Skip to main content
SpringerLink
Log in

Unstable manifold computations for the two-dimensional plane Poiseuille flow

  • Published:
Theoretical and Computational Fluid Dynamics Aims and scope Submit manuscript

Abstract

We follow the unstable manifold of periodic and quasi-periodic solutions in time for the Poiseuille problem, using two formulations: holding a constant flux or mean pressure gradient. By means of a numerical integrator of the Navier–Stokes equations, we let the fluid evolve from an initially perturbed unstable solution until the fluid reaches an attracting state. Thus, we detect several connections among different configurations of the flow such as laminar, periodic, quasi-periodic with two or three basic frequencies, and more complex sets that we have not been able to classify. These connections make possible the location of new families of solutions, usually hard to find by means of numerical continuation of curves, and show the richness of the dynamics of the Poiseuille flow.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. Alavyoon, F., Henningson, D.S., Alfredsson, P.H.: Turbulent spots in plane Poiseuille flow–flow visualization. Phys. Fluids 29, 1328–1331 (1986)

  2. Allgower, E.L., Georg, K.: Numerical Continuation Methods, Vol. 13 of Springer Series in Comp. Maths.. Springer-Verlag (1990)

  3. Canuto, C., Hussaini, M.Y., Quarteroni, A., Zang, T.A.: Spectral methods in fluid dynam.. Springer-Verlag (1988)

  4. Carlson, D.R., Widnall, S.E., Peeters, M.F.: A flow visualization study of transition in plane Poiseuille flow. J. Fluid Mech. 121, 487–505 (1982)

  5. Casas, P.S.: Numerical study of Hopf bifurcations in the two-dimensional plane Poiseuille flow. PhD thesis, Universidad Politécnica de Cataluña. http://www-ma1.upc.es/∼casas/research.html (2002)

  6. Drissi, A., Net, M., Mercader, I.: Subharmonic instabilities of Tollmien-Schlichting waves in two-dimensional Poiseuille flow. Phys. Rev. E 60(2), 1781–1791 (1999)

  7. Ehrenstein, U., Koch, W.: Three-dimensional wavelike equilibrium states in plane Poiseuille flow. J. Fluid Mech. 228, 111–148 (1991)

  8. Herbert, T.: Periodic secondary motions in a plane channel. In Proc. 5th Int. Conf. Numerical Methods in Fluid Dynam., volume 59 of Lec. Notes in Phys., pages 235–240. Springer (1976)

  9. Jiménez, J.: Bifurcations and bursting in two-dimensional Poiseuille flow. Phys. Fluids 30(12), 3644–3646 (1987)

  10. Jiménez, J.: Transition to turbulence in two-dimensional Poiseuille flow. J. Fluid Mech. 218, 265–297 (1990)

  11. Marsden, J.E., McCracken, M.: The Hopf bifurcation and its applications, vol. 10 of Appl. Math. Sci.. Springer-Verlag, NY (1976)

  12. Nishioka, M., Asai, M.: Some observartions of the subcritical transition in plane Poiseuille flow. J. Fluid Mech. 150, 441–450 (1985)

  13. Orszag, S.A.: Accurate solution of the Orr–Sommerfeld stability equation. J. Fluid Mech. 50(4), 689–703 (1971)

  14. Orszag, S.A., Patera, A.T.: Hydrodynamic stability of shear flows. In Iooss, G., Helleman, R.H., and Stora, R., (eds)., Chaotic behavior of deterministic systems (Les Houches, 1981), volume XXXVI, pp. 621–662. North-Holland, Amsterdam (1983)

  15. Perko, L.: Differential equations and dynamical systems. Springer-Verlag, NY, 2nd ed. (1998)

  16. Pugh, J.D., Saffman, P.G.: Two-dimensional superharmonic stability of finite-amplitude waves in plane Poiseuille flow. J. Fluid Mech. 194, 295–307 (1988)

  17. Rand, D.: Dynamics and symmetry predictions for modulated waves in rotating fluids. Arch. Rat. Mech. Anal. 79, 1–37 (1982)

  18. Rozhdestvensky, B.L., Simakin, I.N.: Secondary flows in a plane channel: their relationship and comparison with turbulent flows. J. Fluid Mech. 147, 261–289 (1984)

  19. Saffman, P.G.: Vortices, stability, and turbulence. Ann. N. Y. Acad. Sci. 404, 12–24 (1983)

  20. Soibelman, I., Meiron, D.I.: Finite-amplitude bifurcations in plane Poiseuille flow: two-dimensional Hopf bifurcation. J. Fluid Mech. 229, 389–416 (1991)

  21. Zahn, J.-P., Toomre, J., Spiegel, E.A., Gough, D.O.: Nonlinear cellular motions in Poiseuille channel flow. J. Fluid Mech. 64, 319–345 (1974)

Download references

Author information

Authors and Affiliations

  1. Departamento de Matemática Aplicada I, Universidad Politécnica de Cataluña, Diagonal, 647, 08028, Barcelona, Spain

    Pablo S. Casas

  2. Departamento de Matemática Aplicada y Análisis, Universidad de Barcelona, Gran Via, 585, 08007, Barcelona, Spain

    Àngel Jorba

Authors
  1. Pablo S. Casas
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Àngel Jorba
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Pablo S. Casas.

Additional information

Communicated by

H.J.S. Fernando

PACS

05.45.-a, 47.11.+j, 47.20.-k, 47.20.Ft

Rights and permissions

Reprints and permissions

About this article

Cite this article

Casas, P., Jorba, À. Unstable manifold computations for the two-dimensional plane Poiseuille flow. Theor. Comput. Fluid Dyn. 18, 285–299 (2004). https://doi.org/10.1007/s00162-004-0138-0

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s00162-004-0138-0

Keywords

Search

Navigation