proc. R. Soc. Lond. A 142, 621–628.Google Scholar
Stuart, J. T. 1960 On the nonlinear 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.Google Scholar
Stuart, J. T. 1971 Nonlinear stability theory. Ann. Rev. Fluid Mech. 3, 347–370.Google Scholar
Tani, I. 1969 Boundary-layer transition. Ann. Rev. Fluid Mech. 1, 169–196.Google Scholar
Tollmien, W. 1929 Über die Entstehung der Turbulenz. Nachr. Ges. Wiss. Göttingen, Math. Phys. Kl. 21–44.Google Scholar
Watson, J. 1960 On the nonlinear 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, 371–389.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